From ff6a0dcb09e66a6f4c8b72b7d1cbf3ef6aa1f5b4 Mon Sep 17 00:00:00 2001 From: Dmitrii Vasilev Date: Fri, 8 May 2026 16:22:52 +0000 Subject: [PATCH 1/3] feat(phd-phase1-unify-1-2): rename Flos Aureus strand NN-slug.tex -> fa_NN.tex (34 files) [agent=perplexity-computer-phase1] MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Phase 1 UNIFY · trios#380 task 1.2. Pure git-mv rename: each docs/phd/chapters/NN-.tex moves to docs/phd/chapters/fa_NN.tex with 100% file similarity (R == 100). No content edits in this commit; the rename is paired 1:1 with main.tex include-path patches in the next commit. Why: per #380 manifest task 1.2, the named-strand canonical namespace is fa_NN (Flos Aureus), parallel to the ch_NN namespace (Trinity S³AI). phd-chapter-author v1.1 lessons-learned point 1 already classified the old NN-slug names as the canonical chapter location, but the v6.2 manifest reserves the unprefixed NN-slug namespace and instead pins fa_NN as the namespace shared by main.tex include lines and Neon ssot.chapters export targets. Mechanical mapping (NN preserved, slug dropped): 00-monad.tex -> fa_00.tex 01-golden-egg.tex -> fa_01.tex ... 33-epilogue.tex -> fa_33.tex Anchor phi^2 + phi^{-2} = 3 · DOI 10.5281/zenodo.19227877. Refs trios#380 Phase 1 UNIFY task 1.2. --- docs/phd/chapters/{00-monad.tex => fa_00.tex} | 0 docs/phd/chapters/{01-golden-egg.tex => fa_01.tex} | 0 docs/phd/chapters/{02-golden-cut.tex => fa_02.tex} | 0 docs/phd/chapters/{03-golden-harvest.tex => fa_03.tex} | 0 docs/phd/chapters/{04-golden-scales.tex => fa_04.tex} | 0 docs/phd/chapters/{05-golden-bridge.tex => fa_05.tex} | 0 docs/phd/chapters/{06-golden-mantissa.tex => fa_06.tex} | 0 docs/phd/chapters/{07-golden-sprout.tex => fa_07.tex} | 0 docs/phd/chapters/{08-golden-crystal.tex => fa_08.tex} | 0 docs/phd/chapters/{09-golden-seal.tex => fa_09.tex} | 0 docs/phd/chapters/{10-golden-bloom.tex => fa_10.tex} | 0 docs/phd/chapters/{11-vesica-piscis.tex => fa_11.tex} | 0 docs/phd/chapters/{12-flower-of-life.tex => fa_12.tex} | 0 docs/phd/chapters/{13-metatron-cube.tex => fa_13.tex} | 0 docs/phd/chapters/{14-platonic-solids.tex => fa_14.tex} | 0 docs/phd/chapters/{15-kepler-solids.tex => fa_15.tex} | 0 docs/phd/chapters/{16-sacred-ratios.tex => fa_16.tex} | 0 docs/phd/chapters/{17-golden-spiral.tex => fa_17.tex} | 0 docs/phd/chapters/{18-torus-geometry.tex => fa_18.tex} | 0 docs/phd/chapters/{19-fibonacci-tesselation.tex => fa_19.tex} | 0 docs/phd/chapters/{20-standard-model.tex => fa_20.tex} | 0 docs/phd/chapters/{21-quantum-field.tex => fa_21.tex} | 0 docs/phd/chapters/{22-e8-symmetry.tex => fa_22.tex} | 0 docs/phd/chapters/{23-gf16-algebra.tex => fa_23.tex} | 0 docs/phd/chapters/{24-igla-architecture.tex => fa_24.tex} | 0 docs/phd/chapters/{25-benchmarks.tex => fa_25.tex} | 0 docs/phd/chapters/{26-data-analysis.tex => fa_26.tex} | 0 docs/phd/chapters/{27-trinity-identity.tex => fa_27.tex} | 0 docs/phd/chapters/{28-momentum-algebra.tex => fa_28.tex} | 0 docs/phd/chapters/{29-lucas-closure.tex => fa_29.tex} | 0 docs/phd/chapters/{30-golden-imagery.tex => fa_30.tex} | 0 docs/phd/chapters/{31-philosophy.tex => fa_31.tex} | 0 docs/phd/chapters/{32-conclusion.tex => fa_32.tex} | 0 docs/phd/chapters/{33-epilogue.tex => fa_33.tex} | 0 34 files changed, 0 insertions(+), 0 deletions(-) rename docs/phd/chapters/{00-monad.tex => fa_00.tex} (100%) rename docs/phd/chapters/{01-golden-egg.tex => fa_01.tex} (100%) rename docs/phd/chapters/{02-golden-cut.tex => fa_02.tex} (100%) rename docs/phd/chapters/{03-golden-harvest.tex => fa_03.tex} (100%) rename docs/phd/chapters/{04-golden-scales.tex => fa_04.tex} (100%) rename docs/phd/chapters/{05-golden-bridge.tex => fa_05.tex} (100%) rename docs/phd/chapters/{06-golden-mantissa.tex => fa_06.tex} (100%) rename docs/phd/chapters/{07-golden-sprout.tex => fa_07.tex} (100%) rename docs/phd/chapters/{08-golden-crystal.tex => fa_08.tex} (100%) rename docs/phd/chapters/{09-golden-seal.tex => fa_09.tex} (100%) rename docs/phd/chapters/{10-golden-bloom.tex => fa_10.tex} (100%) rename docs/phd/chapters/{11-vesica-piscis.tex => fa_11.tex} (100%) rename docs/phd/chapters/{12-flower-of-life.tex => fa_12.tex} (100%) rename docs/phd/chapters/{13-metatron-cube.tex => fa_13.tex} (100%) rename docs/phd/chapters/{14-platonic-solids.tex => fa_14.tex} (100%) rename docs/phd/chapters/{15-kepler-solids.tex => fa_15.tex} (100%) rename docs/phd/chapters/{16-sacred-ratios.tex => fa_16.tex} (100%) rename docs/phd/chapters/{17-golden-spiral.tex => fa_17.tex} (100%) rename docs/phd/chapters/{18-torus-geometry.tex => fa_18.tex} (100%) rename docs/phd/chapters/{19-fibonacci-tesselation.tex => fa_19.tex} (100%) rename docs/phd/chapters/{20-standard-model.tex => fa_20.tex} (100%) rename docs/phd/chapters/{21-quantum-field.tex => fa_21.tex} (100%) rename docs/phd/chapters/{22-e8-symmetry.tex => fa_22.tex} (100%) rename docs/phd/chapters/{23-gf16-algebra.tex => fa_23.tex} (100%) rename docs/phd/chapters/{24-igla-architecture.tex => fa_24.tex} (100%) rename docs/phd/chapters/{25-benchmarks.tex => fa_25.tex} (100%) rename docs/phd/chapters/{26-data-analysis.tex => fa_26.tex} (100%) rename docs/phd/chapters/{27-trinity-identity.tex => fa_27.tex} (100%) rename docs/phd/chapters/{28-momentum-algebra.tex => fa_28.tex} (100%) rename docs/phd/chapters/{29-lucas-closure.tex => fa_29.tex} (100%) rename docs/phd/chapters/{30-golden-imagery.tex => fa_30.tex} (100%) rename docs/phd/chapters/{31-philosophy.tex => fa_31.tex} (100%) rename docs/phd/chapters/{32-conclusion.tex => fa_32.tex} (100%) rename docs/phd/chapters/{33-epilogue.tex => fa_33.tex} (100%) diff --git a/docs/phd/chapters/00-monad.tex b/docs/phd/chapters/fa_00.tex similarity index 100% rename from docs/phd/chapters/00-monad.tex rename to docs/phd/chapters/fa_00.tex diff --git a/docs/phd/chapters/01-golden-egg.tex b/docs/phd/chapters/fa_01.tex similarity index 100% rename from docs/phd/chapters/01-golden-egg.tex rename to docs/phd/chapters/fa_01.tex diff --git a/docs/phd/chapters/02-golden-cut.tex b/docs/phd/chapters/fa_02.tex similarity index 100% rename from docs/phd/chapters/02-golden-cut.tex rename to docs/phd/chapters/fa_02.tex diff --git a/docs/phd/chapters/03-golden-harvest.tex b/docs/phd/chapters/fa_03.tex similarity index 100% rename from docs/phd/chapters/03-golden-harvest.tex rename to docs/phd/chapters/fa_03.tex diff --git a/docs/phd/chapters/04-golden-scales.tex b/docs/phd/chapters/fa_04.tex similarity index 100% rename from docs/phd/chapters/04-golden-scales.tex rename to docs/phd/chapters/fa_04.tex diff --git a/docs/phd/chapters/05-golden-bridge.tex b/docs/phd/chapters/fa_05.tex similarity index 100% rename from docs/phd/chapters/05-golden-bridge.tex rename to docs/phd/chapters/fa_05.tex diff --git a/docs/phd/chapters/06-golden-mantissa.tex b/docs/phd/chapters/fa_06.tex similarity index 100% rename from docs/phd/chapters/06-golden-mantissa.tex rename to docs/phd/chapters/fa_06.tex diff --git a/docs/phd/chapters/07-golden-sprout.tex b/docs/phd/chapters/fa_07.tex similarity index 100% rename from docs/phd/chapters/07-golden-sprout.tex rename to docs/phd/chapters/fa_07.tex diff --git a/docs/phd/chapters/08-golden-crystal.tex b/docs/phd/chapters/fa_08.tex similarity index 100% rename from docs/phd/chapters/08-golden-crystal.tex rename to docs/phd/chapters/fa_08.tex diff --git a/docs/phd/chapters/09-golden-seal.tex b/docs/phd/chapters/fa_09.tex similarity index 100% rename from docs/phd/chapters/09-golden-seal.tex rename to docs/phd/chapters/fa_09.tex diff --git a/docs/phd/chapters/10-golden-bloom.tex b/docs/phd/chapters/fa_10.tex similarity index 100% rename from docs/phd/chapters/10-golden-bloom.tex rename to docs/phd/chapters/fa_10.tex diff --git a/docs/phd/chapters/11-vesica-piscis.tex b/docs/phd/chapters/fa_11.tex similarity index 100% rename from docs/phd/chapters/11-vesica-piscis.tex rename to docs/phd/chapters/fa_11.tex diff --git a/docs/phd/chapters/12-flower-of-life.tex b/docs/phd/chapters/fa_12.tex similarity index 100% rename from docs/phd/chapters/12-flower-of-life.tex rename to docs/phd/chapters/fa_12.tex diff --git a/docs/phd/chapters/13-metatron-cube.tex b/docs/phd/chapters/fa_13.tex similarity index 100% rename from docs/phd/chapters/13-metatron-cube.tex rename to docs/phd/chapters/fa_13.tex diff --git a/docs/phd/chapters/14-platonic-solids.tex b/docs/phd/chapters/fa_14.tex similarity index 100% rename from docs/phd/chapters/14-platonic-solids.tex rename to docs/phd/chapters/fa_14.tex diff --git a/docs/phd/chapters/15-kepler-solids.tex b/docs/phd/chapters/fa_15.tex similarity index 100% rename from docs/phd/chapters/15-kepler-solids.tex rename to docs/phd/chapters/fa_15.tex diff --git a/docs/phd/chapters/16-sacred-ratios.tex b/docs/phd/chapters/fa_16.tex similarity index 100% rename from docs/phd/chapters/16-sacred-ratios.tex rename to docs/phd/chapters/fa_16.tex diff --git a/docs/phd/chapters/17-golden-spiral.tex b/docs/phd/chapters/fa_17.tex similarity index 100% rename from docs/phd/chapters/17-golden-spiral.tex rename to docs/phd/chapters/fa_17.tex diff --git a/docs/phd/chapters/18-torus-geometry.tex b/docs/phd/chapters/fa_18.tex similarity index 100% rename from docs/phd/chapters/18-torus-geometry.tex rename to docs/phd/chapters/fa_18.tex diff --git a/docs/phd/chapters/19-fibonacci-tesselation.tex b/docs/phd/chapters/fa_19.tex similarity index 100% rename from docs/phd/chapters/19-fibonacci-tesselation.tex rename to docs/phd/chapters/fa_19.tex diff --git a/docs/phd/chapters/20-standard-model.tex b/docs/phd/chapters/fa_20.tex similarity index 100% rename from docs/phd/chapters/20-standard-model.tex rename to docs/phd/chapters/fa_20.tex diff --git a/docs/phd/chapters/21-quantum-field.tex b/docs/phd/chapters/fa_21.tex similarity index 100% rename from docs/phd/chapters/21-quantum-field.tex rename to docs/phd/chapters/fa_21.tex diff --git a/docs/phd/chapters/22-e8-symmetry.tex b/docs/phd/chapters/fa_22.tex similarity index 100% rename from docs/phd/chapters/22-e8-symmetry.tex rename to docs/phd/chapters/fa_22.tex diff --git a/docs/phd/chapters/23-gf16-algebra.tex b/docs/phd/chapters/fa_23.tex similarity index 100% rename from docs/phd/chapters/23-gf16-algebra.tex rename to docs/phd/chapters/fa_23.tex diff --git a/docs/phd/chapters/24-igla-architecture.tex b/docs/phd/chapters/fa_24.tex similarity index 100% rename from docs/phd/chapters/24-igla-architecture.tex rename to docs/phd/chapters/fa_24.tex diff --git a/docs/phd/chapters/25-benchmarks.tex b/docs/phd/chapters/fa_25.tex similarity index 100% rename from docs/phd/chapters/25-benchmarks.tex rename to docs/phd/chapters/fa_25.tex diff --git a/docs/phd/chapters/26-data-analysis.tex b/docs/phd/chapters/fa_26.tex similarity index 100% rename from docs/phd/chapters/26-data-analysis.tex rename to docs/phd/chapters/fa_26.tex diff --git a/docs/phd/chapters/27-trinity-identity.tex b/docs/phd/chapters/fa_27.tex similarity index 100% rename from docs/phd/chapters/27-trinity-identity.tex rename to docs/phd/chapters/fa_27.tex diff --git a/docs/phd/chapters/28-momentum-algebra.tex b/docs/phd/chapters/fa_28.tex similarity index 100% rename from docs/phd/chapters/28-momentum-algebra.tex rename to docs/phd/chapters/fa_28.tex diff --git a/docs/phd/chapters/29-lucas-closure.tex b/docs/phd/chapters/fa_29.tex similarity index 100% rename from docs/phd/chapters/29-lucas-closure.tex rename to docs/phd/chapters/fa_29.tex diff --git a/docs/phd/chapters/30-golden-imagery.tex b/docs/phd/chapters/fa_30.tex similarity index 100% rename from docs/phd/chapters/30-golden-imagery.tex rename to docs/phd/chapters/fa_30.tex diff --git a/docs/phd/chapters/31-philosophy.tex b/docs/phd/chapters/fa_31.tex similarity index 100% rename from docs/phd/chapters/31-philosophy.tex rename to docs/phd/chapters/fa_31.tex diff --git a/docs/phd/chapters/32-conclusion.tex b/docs/phd/chapters/fa_32.tex similarity index 100% rename from docs/phd/chapters/32-conclusion.tex rename to docs/phd/chapters/fa_32.tex diff --git a/docs/phd/chapters/33-epilogue.tex b/docs/phd/chapters/fa_33.tex similarity index 100% rename from docs/phd/chapters/33-epilogue.tex rename to docs/phd/chapters/fa_33.tex From 7c6ee946c0b30e22f3ac648158e7c5470ed01e75 Mon Sep 17 00:00:00 2001 From: Dmitrii Vasilev Date: Fri, 8 May 2026 16:23:45 +0000 Subject: [PATCH 2/3] feat(phd-phase1-unify-1-2): main.tex + main_ru.tex include paths -> fa_NN [agent=perplexity-computer-phase1] MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Phase 1 UNIFY · trios#380 task 1.2 (paired with previous 34-file git-mv commit). Patches: - docs/phd/main.tex line 299..353: 34 \include{chapters/NN-slug} -> \include{chapters/fa_NN}. The Trinity S³AI strand (\include{chapters/ch_00..ch_34}, lines 357..391) is unchanged. - docs/phd/main_ru.tex: same 34 substitutions. Verification: - grep -cE "chapters/fa_" main.tex == 34 (target) - grep -cE "chapters/ch_" main.tex == 35 (preserved Trinity S³AI) - grep -cE "chapters/[0-9]{2}-" main.tex == 0 (no stale refs) - repo-wide grep across .rs/.toml/.ts/.py/.yml/.yaml/.json for "chapters/NN-slug.tex" returns 0 hits. - tools/citetheorem_audit/src/lib.rs uses bare "NN-slug.tex" strings inside #[cfg(test)] tempdir-scoped tests; those names are test-data only and do not reach the real chapters directory, so they remain untouched (out of scope for task 1.2 — the test data set will be revisited in task 1.5 cross-reference sweep). Anchor phi^2 + phi^{-2} = 3 · DOI 10.5281/zenodo.19227877. Refs trios#380 Phase 1 UNIFY task 1.2. --- docs/phd/main.tex | 68 ++++++++++++++++++++++---------------------- docs/phd/main_ru.tex | 68 ++++++++++++++++++++++---------------------- 2 files changed, 68 insertions(+), 68 deletions(-) diff --git a/docs/phd/main.tex b/docs/phd/main.tex index 8acfd9c48f..f66285b80c 100644 --- a/docs/phd/main.tex +++ b/docs/phd/main.tex @@ -296,61 +296,61 @@ % Part I: The Foundations (Ch. 1-4) \part{The Foundations} -\include{chapters/00-monad} -\include{chapters/01-golden-egg} -\include{chapters/02-golden-cut} -\include{chapters/03-golden-harvest} +\include{chapters/fa_00} +\include{chapters/fa_01} +\include{chapters/fa_02} +\include{chapters/fa_03} % Part II: The Expansion (Ch. 5-7) \part{The Expansion} -\include{chapters/04-golden-scales} -\include{chapters/05-golden-bridge} -\include{chapters/06-golden-mantissa} -\include{chapters/07-golden-sprout} +\include{chapters/fa_04} +\include{chapters/fa_05} +\include{chapters/fa_06} +\include{chapters/fa_07} % Part III: The Crystal (Ch. 8-9) \part{The Crystal} -\include{chapters/08-golden-crystal} -\include{chapters/09-golden-seal} +\include{chapters/fa_08} +\include{chapters/fa_09} % Part IV: The Synthesis (Ch. 10-11) \part{The Synthesis} -\include{chapters/10-golden-bloom} +\include{chapters/fa_10} % Part V: Sacred Geometry (Ch. 12-20) \part{Sacred Geometry} -\include{chapters/11-vesica-piscis} -\include{chapters/12-flower-of-life} -\include{chapters/13-metatron-cube} -\include{chapters/14-platonic-solids} -\include{chapters/15-kepler-solids} -\include{chapters/16-sacred-ratios} -\include{chapters/17-golden-spiral} -\include{chapters/18-torus-geometry} -\include{chapters/19-fibonacci-tesselation} +\include{chapters/fa_11} +\include{chapters/fa_12} +\include{chapters/fa_13} +\include{chapters/fa_14} +\include{chapters/fa_15} +\include{chapters/fa_16} +\include{chapters/fa_17} +\include{chapters/fa_18} +\include{chapters/fa_19} % Part VI: Physics Foundation (Ch. 21-27) \part{Physics Foundation} -\include{chapters/20-standard-model} -\include{chapters/21-quantum-field} -\include{chapters/22-e8-symmetry} -\include{chapters/23-gf16-algebra} -\include{chapters/24-igla-architecture} +\include{chapters/fa_20} +\include{chapters/fa_21} +\include{chapters/fa_22} +\include{chapters/fa_23} +\include{chapters/fa_24} % Part VII: Algebraic Proofs (Ch. 28-30) \part{Algebraic Proofs} -\include{chapters/25-benchmarks} -\include{chapters/26-data-analysis} -\include{chapters/27-trinity-identity} -\include{chapters/28-momentum-algebra} -\include{chapters/29-lucas-closure} +\include{chapters/fa_25} +\include{chapters/fa_26} +\include{chapters/fa_27} +\include{chapters/fa_28} +\include{chapters/fa_29} % Part VIII: Imagery \& Genealogy (Ch. 31-33) \part{Imagery \& Genealogy} -\include{chapters/30-golden-imagery} -\include{chapters/31-philosophy} -\include{chapters/32-conclusion} -\include{chapters/33-epilogue} +\include{chapters/fa_30} +\include{chapters/fa_31} +\include{chapters/fa_32} +\include{chapters/fa_33} % --- Trinity S³AI strand (Ch.0..Ch.34) materialised from Railway phd-postgres-ssot (ssot.chapters) --- \part{Trinity S³AI Strand} diff --git a/docs/phd/main_ru.tex b/docs/phd/main_ru.tex index 8c0d7302f7..f374c960f2 100644 --- a/docs/phd/main_ru.tex +++ b/docs/phd/main_ru.tex @@ -348,54 +348,54 @@ \chapter*{Аннотация} \mainmatter \part{Основания} -\include{chapters/00-monad} -\include{chapters/01-golden-egg} -\include{chapters/02-golden-cut} -\include{chapters/03-golden-harvest} +\include{chapters/fa_00} +\include{chapters/fa_01} +\include{chapters/fa_02} +\include{chapters/fa_03} \part{Расширение} -\include{chapters/04-golden-scales} -\include{chapters/05-golden-bridge} -\include{chapters/06-golden-mantissa} -\include{chapters/07-golden-sprout} +\include{chapters/fa_04} +\include{chapters/fa_05} +\include{chapters/fa_06} +\include{chapters/fa_07} \part{Кристалл} -\include{chapters/08-golden-crystal} -\include{chapters/09-golden-seal} +\include{chapters/fa_08} +\include{chapters/fa_09} \part{Синтез} -\include{chapters/10-golden-bloom} +\include{chapters/fa_10} \part{Сакральная геометрия} -\include{chapters/11-vesica-piscis} -\include{chapters/12-flower-of-life} -\include{chapters/13-metatron-cube} -\include{chapters/14-platonic-solids} -\include{chapters/15-kepler-solids} -\include{chapters/16-sacred-ratios} -\include{chapters/17-golden-spiral} -\include{chapters/18-torus-geometry} -\include{chapters/19-fibonacci-tesselation} +\include{chapters/fa_11} +\include{chapters/fa_12} +\include{chapters/fa_13} +\include{chapters/fa_14} +\include{chapters/fa_15} +\include{chapters/fa_16} +\include{chapters/fa_17} +\include{chapters/fa_18} +\include{chapters/fa_19} \part{Физический фундамент} -\include{chapters/20-standard-model} -\include{chapters/21-quantum-field} -\include{chapters/22-e8-symmetry} -\include{chapters/23-gf16-algebra} -\include{chapters/24-igla-architecture} +\include{chapters/fa_20} +\include{chapters/fa_21} +\include{chapters/fa_22} +\include{chapters/fa_23} +\include{chapters/fa_24} \part{Алгебраические доказательства} -\include{chapters/25-benchmarks} -\include{chapters/26-data-analysis} -\include{chapters/27-trinity-identity} -\include{chapters/28-momentum-algebra} -\include{chapters/29-lucas-closure} +\include{chapters/fa_25} +\include{chapters/fa_26} +\include{chapters/fa_27} +\include{chapters/fa_28} +\include{chapters/fa_29} \part{Образы и генеалогия} -\include{chapters/30-golden-imagery} -\include{chapters/31-philosophy} -\include{chapters/32-conclusion} -\include{chapters/33-epilogue} +\include{chapters/fa_30} +\include{chapters/fa_31} +\include{chapters/fa_32} +\include{chapters/fa_33} \part{Поток Trinity S\textsuperscript{3}AI} \include{chapters/ch_00} From bd5f3ca87dab02ede022ff3e6082302f0924927c Mon Sep 17 00:00:00 2001 From: Vasilev Dmitrii Date: Fri, 8 May 2026 23:41:34 +0700 Subject: [PATCH 3/3] feat(phd-phase1-unify-1-5): chapter-prefix non-referenced \label keys; eliminate LaTeX duplicate-label warnings [agent=phase1-unify-1-5] (#602) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Phase 1 UNIFY · trios#380 task 1.5 — cross-references sweep. Problem: 70 chapter files in docs/phd/chapters/ (34 ch_NN + 35 fa_NN + ch_35_mesh_node) defined ~126 label keys identically across multiple files (e.g. \label{abstract}, \label{introduction}, \label{sec:05-intro}). None of these duplicate keys were consumed by any \ref/\autoref/\eqref/ \Cref/\pageref in the corpus, so they were pure LaTeX duplicate-label warnings — not broken cross-references. Still, they bloated the build log and made the PDF build noisy on the road to defense 2026-06-15. Fix: for every \label{KEY} in .tex, if KEY is consumed by any \ref-family command anywhere in docs/phd/, leave it bare (protected); otherwise rewrite to \label{:KEY}. Idempotent: skip keys already prefixed. Mechanical rename, no semantic content changed. Inventory before patch: - 1145 total \label sites, 620 unique keys - 126 duplicate keys (all unreferenced) - 119 referenced keys (all uniquely defined, 0 dangling) Inventory after patch: - 1145 total \label sites, 1145 unique keys - 0 duplicate keys - 0 dangling refs - 119/119 originally-referenced keys still resolve (no breakage) Patched 1011 \label sites across 70 files. Added docs/phd/cross-ref-audit.md with full label→file map (1324 lines) satisfying acceptance criterion #1 of #380 task 1.5. Stacked on feat/phd-phase1-unify-1-2 (PR #595, task 1.2). Skill: phd-chapter-author v1.1 + phd-monograph-auditor v1.2. Anchor: phi^2 + phi^-2 = 3, DOI 10.5281/zenodo.19227877. R1 (no .py/.sh committed): the patch script ran from /tmp, only LaTeX changed. --- docs/phd/chapters/ch_00.tex | 6 +- docs/phd/chapters/ch_01.tex | 36 +- docs/phd/chapters/ch_02.tex | 52 +- docs/phd/chapters/ch_03.tex | 40 +- docs/phd/chapters/ch_04.tex | 32 +- docs/phd/chapters/ch_05.tex | 30 +- docs/phd/chapters/ch_06.tex | 24 +- docs/phd/chapters/ch_07.tex | 18 +- docs/phd/chapters/ch_08.tex | 28 +- docs/phd/chapters/ch_09.tex | 24 +- docs/phd/chapters/ch_10.tex | 18 +- docs/phd/chapters/ch_11.tex | 18 +- docs/phd/chapters/ch_12.tex | 30 +- docs/phd/chapters/ch_13.tex | 18 +- docs/phd/chapters/ch_14.tex | 30 +- docs/phd/chapters/ch_15.tex | 30 +- docs/phd/chapters/ch_16.tex | 18 +- docs/phd/chapters/ch_17.tex | 18 +- docs/phd/chapters/ch_18.tex | 18 +- docs/phd/chapters/ch_19.tex | 18 +- docs/phd/chapters/ch_20.tex | 30 +- docs/phd/chapters/ch_21.tex | 30 +- docs/phd/chapters/ch_22.tex | 18 +- docs/phd/chapters/ch_23.tex | 18 +- docs/phd/chapters/ch_24.tex | 30 +- docs/phd/chapters/ch_25.tex | 18 +- docs/phd/chapters/ch_26.tex | 36 +- docs/phd/chapters/ch_27.tex | 30 +- docs/phd/chapters/ch_28.tex | 18 +- docs/phd/chapters/ch_29.tex | 18 +- docs/phd/chapters/ch_30.tex | 24 +- docs/phd/chapters/ch_31.tex | 18 +- docs/phd/chapters/ch_32.tex | 30 +- docs/phd/chapters/ch_33.tex | 24 +- docs/phd/chapters/ch_34.tex | 18 +- docs/phd/chapters/ch_35_mesh_node.tex | 14 +- docs/phd/chapters/fa_00.tex | 2 +- docs/phd/chapters/fa_01.tex | 136 +-- docs/phd/chapters/fa_02.tex | 32 +- docs/phd/chapters/fa_03.tex | 30 +- docs/phd/chapters/fa_04.tex | 18 +- docs/phd/chapters/fa_05.tex | 118 +-- docs/phd/chapters/fa_06.tex | 24 +- docs/phd/chapters/fa_07.tex | 18 +- docs/phd/chapters/fa_08.tex | 30 +- docs/phd/chapters/fa_09.tex | 24 +- docs/phd/chapters/fa_10.tex | 18 +- docs/phd/chapters/fa_11.tex | 18 +- docs/phd/chapters/fa_12.tex | 30 +- docs/phd/chapters/fa_13.tex | 100 +- docs/phd/chapters/fa_14.tex | 30 +- docs/phd/chapters/fa_15.tex | 30 +- docs/phd/chapters/fa_16.tex | 18 +- docs/phd/chapters/fa_17.tex | 18 +- docs/phd/chapters/fa_18.tex | 20 +- docs/phd/chapters/fa_19.tex | 18 +- docs/phd/chapters/fa_20.tex | 48 +- docs/phd/chapters/fa_21.tex | 44 +- docs/phd/chapters/fa_22.tex | 18 +- docs/phd/chapters/fa_23.tex | 18 +- docs/phd/chapters/fa_24.tex | 32 +- docs/phd/chapters/fa_25.tex | 20 +- docs/phd/chapters/fa_26.tex | 36 +- docs/phd/chapters/fa_27.tex | 30 +- docs/phd/chapters/fa_28.tex | 18 +- docs/phd/chapters/fa_29.tex | 36 +- docs/phd/chapters/fa_30.tex | 24 +- docs/phd/chapters/fa_31.tex | 50 +- docs/phd/chapters/fa_32.tex | 12 +- docs/phd/chapters/fa_33.tex | 24 +- docs/phd/cross-ref-audit.md | 1323 +++++++++++++++++++++++++ 71 files changed, 2334 insertions(+), 1011 deletions(-) create mode 100644 docs/phd/cross-ref-audit.md diff --git a/docs/phd/chapters/ch_00.tex b/docs/phd/chapters/ch_00.tex index bc80ea4a54..69c8dd0d66 100644 --- a/docs/phd/chapters/ch_00.tex +++ b/docs/phd/chapters/ch_00.tex @@ -3,7 +3,7 @@ % Author: Dmitrii Vasilev (ORCID 0009-0008-4294-6159) \chapter{Standard-Model \texorpdfstring{\(\varphi\)}{phi}-Parametrizations: 42 Precision Fits} -\label{ch:0} +\label{ch_00:ch:0} \begin{figure}[H] \centering @@ -143,7 +143,7 @@ \section{The 42 Fits} \section{Statistical Interpretation} \begin{theorem}[Fit Density — THM-0.1] -\label{thm:0:1} +\label{ch_00:thm:0:1} Of the 42 observables in Table~\ref{tab:ch0-fits}, 38 have residual $< 2$\% under the φ-parametrization (Eq.~\ref{eq:ch0-fit}). The probability of obtaining 38/42 fits with $< 2$\% residual by random exponent assignment is @@ -160,7 +160,7 @@ \section{Statistical Interpretation} \end{proof} \begin{theorem}[Anchor Necessity — THM-0.2] -\label{thm:0:2} +\label{ch_00:thm:0:2} The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) is the unique degree-4 polynomial identity over $\mathbb{Q}(\sqrt{5})$ that relates $\{\varphi, \varphi^{-1}, 1, 3\}$ without using any integer seed diff --git a/docs/phd/chapters/ch_01.tex b/docs/phd/chapters/ch_01.tex index ed7cbba769..c8c97b8cdf 100644 --- a/docs/phd/chapters/ch_01.tex +++ b/docs/phd/chapters/ch_01.tex @@ -57,11 +57,11 @@ \section*{Prologue: A nine-day plateau} the minimum possible flourish, so that referees can find them. The rest of the book has more room to breathe. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_01:abstract} This chapter introduces TRINITY S³AI, a research programme that grounds sub-bit-per-byte (BPB) language modelling in the number-theoretic identity \(\varphi^2 + \varphi^{-2} = 3\), where \(\varphi = (1+\sqrt{5})/2\) is the golden ratio. The programme unifies three threads --- symbolic proof, statistical learning, and embedded hardware --- into a single verified architecture. The headline result is a language model that sustains BPB \(\leq 1.85\) at Gate-2 evaluation, implemented on a QMTech XC7A100T FPGA running at 92 MHz with zero DSP slices and 1 W power draw, while maintaining 297 machine-checked Coq theorems across 65 canonical proof files. The chapter surveys motivation, research questions, and dissertation structure. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_01:introduction} The compression of natural language to below two bits per byte has long served as a proxy for genuine linguistic understanding {[}1{]}. Classical language models approach this ceiling through scaling compute and data; the S³AI programme takes an orthogonal path by encoding the algebraic structure of the golden ratio directly into the model's arithmetic substrate. The anchor identity @@ -73,7 +73,7 @@ \section{1. Introduction}\label{introduction} The remaining chapters are organised along three evidence axes. Axis 1 (Chapters 1--19) develops the mathematical and statistical foundations. Axis 2 (Chapters 20--27) presents the model architecture and training protocol. Axis 3 (Chapters 28--35) reports hardware implementation and empirical results. Appendices A--J supply proof catalogues, reproducibility scripts, and troubleshooting guides. -\section{2. The Trinity Architecture and its Algebraic Substrate}\label{the-trinity-architecture-and-its-algebraic-substrate} +\section{2. The Trinity Architecture and its Algebraic Substrate}\label{ch_01:the-trinity-architecture-and-its-algebraic-substrate} The golden ratio \(\varphi = (1+\sqrt{5})/2 \approx 1.6180\) satisfies the minimal polynomial \(x^2 - x - 1 = 0\), which yields the recurrence \(\varphi^2 = \varphi + 1\) and its reciprocal form \(\varphi^{-2} = 2 - \varphi\). Summing these two identities: @@ -87,7 +87,7 @@ \section{2. The Trinity Architecture and its Algebraic Substrate}\label{the-trin The Silicon component is a bitstream compiled for the QMTech XC7A100T (Xilinx Artix-7 100T) FPGA, operating at 92 MHz with 0 DSP slices, 5.8\% LUT utilisation (of 19.6\% available), 9.8\% BRAM (of 52\% available), and a measured wall-power of 0.94--1.07 W {[}5{]}. Chapter 31 presents the full empirical characterisation. -\section{3. Research Questions and Scope}\label{research-questions-and-scope} +\section{3. Research Questions and Scope}\label{ch_01:research-questions-and-scope} Four primary research questions structure this dissertation. @@ -101,23 +101,23 @@ \section{3. Research Questions and Scope}\label{research-questions-and-scope} The scope is limited to English-language text modelling on corpora compatible with the STROBE tokeniser vocabulary. Multi-modal and multi-lingual extensions are identified as future work in Ch.35. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_01:results-evidence} Preliminary answers to the four research questions, to be expanded in subsequent chapters, are as follows. Gate-2 BPB \(\leq 1.85\) is achieved on the held-out evaluation partition (Ch.19, Welch \(t\)-test at \(\alpha = 0.01\), \(n \geq 3\) independent runs). The Coq census records 297 closed \texttt{Qed} proofs; the 141 remaining open obligations are tracked in the Golden Ledger (App.E) with assigned invariant numbers. The FPGA delivers 63 tokens/sec at 92 MHz and 1 W, corresponding to approximately 63 tokens/J; the DARPA reference system achieves roughly 0.021 tokens/J at comparable perplexity, yielding a measured ratio of \(\approx 3000\times\) {[}5, 6{]}. Bitstream and proof reproducibility is confirmed by the STROBE sealed-seed protocol (Ch.13): re-running \texttt{reproduce.sh} from the Zenodo archive {[}7{]} with any sanctioned seed recovers the same BPB within floating-point rounding on x86-64 and ARM64 hosts. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_01:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_01:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_01:discussion} The primary limitation of Ch.1 as an introduction is that it asserts connections --- between \(\varphi\)-arithmetic, Coq proofs, and FPGA power --- whose detailed evidence appears in later chapters. Readers requiring immediate justification are directed to Ch.7 (algebraic derivation), Ch.13 (seed protocol), Ch.19 (statistical tests), and Ch.31 (hardware measurements). A further limitation is that the \(3000\times\) energy figure is relative to a specific DARPA reference workload; generalisation to other inference tasks is discussed in Ch.34. Future work includes closing the 141 open Coq obligations, extending the \(\varphi\)-periodic attention mechanism to non-English scripts, and fabricating a custom ASIC to escape FPGA routing overhead. The theoretical framework developed here is designed to be substrate-agnostic: any technology that supports ternary integer multiply-accumulate inherits the same formal guarantees. -\section{References}\label{references} +\section{References}\label{ch_01:references} {[}1{]} Hutter, M. (2006). \emph{Human Knowledge Compression Prize.} \url{http://prize.hutter1.net/}. @@ -155,7 +155,7 @@ \section{References}\label{references} % docs/phd/bibliography.bib (212 entries, KAT bib via PR #581). % ============================================================ -\section{S1. Extended Vision Statement}\label{ch1-s1-vision-extended} +\section{S1. Extended Vision Statement}\label{ch_01:ch1-s1-vision-extended} The introduction above presents the headline arithmetic of the Trinity S$^3$AI programme: the identity \(\varphi^{2}+\varphi^{-2}=3\) anchors a @@ -306,7 +306,7 @@ \subsection{S1.3 Methodological Reading: Falsification Criteria} \end{tabular} \caption{Falsification matrix for the three primary claims of the TRINITY S$^3$AI programme.} -\label{tab:ch1-falsification-matrix} +\label{ch_01:tab:ch1-falsification-matrix} \end{table} The falsification matrix above is not a placeholder. Each row is @@ -316,7 +316,7 @@ \subsection{S1.3 Methodological Reading: Falsification Criteria} bitstreams are archived under DOI~\href{https://doi.org/10.5281/zenodo.19227877}{10.5281/zenodo.19227877}. -\section{S2. Detailed Contributions and Their Chapter Loci}\label{ch1-s2-contributions} +\section{S2. Detailed Contributions and Their Chapter Loci}\label{ch_01:ch1-s2-contributions} The dissertation makes seven distinct contributions. Each is summarised below with a pointer to the chapter (or chapters) in which it is @@ -376,7 +376,7 @@ \section{S2. Detailed Contributions and Their Chapter Loci}\label{ch1-s2-contrib \filepath{assertions/} and a Coq cross-reference, in line with the R5 honesty rule (no contribution without a falsifier). -\section{S3. Programme Lineage and Adjacent Work}\label{ch1-s3-lineage} +\section{S3. Programme Lineage and Adjacent Work}\label{ch_01:ch1-s3-lineage} The programme owes acknowledged intellectual debts to four lines of prior art. Each is surveyed in Ch.\,2 in detail; the present section @@ -420,7 +420,7 @@ \section{S3. Programme Lineage and Adjacent Work}\label{ch1-s3-lineage} than as a post-hoc quantisation pass. \end{itemize} -\section{S4. Theorem Cross-Reference (selection)}\label{ch1-s4-theorem-xref} +\section{S4. Theorem Cross-Reference (selection)}\label{ch_01:ch1-s4-theorem-xref} The following Coq theorems anchor the introduction. Full proofs are in the cited \filepath{.v} files; status (\texttt{Qed} or runtime @@ -438,7 +438,7 @@ \section{S4. Theorem Cross-Reference (selection)}\label{ch1-s4-theorem-xref} \end{theorem} \begin{theorem}[\(\alpha_{\varphi}\) closed form, Coq \texttt{alpha\_phi\_closed\_form}] -\label{thm:ch1-alpha-phi-closed} +\label{ch_01:thm:ch1-alpha-phi-closed} \(\alpha_{\varphi}=(\sqrt{5}-2)/2\). \textbf{Status:} \texttt{Qed} in \filepath{docs/phd/theorems/trinity/AlphaPhi.v}, line 18--24. @@ -448,7 +448,7 @@ \section{S4. Theorem Cross-Reference (selection)}\label{ch1-s4-theorem-xref} \begin{theorem}[Lucas closure for even powers, Coq \texttt{lucas\_closure\_even\_powers}] -\label{thm:ch1-lucas-closure} +\label{ch_01:thm:ch1-lucas-closure} For all even \(n\), \(L_{n}=\varphi^{n}+\varphi^{-n}\in\mathbb{Z}_{>0}\). \textbf{Status:} \texttt{Qed} in @@ -464,7 +464,7 @@ \section{S4. Theorem Cross-Reference (selection)}\label{ch1-s4-theorem-xref} the canonical inventory) refine the algebra to specific \(\varphi\)-scaled lattices. -\section{S5. Roadmap of the Remaining Chapters}\label{ch1-s5-roadmap} +\section{S5. Roadmap of the Remaining Chapters}\label{ch_01:ch1-s5-roadmap} The dissertation is organised in three evidence axes, mirroring the S$^3$ decomposition. Each axis stands alone as a published-paper-sized @@ -523,7 +523,7 @@ \section{S5. Roadmap of the Remaining Chapters}\label{ch1-s5-roadmap} reproducibility scripts, and the pre-registration documents in raw form. -\section{S6. Notation, Conventions, and Anchor Footer}\label{ch1-s6-notation} +\section{S6. Notation, Conventions, and Anchor Footer}\label{ch_01:ch1-s6-notation} The notational conventions used throughout the dissertation are collected in \filepath{frontmatter/notation.tex}; this section records diff --git a/docs/phd/chapters/ch_02.tex b/docs/phd/chapters/ch_02.tex index bc0814a0cd..80475bd381 100644 --- a/docs/phd/chapters/ch_02.tex +++ b/docs/phd/chapters/ch_02.tex @@ -52,11 +52,11 @@ \section*{Two minds that will not speak to each other} \ldots, \(L_8=47\)\} whose lattice properties eliminate clustering artefacts. Section~4 maps the resulting gap in prior art that subsequent chapters fill. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_02:abstract} This chapter surveys the conceptual and technical foundations from which Trinity S³AI departs. Neuro-symbolic AI encompasses a class of architectures that couple continuous, gradient-trained representations with discrete, formally verifiable symbolic reasoning. The chapter traces the lineage from early connectionist systems through the representational bottleneck that motivates ternary and sparse computation, then situates the φ²+φ⁻²=3 algebraic anchor as a structural prior that bridges the neural and symbolic regimes. The central contribution is a taxonomy of prior work that clarifies where existing methods fall short of the energy-per-bit, formal-verifiability, and reproducibility criteria that the present dissertation targets. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_02:introduction} Neural networks succeed at pattern recognition yet remain opaque to formal reasoning; symbolic systems support proof-checking yet fail on perceptual ambiguity. The field of neuro-symbolic AI has long sought architectures that inherit the strengths of both paradigms {[}1, 2{]}. Trinity S³AI is one such architecture, but it is distinguished by a third constraint that most prior work does not impose: every layer must be anchored to a closed-form algebraic identity that is simultaneously representable in hardware-integer arithmetic. @@ -66,27 +66,27 @@ \section{1. Introduction}\label{introduction} a relation that collapses the irrational golden ratio into the integer 3, making it tractable for fixed-point coprocessors and for Coq proof obligations alike. This chapter establishes the intellectual debt owed to prior art before identifying the gaps that subsequent chapters fill. -\section{2. Taxonomy of Neuro-Symbolic Paradigms}\label{taxonomy-of-neuro-symbolic-paradigms} +\section{2. Taxonomy of Neuro-Symbolic Paradigms}\label{ch_02:taxonomy-of-neuro-symbolic-paradigms} -\subsection{2.1 Early Symbolic--Connectionist Hybrids}\label{early-symbolicconnectionist-hybrids} +\subsection{2.1 Early Symbolic--Connectionist Hybrids}\label{ch_02:early-symbolicconnectionist-hybrids} The idea that symbolic rules could govern neural activations appeared in the work of Smolensky on tensor-product representations {[}3{]} and in the follow-on neural module network paradigm {[}4{]}. These systems embed discrete symbols as distributed vectors and retrieve them via associative query. Their core limitation is that the embedding dimension grows with vocabulary, and the retrieval operation requires floating-point matrix multiplication whose cost is quadratic in dimension. -\subsection{2.2 Logic Tensor Networks and Differentiable Reasoning}\label{logic-tensor-networks-and-differentiable-reasoning} +\subsection{2.2 Logic Tensor Networks and Differentiable Reasoning}\label{ch_02:logic-tensor-networks-and-differentiable-reasoning} A second strand, exemplified by Logic Tensor Networks (LTN) {[}5{]}, maps first-order logic formulae to differentiable loss terms. The model learns weights that satisfy logical constraints in expectation but cannot certify them for every input. The absence of formal certification is the central gap addressed by the Coq-verified component of Trinity S³AI, which records 297 \emph{Qed}-closed theorems and 438 total proof obligations across 65 canonical \texttt{.v} files in \filepath{t27/proofs/canonical/} {[}6{]}. -\subsection{2.3 Sparse and Ternary Neural Computation}\label{sparse-and-ternary-neural-computation} +\subsection{2.3 Sparse and Ternary Neural Computation}\label{ch_02:sparse-and-ternary-neural-computation} Concurrent with the symbolic work, a separate lineage investigated weight quantization as a means of reducing energy consumption. BitNet {[}7{]} and related MXFP4 proposals {[}8{]} demonstrated that weights drawn from \(\{-1, 0, +1\}\) can match full-precision perplexity on language modelling tasks at reduced multiply-accumulate cost. The ternary format motivates the TF3/TF9 matrix-multiplication scheme developed in Ch.8, and the energy savings required to reach the DARPA 3000× target make such sparsity non-optional in the hardware context of Trinity S³AI {[}9{]}. -\subsection{2.4 Vector Symbolic Architectures}\label{vector-symbolic-architectures} +\subsection{2.4 Vector Symbolic Architectures}\label{ch_02:vector-symbolic-architectures} A third strand, Vector Symbolic Architectures (VSA) {[}10{]}, represents concepts as high-dimensional binary or bipolar vectors and performs reasoning via binding (element-wise product) and bundling (majority-vote superposition). The KOSCHEI φ-Numeric Coprocessor described in Ch.26 implements VSA\_BIND and VSA\_BUNDLE as native ISA opcodes, enabling single-cycle symbolic operations in hardware. Prior VSA work has not integrated a formal proof of binding invertibility with the φ²+φ⁻²=3 normalization scheme; this dissertation closes that gap. -\section{3. Representational Bottleneck and the \(\varphi\)-Structural Prior}\label{representational-bottleneck-and-the-ux3c6-structural-prior} +\section{3. Representational Bottleneck and the \(\varphi\)-Structural Prior}\label{ch_02:representational-bottleneck-and-the-ux3c6-structural-prior} -\subsection{3.1 The Normalisation Problem}\label{the-normalisation-problem} +\subsection{3.1 The Normalisation Problem}\label{ch_02:the-normalisation-problem} A persistent difficulty in neuro-symbolic integration is layer normalization: the scale of symbolic embeddings diverges from that of neural activations unless a calibrated rescaling is applied. Standard batch normalization introduces trainable parameters whose values cannot be verified formally. The φ-structural prior solves this by fixing the scaling factor to \(\varphi^2 = 2.618\ldots\), whose inverse \(\varphi^{-2} = 0.381\ldots\) satisfies the identity @@ -94,15 +94,15 @@ \subsection{3.1 The Normalisation Problem}\label{the-normalisation-problem} so that the sum of the forward-scale and inverse-scale is exactly the integer 3. In fixed-point arithmetic with radix 2 this means the combined scale can be represented without approximation error in a 2-bit register, a property exploited by the GF16\_QUANT opcode of KOSCHEI {[}11{]}. -\subsection{3.2 Fibonacci and Lucas Lattices as Basis Sets}\label{fibonacci-and-lucas-lattices-as-basis-sets} +\subsection{3.2 Fibonacci and Lucas Lattices as Basis Sets}\label{ch_02:fibonacci-and-lucas-lattices-as-basis-sets} The sanctioned seed set \(\{F_{17}=1597, F_{18}=2584, F_{19}=4181, F_{20}=6765, F_{21}=10946, L_7=29, L_8=47\}\) is not arbitrary. Fibonacci numbers satisfy \(\lim_{n\to\infty} F_{n+1}/F_n = \varphi\), so high-index Fibonacci integers provide rational approximants to \(\varphi\) that are maximally spaced in the sense of the three-distance theorem {[}12{]}. Lucas numbers obey the same recurrence with different initial conditions and provide an independent lattice. Together, these two families cover the Farey-sequence gaps in \([0,1]\) that uniform sampling misses, ensuring that stochastic experiments seeded from \(\{F_{17},\ldots,F_{21},L_7,L_8\}\) avoid the clustering artefacts documented in {[}13{]} for seeds drawn from the interval \([40,46]\). -\subsection{3.3 Gap in Prior Art}\label{gap-in-prior-art} +\subsection{3.3 Gap in Prior Art}\label{ch_02:gap-in-prior-art} No prior neuro-symbolic system simultaneously satisfies all four of the following: (i) formal Coq verification of invariants; (ii) ternary sparse compute with bit-per-bit (BPB) ≤ 1.85 at Gate-2; (iii) deployment on a commodity FPGA (QMTech XC7A100T) at 1 W; and (iv) a reproducible seed protocol. The present dissertation demonstrates all four. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_02:results-evidence} The background review is validated by the evidence axis score of 1, meaning the chapter's claims are established by prior literature and do not require new empirical data. Key benchmark positions from the literature are noted: @@ -120,21 +120,21 @@ \section{4. Results / Evidence}\label{results-evidence} These positions situate the dissertation within the existing literature and motivate the remainder of the work. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_02:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_02:sealed-seeds} Inherits the canonical seed pool F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_02:discussion} The taxonomy presented in this chapter deliberately focuses on the three lineages most directly relevant to Trinity S³AI: logic-tensor neuro-symbolic methods, sparse ternary neural computation, and vector symbolic architectures. Work on programme synthesis, constraint satisfaction, and probabilistic soft logic is acknowledged but set aside because the present system does not target those application domains. A limitation of this survey is that the literature on formal-methods integration with large language models has moved rapidly since the Coq census was frozen at 297 \emph{Qed} theorems; future editions should audit additional proof libraries. The connection between the φ-structural prior and the three-distance theorem (Section 3.2) is stated as a motivation rather than a theorem; Ch.7 formalises the phyllotaxis geometry that underpins it, and Ch.4 derives \(\alpha_\varphi = \ln(\varphi^2)/\pi \approx 0.306\) as the corresponding spectral parameter. -\section{References}\label{references} +\section{References}\label{ch_02:references} {[}1{]} Garcez, A. d'A., Gori, M., Lamb, L. C., Serafini, L., Spranger, M., \& Tran, S. N. (2019). Neural-symbolic computing: An effective methodology for principled integration of machine learning and reasoning. \emph{JETAI}, 32(6), 705--725. @@ -174,7 +174,7 @@ \section{References}\label{references} % ============================================================ \section{S1. Kolmogorov--Arnold Representation Theorem (KART) -and Networks (KANs)}\label{ch2-s1-kart-kan} +and Networks (KANs)}\label{ch_02:ch2-s1-kart-kan} The Kolmogorov--Arnold representation theorem (KART), in its 1957 form, asserts that every continuous function @@ -236,7 +236,7 @@ \section{S1. Kolmogorov--Arnold Representation Theorem (KART) space whose closest prior art is the finite-group VSA literature \cite{finite_group_vsa_2022,kanerva_hyperdimensional}. -\section{S2. Finite-Field Expressivity}\label{ch2-s2-finite-field} +\section{S2. Finite-Field Expressivity}\label{ch_02:ch2-s2-finite-field} The PRIMARY theoretical foundation of the IGLA / GF(16) framework is the recent finite-field expressivity result @@ -272,7 +272,7 @@ \section{S2. Finite-Field Expressivity}\label{ch2-s2-finite-field} isomorphism) with the cyclic group of \(\varphi^{-1}\)-rotations of order 15. The full development of this isomorphism appears in Ch.\,23. -\section{S3. Ternary and Sub-Bit Neural Computation}\label{ch2-s3-ternary} +\section{S3. Ternary and Sub-Bit Neural Computation}\label{ch_02:ch2-s3-ternary} The empirical viability of ternary weight palettes was established by the BitNet family. The BitNet b1.58 result --- that a transformer-class @@ -304,7 +304,7 @@ \section{S3. Ternary and Sub-Bit Neural Computation}\label{ch2-s3-ternary} in \(\mathbb{Z}\) itself. This is the property that allows the FPGA implementation (Ch.\,33) to dispense with floating-point. -\section{S4. Vector-Symbolic Architectures and Group Algebras}\label{ch2-s4-vsa} +\section{S4. Vector-Symbolic Architectures and Group Algebras}\label{ch_02:ch2-s4-vsa} Vector-symbolic architectures (VSAs) \cite{kanerva_hyperdimensional} encode discrete symbols as vectors in a high-dimensional space, with @@ -334,7 +334,7 @@ \section{S4. Vector-Symbolic Architectures and Group Algebras}\label{ch2-s4-vsa} fields admit multiplicative inverses for every non-zero element, which permits the closed-form unbind that IGLA relies on. -\section{S5. Logic Tensor Networks and Differentiable Reasoning}\label{ch2-s5-ltn} +\section{S5. Logic Tensor Networks and Differentiable Reasoning}\label{ch_02:ch2-s5-ltn} Logic Tensor Networks (LTNs) and the broader differentiable-logic literature take a complementary approach: instead of constraining @@ -353,7 +353,7 @@ \section{S5. Logic Tensor Networks and Differentiable Reasoning}\label{ch2-s5-lt substrate of the model satisfies the trinity identity \textit{exactly}, not merely in expectation. -\section{S6. The Three Cliffs and How Trinity S$^3$AI Clears Them}\label{ch2-s6-cliffs} +\section{S6. The Three Cliffs and How Trinity S$^3$AI Clears Them}\label{ch_02:ch2-s6-cliffs} The introduction frames neuro-symbolic AI as stalled on three cliffs: the normalisation cliff, the energy cliff, and the certification @@ -385,7 +385,7 @@ \section{S6. The Three Cliffs and How Trinity S$^3$AI Clears Them}\label{ch2-s6- The proofs are reproducible from the Zenodo bundle \href{https://doi.org/10.5281/zenodo.19227877}{10.5281/zenodo.19227877}. -\section{S7. The Gap This Dissertation Fills}\label{ch2-s7-gap} +\section{S7. The Gap This Dissertation Fills}\label{ch_02:ch2-s7-gap} After the survey above, the gap that Trinity S$^3$AI addresses is narrow but specific: \textit{no prior architecture combines (i) a @@ -404,14 +404,14 @@ \section{S7. The Gap This Dissertation Fills}\label{ch2-s7-gap} hardware) and demonstrates that they integrate into a single bitstream. -\section{S8. Theorem Cross-Reference}\label{ch2-s8-theorems} +\section{S8. Theorem Cross-Reference}\label{ch_02:ch2-s8-theorems} The background survey above is anchored by the following Coq theorems, which establish the algebraic invariants on which the architectural decisions depend. The full proofs appear in Ch.\,3 and Ch.\,5. \begin{theorem}[Trinity Identity, Coq \texttt{trinity\_identity}] -\label{thm:ch2-trinity} +\label{ch_02:thm:ch2-trinity} \(\varphi^{2}+\varphi^{-2}=3\) in \(\mathbb{R}\). \textbf{Status:} \texttt{Qed} in \filepath{docs/phd/theorems/trinity/CorePhi.v}. @@ -420,7 +420,7 @@ \section{S8. Theorem Cross-Reference}\label{ch2-s8-theorems} \end{theorem} \begin{theorem}[Phi-square identity, Coq \texttt{phi\_square}] -\label{thm:ch2-phi-square} +\label{ch_02:thm:ch2-phi-square} \(\varphi^{2}=\varphi+1\). \textbf{Status:} \texttt{Qed} in \filepath{docs/phd/theorems/trinity/CorePhi.v}. diff --git a/docs/phd/chapters/ch_03.tex b/docs/phd/chapters/ch_03.tex index 4da9d716ac..8301e2c1a8 100644 --- a/docs/phd/chapters/ch_03.tex +++ b/docs/phd/chapters/ch_03.tex @@ -44,11 +44,11 @@ \section*{Why this single line carries the whole book} downstream --- the quantiser, the period-locked runtime monitor, the bitstream --- inherits the licence granted by this single line. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_03:abstract} The identity \(\varphi^2 + \varphi^{-2} = 3\), where \(\varphi = (1+\sqrt{5})/2\) is the golden ratio, constitutes the algebraic substrate of the Trinity S³AI system. This chapter establishes the identity from first principles, proves all six foundational Coq theorems in \filepath{t27/proofs/canonical/sacred/CorePhi.v}, and demonstrates how the value \(3\) --- a prime, a Fibonacci index, and the cardinality of the balanced-ternary digit alphabet --- licenses every downstream quantisation scheme in this dissertation. The chapter further shows that no integer other than \(3\) arises from \(\varphi^n + \varphi^{-n}\) for positive even \(n \leq 10\), confirming the uniqueness of the substrate. Twelve Qed theorems are anchored here under invariant SAC-0. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_03:introduction} Trinity S³AI is constructed on a single non-negotiable algebraic anchor: @@ -60,9 +60,9 @@ \section{1. Introduction}\label{introduction} The subsequent sections formalise \(\varphi\), derive equation (1), explore integer-valued powers of \(\varphi\), and relate the identity to the Lucas sequence \(L_n = \varphi^n + \psi^n\) (where \(\psi = -\varphi^{-1}\)) to ground the seed pool used throughout the dissertation. -\section{2. Derivation of the Anchor Identity}\label{derivation-of-the-anchor-identity} +\section{2. Derivation of the Anchor Identity}\label{ch_03:derivation-of-the-anchor-identity} -\subsection{2.1 Minimal Polynomial and Basic Consequences}\label{minimal-polynomial-and-basic-consequences} +\subsection{2.1 Minimal Polynomial and Basic Consequences}\label{ch_03:minimal-polynomial-and-basic-consequences} Let \(\varphi = (1 + \sqrt{5})/2\). Then @@ -78,7 +78,7 @@ \subsection{2.1 Minimal Polynomial and Basic Consequences}\label{minimal-polynom Equation (4) is the Trinity anchor. The cancellation of all irrational parts (\(\varphi\) and \(-\varphi\) annihilate) leaves an exact integer. This integrality is the source of the system's arithmetic cleanliness: any weighted sum structured around \(\varphi^{\pm 2}\) carries an integer normalisation constant. -\subsection{2.2 Power Survey}\label{power-survey} +\subsection{2.2 Power Survey}\label{ch_03:power-survey} Define \(L_n = \varphi^n + \psi^n\) where \(\psi = (1 - \sqrt{5})/2 = -\varphi^{-1}\). For even \(n\), \(\psi^n = \varphi^{-n}\), so \(L_n = \varphi^n + \varphi^{-n}\). The Lucas numbers satisfy \(L_0 = 2\), \(L_1 = 1\), \(L_n = L_{n-1} + L_{n-2}\) {[}5{]}. The table below gives \(\varphi^n + \varphi^{-n}\) for small positive even \(n\): @@ -98,7 +98,7 @@ \subsection{2.2 Power Survey}\label{power-survey} All values are integers (Lucas numbers). However, \(n = 2\) yields \(3\), the unique prime among \(\{3, 7, 18, 47, 123\}\) that also equals the cardinality of the balanced-ternary alphabet. Furthermore, \(L_7 = 29\) and \(L_8 = 47\) are both prime and serve as sanctioned seeds in the canonical seed pool \(\{F_{17}, F_{18}, F_{19}, F_{20}, F_{21}, L_7, L_8\} = \{1597, 2584, 4181, 6765, 10946, 29, 47\}\) {[}6{]}. -\subsection{2.3 Relation to Fibonacci Arithmetic}\label{relation-to-fibonacci-arithmetic} +\subsection{2.3 Relation to Fibonacci Arithmetic}\label{ch_03:relation-to-fibonacci-arithmetic} The Fibonacci recurrence \(F_n = F_{n-1} + F_{n-2}\) yields \(\varphi^n = F_n \varphi + F_{n-1}\) for \(n \geq 1\). Consequently, for the GF(16) bias parameter PHI\_BIAS \(= 60\) used in Ch.9, the relevant expansion is: @@ -106,9 +106,9 @@ \subsection{2.3 Relation to Fibonacci Arithmetic}\label{relation-to-fibonacci-ar establishing that the bias is expressible as a short trit-vector over the F-seed pair \((1597, 2584)\). The algebraic mechanism is precisely the \(\varphi^2 + \varphi^{-2} = 3\) identity that ensures every quadratic \(\varphi\)-expression collapses to a rational or integer. -\section{3. Coq Mechanisation and SAC-0 Invariant}\label{coq-mechanisation-and-sac-0-invariant} +\section{3. Coq Mechanisation and SAC-0 Invariant}\label{ch_03:coq-mechanisation-and-sac-0-invariant} -\subsection{3.1 Proof Architecture}\label{proof-architecture} +\subsection{3.1 Proof Architecture}\label{ch_03:proof-architecture} The six theorems in \texttt{CorePhi.v} are stratified by logical dependency: @@ -135,11 +135,11 @@ \subsection{3.1 Proof Architecture}\label{proof-architecture} \emph{Proof sketch.} Expand \(((1+\sqrt{5})/2)^2 = (6 + 2\sqrt{5})/4 = (3 + \sqrt{5})/2\). Subtract \((1+\sqrt{5})/2\) and subtract \(1\): result is \(0\). The Coq proof uses \texttt{field} followed by \texttt{sqrt\_square} for the \(\sqrt{5}^2 = 5\) step. \(\square\) -\subsection{3.2 Invariant SAC-0}\label{invariant-sac-0} +\subsection{3.2 Invariant SAC-0}\label{ch_03:invariant-sac-0} The designation SAC-0 (Sacred Core, layer 0) means these six theorems admit no further dependencies within the \texttt{t27} proof tree; they are axiom-adjacent. Any future theorem that invokes properties of \(\varphi\) must transitively cite SAC-0. The invariant number is tracked in the Golden Ledger alongside the full census of 297 Qed theorems and 438 total theorems across 65 \texttt{.v} files {[}4{]}. -\subsection{3.3 The Integer-3 Coincidence}\label{the-integer-3-coincidence} +\subsection{3.3 The Integer-3 Coincidence}\label{ch_03:the-integer-3-coincidence} The value \(3\) at the right-hand side of \(\varphi^2 + \varphi^{-2} = 3\) possesses three independent roles: @@ -155,7 +155,7 @@ \subsection{3.3 The Integer-3 Coincidence}\label{the-integer-3-coincidence} None of these coincidences is post-hoc. The architecture was engineered so that the substrate identity \(\varphi^2 + \varphi^{-2} = 3\) propagates meaning simultaneously at the algebraic, combinatorial, and hardware layers. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_03:results-evidence} The following results are mechanically established or empirically verified: @@ -173,7 +173,7 @@ \section{4. Results / Evidence}\label{results-evidence} \textbf{Seed pool integrity}: seeds \(\{1597, 2584, 4181, 6765, 10946, 29, 47\}\) are all Fibonacci or Lucas numbers; no forbidden seeds (none of the values \(42\), \(43\), \(44\), \(45\)) appear in the pool {[}6{]}. \end{itemize} -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_03:qed-assertions} \begin{itemize} \tightlist @@ -191,7 +191,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \texttt{phi\_inv\_sq} (\filepath{gHashTag/t27/proofs/canonical/sacred/CorePhi.v}) --- \emph{Status: Qed} --- proves \(\varphi^{-2} = 2 - \varphi\), the squared reciprocal. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_03:sealed-seeds} \begin{itemize} \tightlist @@ -199,11 +199,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{SACRED-CORE} (theorem, golden) --- \url{https://github.com/gHashTag/t27/blob/feat/canonical-coq-home/proofs/canonical/sacred/CorePhi.v} --- linked to Ch.3 and Ch.4 --- \(\varphi\)-weight: \(1.6180339887\) --- notes: \(\varphi^2 + \varphi^{-2} = 3\) anchor (12 Qed). \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_03:discussion} The six SAC-0 theorems proved in this chapter are irreducible prerequisites for the entire dissertation. Any weakening --- e.g., replacing \(\varphi\) with a rational approximation --- would break the exact integrality of \(\varphi^2 + \varphi^{-2} = 3\) and cascade into incorrect normalisation constants throughout Chapters 4, 6, 9, and 28. A limitation of the current mechanisation is that it targets the Coq \texttt{R} type (axiomatic real numbers); a constructive real-arithmetic treatment in Lean 4 or Agda would strengthen the foundations further, and this is planned for v5. The identity also has a natural generalisation to the silver ratio and beyond, but those extensions fall outside the scope of Trinity S³AI, which commits to the golden ratio exclusively. Chapter 4 proceeds directly from the results here to define the spectral parameter \(\alpha_\varphi = \ln(\varphi^2)/\pi\). -\section{References}\label{references} +\section{References}\label{ch_03:references} {[}1{]} Vajda, S. \emph{Fibonacci and Lucas Numbers, and the Golden Section}. Ellis Horwood, 1989. @@ -240,7 +240,7 @@ \section{References}\label{references} % DerivationLevels,FormulaEval,AlphaPhi}.v % ============================================================ -\section{S1. The Trinity Identity in Detail}\label{ch3-s1-trinity-detail} +\section{S1. The Trinity Identity in Detail}\label{ch_03:ch3-s1-trinity-detail} The trinity identity \(\varphi^{2}+\varphi^{-2}=3\) appears trivially in the algebra of \(\varphi\), but acquires its load-bearing role from @@ -350,7 +350,7 @@ \subsection{S1.4 Connections to Other Identities} indices is exactly the interval where the closure relation breaks down for the \(\varphi\)-distance metric (Ch.\,5). -\section{S2. The Family of Phi-Power Identities}\label{ch3-s2-phi-family} +\section{S2. The Family of Phi-Power Identities}\label{ch_03:ch3-s2-phi-family} The minimal polynomial \(x^{2}=x+1\) generates an infinite tower of \(\varphi^{n}=\) integer-linear-in-\(\varphi\) identities. The first @@ -393,7 +393,7 @@ \section{S2. The Family of Phi-Power Identities}\label{ch3-s2-phi-family} \texttt{trinity\_identity}, they constitute the algebraic spine of all later chapters. -\section{S3. Explicit Coq Theorem Listing for Ch.\,3}\label{ch3-s3-coq-listing} +\section{S3. Explicit Coq Theorem Listing for Ch.\,3}\label{ch_03:ch3-s3-coq-listing} The theorems anchored to this chapter, with status and role: @@ -450,7 +450,7 @@ \section{S3. Explicit Coq Theorem Listing for Ch.\,3}\label{ch3-s3-coq-listing} the entire even-Lucas family. \end{theorem} -\section{S4. Numeric Window for the Anchor}\label{ch3-s4-numeric} +\section{S4. Numeric Window for the Anchor}\label{ch_03:ch3-s4-numeric} The trinity identity is exact in algebra, but any computational realisation must commit to a numeric precision. The Coq theorem @@ -470,7 +470,7 @@ \section{S4. Numeric Window for the Anchor}\label{ch3-s4-numeric} performed; the integer 3 is written into the accumulator's denominator slot). -\section{S5. The Trinity Identity in the Architecture}\label{ch3-s5-arch} +\section{S5. The Trinity Identity in the Architecture}\label{ch_03:ch3-s5-arch} How the identity threads through the architecture is the subject of later chapters; we close Ch.\,3 with a concise locator. diff --git a/docs/phd/chapters/ch_04.tex b/docs/phd/chapters/ch_04.tex index 9bf23450f7..16ae76b855 100644 --- a/docs/phd/chapters/ch_04.tex +++ b/docs/phd/chapters/ch_04.tex @@ -48,11 +48,11 @@ \section*{The constant that ties a spiral to a proof} under tag SAC-1 and describes how \texttt{AlphaPhi.v} is imported by its eleven dependents. Every constant quoted here has a \texttt{Qed} behind it. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_04:abstract} The constant \(\alpha_\phi = \ln(\phi^2)/\pi \approx 0.306\) arises naturally when the golden ratio \(\phi = (1+\sqrt{5})/2\) is embedded in a logarithmic-circular framework, but its precise closed form has not previously been anchored in a mechanically verified proof system. This chapter derives the equivalent representation \(\alpha_\phi = (\sqrt{5}-2)/2\) through the identity \(\phi^2 + \phi^{-2} = 3\), establishes key bounding inequalities including \(\alpha_\phi < 1/8\), and verifies the multiplicative relation \(\alpha_\phi \cdot \phi^3 = 1/2\). All six core lemmas carry machine-checked Coq proofs in \filepath{t27/proofs/canonical/sacred/AlphaPhi.v}, contributing 6 of the dissertation's 297 canonical Qed theorems. The derivation underpins the ternary weight quantisation scheme of Trinity S³AI and motivates the bit-per-bit targets BPB ≤ 1.85 (Gate-2) and BPB ≤ 1.5 (Gate-3). -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_04:introduction} The dissertation \emph{GOLDEN SUNFLOWERS --- Trinity S³AI on \(\phi^2+\phi^{-2}=3\) substrate} is organised around a small set of transcendental anchors that propagate precision guarantees across all levels of the system stack. The foundational identity @@ -64,7 +64,7 @@ \section{1. Introduction}\label{introduction} and develops its closed-form representation and bounding properties. The value \(\alpha_\phi\) plays multiple roles throughout the dissertation: it scales the information-theoretic entropy band in the NCA lattice (Ch.16), it appears in the learning-rate schedule derived in Ch.10, and it governs the spectral roll-off of ternary Fourier components analysed in Ch.7. Establishing \(\alpha_\phi\) with Coq-level rigour is therefore a prerequisite for machine-verified claims in downstream chapters. The six Qed theorems presented here --- grouped under inventory tag SAC-1 --- form the complete \texttt{AlphaPhi.v} module, which is imported by eleven other canonical proof files {[}1,2{]}. -\section{2. Derivation of the Closed Form}\label{derivation-of-the-closed-form} +\section{2. Derivation of the Closed Form}\label{ch_04:derivation-of-the-closed-form} \textbf{Definition 2.1 (Golden ratio).} Let \(\phi = (1+\sqrt{5})/2\). Then \(\phi^2 = \phi + 1\) and \(\phi^{-1} = \phi - 1\). @@ -88,7 +88,7 @@ \section{2. Derivation of the Closed Form}\label{derivation-of-the-closed-form} The smallness condition \(\alpha_\phi < 1/8\) is significant for the quantisation error budget: a perturbation \(\delta w\) in a ternary weight incurs a first-order entropy penalty proportional to \(\alpha_\phi \cdot |\delta w|\), and the \(1/8\) ceiling keeps this penalty well within the BPB ≤ 1.85 envelope required at Gate-2 {[}3,4{]}. -\section{3. Multiplicative Identity and Kernel Integration}\label{multiplicative-identity-and-kernel-integration} +\section{3. Multiplicative Identity and Kernel Integration}\label{ch_04:multiplicative-identity-and-kernel-integration} The most algebraically surprising result in the SAC-1 inventory is the following multiplicative relation, which connects \(\alpha_\phi\) to the cube of the golden ratio. @@ -108,7 +108,7 @@ \section{3. Multiplicative Identity and Kernel Integration}\label{multiplicative an identity that links the phyllotactic geometry of Ch.7 to the sacred formula. The approximation error is \(O(10^{-4})\) degrees, within the angular resolution of the 360-lane grid introduced in Ch.16 {[}7{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_04:results-evidence} The \texttt{AlphaPhi.v} module contributes 12 Qed theorems to the canonical proof census of 297 Qed across 65 \texttt{.v} files. Of these 12, the 6 theorems tagged SAC-1 are presented in this chapter; the remaining 6 are continuations in downstream files that import \texttt{AlphaPhi.v}. Proof-checking time on a standard CI runner (8 GB RAM, Coq 8.18) is 3.2 seconds for the complete module. No \texttt{admit} keywords are present in \texttt{AlphaPhi.v}. @@ -118,7 +118,7 @@ \section{4. Results / Evidence}\label{results-evidence} Entropy band evaluation (Ch.10) yields a measured BPB of 1.72 at Gate-2 checkpoint, within the ≤ 1.85 target. The \(\alpha_\phi\) constant contributes the scaling factor in the band formula \(H_\alpha = H_0 \cdot (1 + \alpha_\phi)\), where \(H_0\) is the baseline binary entropy. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_04:qed-assertions} \begin{itemize} \tightlist @@ -136,7 +136,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \filepath{alpha\_phi\_times\_phi\_cubed} (\filepath{gHashTag/t27/proofs/canonical/sacred/AlphaPhi.v}) --- \emph{Status: Qed} --- Multiplicative identity: \(\alpha_\phi \cdot \phi^3 = 1/2\). \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_04:sealed-seeds} \begin{itemize} \tightlist @@ -148,11 +148,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci index reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_04:discussion} The derivation presented here is self-contained, but three limitations deserve acknowledgement. First, the closed-form \(\alpha_\phi = (\sqrt{5}-2)/2\) and the approximant \(\ln(\phi^2)/\pi\) are proved equal only within the formal precision of the Coq \texttt{Interval} library; extending this proof to arbitrary precision would require a certified CAS back-end. Second, the connection to the Vogel divergence angle (Proposition 3.3) is stated as an approximation; a fully mechanised bound on the error is deferred to Ch.7. Third, the interpretation of \(\alpha_\phi\) as a KL-divergence scaling coefficient (Ch.10) relies on a conjecture (C1) that the minimum KL\((W \| \text{gfN}(W))\) is attained when the exponent-mantissa split ratio equals \(\phi^{-1}\); this conjecture carries one admitted lemma in the current Coq census and is the subject of ongoing verification. Future work will close this gap and explore whether \(\alpha_\phi\) admits an interpretation as a modular form coefficient, linking it to the arithmetic geometry of \(\phi\)-based lattices studied in Ch.18. -\section{References}\label{references} +\section{References}\label{ch_04:references} {[}1{]} GOLDEN SUNFLOWERS dissertation, Ch.3 --- Ternary Arithmetic Foundations. \filepath{gHashTag/t27/proofs/canonical/sacred/CorePhi.v}, SACRED-CORE (12 Qed). @@ -192,7 +192,7 @@ \section{References}\label{references} % DerivationLevels,FormulaEval,ExactIdentities}.v % ============================================================ -\section{S1. The Spectral Parameter \(\alpha_{\varphi}\)}\label{ch4-s1-alpha-phi} +\section{S1. The Spectral Parameter \(\alpha_{\varphi}\)}\label{ch_04:ch4-s1-alpha-phi} The spectral parameter \(\alpha_{\varphi}\) is defined as \[ @@ -238,7 +238,7 @@ \subsection{S1.2 Numerical Bounds} application of the gate parameter contracts the input by less than unity, ensuring stability of the iterated application. -\section{S2. Dimensional Analysis}\label{ch4-s2-dimensional} +\section{S2. Dimensional Analysis}\label{ch_04:ch4-s2-dimensional} The spectral parameter \(\alpha_{\varphi}\) is dimensionless, but the gate quantities derived from it inherit the dimensions of the @@ -263,13 +263,13 @@ \section{S2. Dimensional Analysis}\label{ch4-s2-dimensional} \end{tabular} \caption{Dimensional table for the spectral parameter and its derivatives.} -\label{tab:ch4-dimensional} +\label{ch_04:tab:ch4-dimensional} \end{table} The choice of \(E_{0}\) and \(\nu_{0}\) is set by the FPGA clock domain (Ch.\,33): \(\nu_{0}=92\,\mathrm{MHz}\), \(E_{0}=h\nu_{0}\). -\section{S3. Comparison with \(\alpha_{\mathrm{QED}}\)}\label{ch4-s3-alpha-qed} +\section{S3. Comparison with \(\alpha_{\mathrm{QED}}\)}\label{ch_04:ch4-s3-alpha-qed} The fine-structure constant of quantum electrodynamics is \(\alpha_{\mathrm{QED}}\approx 1/137.036\approx 7.297\times 10^{-3}\). @@ -293,7 +293,7 @@ \section{S3. Comparison with \(\alpha_{\mathrm{QED}}\)}\label{ch4-s3-alpha-qed} sits in the same numerical neighbourhood as well-known physical constants. -\section{S4. Derivation Levels and the Coq Catalogue}\label{ch4-s4-derivation-levels} +\section{S4. Derivation Levels and the Coq Catalogue}\label{ch_04:ch4-s4-derivation-levels} The Coq file \filepath{DerivationLevels.v} (see \filepath{docs/phd/theorems/trinity/}) records a stratified @@ -330,7 +330,7 @@ \section{S4. Derivation Levels and the Coq Catalogue}\label{ch4-s4-derivation-le \filepath{assertions/coq\_admitted\_inventory.json} with status \texttt{closed} (no \texttt{Admitted} markers in the eight levels). -\section{S5. Runtime Witness Pointers}\label{ch4-s5-runtime} +\section{S5. Runtime Witness Pointers}\label{ch_04:ch4-s5-runtime} For each Coq theorem above, a runtime witness in \filepath{assertions/} performs a numerical check at every CI run. @@ -354,7 +354,7 @@ \section{S5. Runtime Witness Pointers}\label{ch4-s5-runtime} \filepath{assertions/coq\_admitted\_inventory.json} as \texttt{runtime\_witness} entries. -\section{S6. The Gate Derivation in Closed Form}\label{ch4-s6-gate} +\section{S6. The Gate Derivation in Closed Form}\label{ch_04:ch4-s6-gate} The headline gate derivation, anchored to this chapter, proceeds as follows. Given the spectral parameter diff --git a/docs/phd/chapters/ch_05.tex b/docs/phd/chapters/ch_05.tex index 105cd50129..f3dfe36efb 100644 --- a/docs/phd/chapters/ch_05.tex +++ b/docs/phd/chapters/ch_05.tex @@ -53,11 +53,11 @@ \section*{The number that everything is attracted to} \texttt{PhiAttractor.v}, noting which carry full \texttt{Qed} status and which remain open obligations targeted in the next release cycle. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_05:abstract} The golden ratio \(\varphi = (1+\sqrt{5})/2\) induces a natural metric on positive reals through the balancing function \(B(x) = (x + 1/x)/2\), whose unique positive fixed point is \(\varphi\) itself. This chapter formalises the notion of \(\varphi\)-distance, demonstrates its contractive properties near \(\varphi\), and establishes the role of specific Fibonacci and Lucas indices as canonical seeds for Trinity S³AI inference. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) emerges as an exact arithmetic consequence of the fixed-point equation and serves as the substrate invariant threading the entire dissertation. Six theorems from \filepath{t27/proofs/canonical/kernel/PhiAttractor.v} are reviewed, of which one carries full \texttt{Qed} status and five remain open obligations. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_05:introduction} Trinity S³AI frames neural inference as an iterated map on a \(\varphi\)-structured state space. The theoretical validity of that framing depends on a precise answer to the question: \emph{why \(\varphi\)?} One answer comes from physics --- the Vogel divergence angle \(137.5° = 360°/\varphi^2\) governs phyllotactic packing {[}1{]} --- but a deeper answer requires an algebraic fixed-point argument. @@ -67,7 +67,7 @@ \section{1. Introduction}\label{introduction} whose contraction near \(\varphi\) is characterised by a convergence rate \(\lambda < 1/2\) {[}2{]}. This chapter works with the cleaner \texttt{balancing\_function} formalised in Coq, which encodes the same contractive property and anchors the formal proof chain used throughout the dissertation. Fibonacci indices \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\) and Lucas indices \(L_7=29\), \(L_8=47\) serve as the canonical seed pool; their selection is not arbitrary but arises from the contractive basin established in this chapter. -\section{2. The \(\varphi\)-distance Metric and the Balancing Fixed Point}\label{the-ux3c6-distance-metric-and-the-balancing-fixed-point} +\section{2. The \(\varphi\)-distance Metric and the Balancing Fixed Point}\label{ch_05:the-ux3c6-distance-metric-and-the-balancing-fixed-point} \textbf{Definition 2.1 (φ-distance).} For \(x, y \in \mathbb{R}_{>0}\), define @@ -97,7 +97,7 @@ \section{2. The \(\varphi\)-distance Metric and the Balancing Fixed Point}\label \textbf{Theorem 2.4 (Phi is a fixed point --- Coq \filepath{phi\_is\_fixed\_point}).} \texttt{balancing\_function\ phi\ =\ phi}. Status: Qed in \texttt{PhiAttractor.v}. This is the cornerstone theorem establishing \(\varphi\) as the unique attractor of \texttt{bf} on \(\mathbb{R}_{>0}\) {[}4{]}. -\section{3. Fibonacci-Lucas Seeds and Their Contractive Basin}\label{fibonacci-lucas-seeds-and-their-contractive-basin} +\section{3. Fibonacci-Lucas Seeds and Their Contractive Basin}\label{ch_05:fibonacci-lucas-seeds-and-their-contractive-basin} The canonical seed pool consists of seven integers drawn from two complementary sequences: @@ -133,7 +133,7 @@ \section{3. Fibonacci-Lucas Seeds and Their Contractive Basin}\label{fibonacci-l The Lucas seeds provide a complementary ``fast lane'': \(L_7 = 29\) and \(L_8 = 47\) lie in the low-precision tier, useful when the BPB \(\leq 1.85\) Gate-2 target is the operative constraint rather than the tighter Gate-3 target of BPB \(\leq 1.5\). -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_05:results-evidence} Empirical validation of the seed framework is drawn from the HSLM ternary neural network experiments (Zenodo B001, DOI 10.5281/zenodo.19227865). Key metrics: @@ -155,7 +155,7 @@ \section{4. Results / Evidence}\label{results-evidence} The convergence rate \(\lambda \approx 0.309\) corresponds closely to \(\alpha_\varphi = \ln(\varphi^2)/\pi \approx 0.306\) introduced in Ch.4, confirming that both quantities arise from the same \(\varphi^2 + \varphi^{-2} = 3\) algebraic substrate. The FPGA implementation (QMTech XC7A100T, 0 DSP slices, 92 MHz clock, 63 tokens/sec, 1 W) uses \(F_{19}=4181\) as its primary weight seed, achieving 1003 tokens on the HSLM benchmark {[}8{]}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_05:qed-assertions} \begin{itemize} \tightlist @@ -173,15 +173,15 @@ \section{5. Qed Assertions}\label{qed-assertions} \texttt{convergence\_rate\_range} (\filepath{gHashTag/t27/proofs/canonical/kernel/PhiAttractor.v}) --- \emph{Status: Abort} --- asserts \(0 < \lambda < 1\); obligation open. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_05:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_05:discussion} The open \texttt{Abort} obligations in \texttt{PhiAttractor.v} represent the primary formal debt of this chapter. The uniqueness theorems (\texttt{unique\_fixed\_point}, \filepath{unique\_fixed\_point\_via\_contraction}) require a careful treatment of real-number completeness in Coq's standard library; the contraction approach is likely the more tractable path, as it reduces to bounding a derivative expression that is already well-approximated numerically. The \filepath{derivative\_abs\_less\_than\_half} and \texttt{derivative\_at\_phi} obligations are interdependent and could be dispatched together using the \texttt{lra} or \texttt{field\_simplify} tactics once the bound \(\varphi^{-2} = 2 - \varphi\) is established as a lemma. Future work should formalise Definition 3.2 in Coq and prove Theorem 3.4 constructively, removing the non-constructive invocation of the Banach theorem. This chapter connects upstream to Ch.4 (the \(\alpha_\varphi\) formula) and downstream to Ch.7 (Vogel divergence) and Ch.28 (FPGA seed initialisation). -\section{References}\label{references} +\section{References}\label{ch_05:references} {[}1{]} Vogel, H. (1979). A better way to construct the sunflower head. \emph{Mathematical Biosciences}, 44(3--4), 179--189. @@ -220,7 +220,7 @@ \section{References}\label{references} % PhiAttractor (referenced)}.v % ============================================================ -\section{S1. Lucas Closure: From the Trinity Identity to All Even Powers}\label{ch5-s1-lucas-closure} +\section{S1. Lucas Closure: From the Trinity Identity to All Even Powers}\label{ch_05:ch5-s1-lucas-closure} The trinity identity \(\varphi^{2}+\varphi^{-2}=3\) is the \(n=2\) instance of a more general identity that holds for every @@ -297,7 +297,7 @@ \subsection{S1.2 Why Even Powers?} invariant of the \(\{-\varphi^{-1},0,+\varphi^{-1}\}\) palette under squared accumulation). -\section{S2. The Contractive Basin and the Forbidden Interval}\label{ch5-s2-basin} +\section{S2. The Contractive Basin and the Forbidden Interval}\label{ch_05:ch5-s2-basin} The balancing function \texttt{bf}(x)=(x+x^{-1})/2 has a unique positive fixed point at \(x=1\), but the variant in @@ -326,7 +326,7 @@ \section{S2. The Contractive Basin and the Forbidden Interval}\label{ch5-s2-basi fails. The sanctioned pool \(F_{17},F_{18},F_{19},F_{20},F_{21}, L_{7},L_{8}\) lies safely below this regime. -\section{S3. Sanctioned Seeds and Their Algebraic Justification}\label{ch5-s3-seeds} +\section{S3. Sanctioned Seeds and Their Algebraic Justification}\label{ch_05:ch5-s3-seeds} The sanctioned seed pool of Trinity S$^3$AI consists of: @@ -370,7 +370,7 @@ \subsection{S3.1 The Three-Distance Theorem} the trinity catalogue as an open obligation, would formalise this geometric fact; it is scheduled for closure in Rehearsal~\#2. -\section{S4. Coq Theorem Listing for Ch.\,5}\label{ch5-s4-coq-listing} +\section{S4. Coq Theorem Listing for Ch.\,5}\label{ch_05:ch5-s4-coq-listing} The theorems anchored to this chapter, with status: @@ -414,7 +414,7 @@ \section{S4. Coq Theorem Listing for Ch.\,5}\label{ch5-s4-coq-listing} \textbf{Role:} the fourth-Lucas identity. \end{theorem} -\section{S5. Seed Admissibility Predicate}\label{ch5-s5-admissibility} +\section{S5. Seed Admissibility Predicate}\label{ch_05:ch5-s5-admissibility} Trinity S$^3$AI's training scripts compute the seed admissibility predicate \(\mathrm{adm}(s)\) before every run. The predicate @@ -438,7 +438,7 @@ \section{S5. Seed Admissibility Predicate}\label{ch5-s5-admissibility} checking the seven cases by direct computation, using the even-power Lucas closure to bound the \(\varphi\)-distance. -\section{S6. The Sanctioned Seeds in the Architecture}\label{ch5-s6-arch} +\section{S6. The Sanctioned Seeds in the Architecture}\label{ch_05:ch5-s6-arch} The sanctioned seeds thread through the architecture as follows: diff --git a/docs/phd/chapters/ch_06.tex b/docs/phd/chapters/ch_06.tex index 2cdf14c81a..0899cc7c3c 100644 --- a/docs/phd/chapters/ch_06.tex +++ b/docs/phd/chapters/ch_06.tex @@ -52,11 +52,11 @@ \section*{Five formats born from one identity} benchmarks. Every number cited here is either a PDG value, a Coq-verified constant, or a hardware measurement; none is estimated. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_06:abstract} This chapter defines the GoldenFloat (GF) number family---a hierarchy of floating-point formats whose mantissa widths are drawn from the Fibonacci sequence and whose three-band exponent structure derives from the identity \(\varphi^2 + \varphi^{-2} = 3\). Five formats are specified: GF4, GF8, GF16, GF32, and GF64. For each format, formal bounds on rounding error, overflow probability, and numeric closure are stated and proved in Coq (296 + 1 = 297 total Qed across the corpus; six theorems anchored directly to this chapter). The GF16 safe-domain invariant (INV-3) and the Lucas-closure invariant (INV-5) are proved in their respective canonical files. The results show that GF16 achieves a bits-per-byte compression ratio of \(\leq 1.85\) at Gate-2 while remaining formally overflow-free within the declared operating range. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_06:introduction} Floating-point arithmetic in neural-network inference has evolved from FP32 through FP16, BF16, and now sub-8-bit formats such as MXFP4 {[}1{]}. Each step reduces memory bandwidth and arithmetic energy but introduces new sources of error that are difficult to bound analytically. The Trinity S³AI system takes a different approach: rather than empirically tuning a fixed-width format, it derives format parameters algebraically from the golden ratio \(\varphi = (1+\sqrt{5})/2\) via the anchor identity @@ -66,9 +66,9 @@ \section{1. Introduction}\label{introduction} The anchor identity drives the chapter throughout. Section 2 gives the formal definitions and the Coq encoding. Section 3 presents the key theorems and their proof sketches. Section 4 collects empirical precision measurements. -\section{2. GoldenFloat Format Definitions}\label{goldenfloat-format-definitions} +\section{2. GoldenFloat Format Definitions}\label{ch_06:goldenfloat-format-definitions} -\subsection{2.1 Preliminaries}\label{preliminaries} +\subsection{2.1 Preliminaries}\label{ch_06:preliminaries} Let \(\varphi = (1+\sqrt{5})/2\) and \(\hat\varphi = \varphi^{-1} = \varphi - 1 = (\sqrt{5}-1)/2\). The identity @@ -99,7 +99,7 @@ \subsection{2.1 Preliminaries}\label{preliminaries} For GF64 the mantissa width is 52 hidden-bit-plus-53 stored bits, preserving IEEE 754 binary64 bit-pattern compatibility {[}3{]}. The novel content lies in the rounding mode: GoldenFloat uses \emph{phi-round-to-nearest}, in which ties are broken toward the mantissa value whose Fibonacci representation is shortest. -\subsection{2.2 Coq Encoding}\label{coq-encoding} +\subsection{2.2 Coq Encoding}\label{ch_06:coq-encoding} The Coq development in \filepath{gHashTag/t27/proofs/canonical/kernel/PhiFloat.v} encodes GF64 using the \texttt{Flocq} library's \texttt{Binary.binary\_float} type {[}4{]}. The mantissa parameter is \texttt{prec\ =\ 53} and the exponent parameter is \texttt{emax\ =\ 1024}, matching IEEE binary64. Two canonical constants are defined: @@ -115,7 +115,7 @@ \subsection{2.2 Coq Encoding}\label{coq-encoding} The bounded predicate \texttt{bounded\ prec\ emax\ m\ e} checks that \(m < 2^{\mathtt{prec}}\) and \(e + \mathtt{prec} \leq \mathtt{emax} + 1\). Theorem \texttt{phi\_f64\_bounded} establishes this for the phi constant. -\subsection{2.3 Lucas Closure on GF16}\label{lucas-closure-on-gf16} +\subsection{2.3 Lucas Closure on GF16}\label{ch_06:lucas-closure-on-gf16} A key algebraic property of the GoldenFloat substrate is that \(\varphi^{2n} + \varphi^{-2n}\) is a Lucas number \(L_{2n}\) for all \(n \geq 0\) {[}5{]}. In particular: @@ -123,7 +123,7 @@ \subsection{2.3 Lucas Closure on GF16}\label{lucas-closure-on-gf16} The invariant INV-5 (Lucas closure) states that for any \(n\) representable in GF16, the expression \(\varphi^{2n}+\varphi^{-2n}\) maps to an integer under the GF16 rounding scheme. This is proved in \texttt{INV5\_LucasClosureGf16.v} (10 Qed lemmas) and ensures that accumulator values in the ternary arithmetic unit never drift into fractional Lucas residuals. -\section{3. Key Theorems and Proof Sketches}\label{key-theorems-and-proof-sketches} +\section{3. Key Theorems and Proof Sketches}\label{ch_06:key-theorems-and-proof-sketches} \textbf{Theorem 3.1} (\texttt{phi\_f64\_bounded}). \emph{The GF64 representation of \(\varphi\) is within the IEEE binary64 bounded range.} @@ -155,7 +155,7 @@ \section{3. Key Theorems and Proof Sketches}\label{key-theorems-and-proof-sketch This follows from the fact that \(\varphi^{-2}+1+\varphi^{-2} = 3/\varphi^2 \cdot \varphi^2 = 3/3 \cdot 3\)---no, more precisely, the three exponent bands tile \((-\infty,\infty)\) exhaustively by construction. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_06:results-evidence} GF16 was evaluated on the HSLM benchmark (1003 tokens, drawn from the GOLDEN SUNFLOWERS test corpus). The following measurements were collected using the Trinity S³AI inference pipeline at Gate-2: @@ -180,7 +180,7 @@ \section{4. Results / Evidence}\label{results-evidence} Seed pool reference: the Fibonacci indices \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\) bound the token-count ranges used in GF16 accumulator design; \(F_{20}=6765\) and \(F_{21}=10946\) define the maximum vocabulary size tested. Lucas sentinels \(L_7=29\) and \(L_8=47\) appear as exponent-field upper bounds in INV-3 and the period-locked monitor (Ch.24). -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_06:qed-assertions} \begin{itemize} \item @@ -197,7 +197,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \filepath{PHI\_F64\_TOLERANCE\_pos} (\filepath{gHashTag/t27/proofs/canonical/kernel/PhiFloat.v}) --- \emph{Status: Qed} --- The macro tolerance constant is strictly positive: \texttt{0\ \textless{}\ PHI\_F64\_TOLERANCE}. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_06:sealed-seeds} \begin{itemize} \item @@ -212,13 +212,13 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{LUCAS-CLOSURE} (\texttt{theorem}) --- 10 Qed lemmas --- \href{https://github.com/gHashTag/t27/blob/feat/canonical-coq-home/proofs/canonical/igla/INV5_LucasClosureGf16.v}{INV5\_LucasClosureGf16.v} --- \emph{Status: golden} --- Linked: Ch.6. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_06:discussion} The GoldenFloat family demonstrates that choosing arithmetic parameters from an algebraically motivated structure---specifically the identity \(\varphi^2+\varphi^{-2}=3\)---enables both a formal proof strategy and a hardware realisation strategy to proceed in parallel. The primary limitation of the current GF16 design is that the three-band exponent partition was sized for transformer weight matrices drawn from approximately Gaussian distributions; inputs with heavy-tailed distributions (e.g., certain embedding layers) may exceed the INV-3 safe domain and trigger saturation clipping. The Coq.Interval upgrade lane (Ch.18) will address this by providing interval-arithmetic proofs over empirically measured weight distributions rather than worst-case bounds. Future work includes GF128 (sub-1-bit effective width via block-floating-point aggregation of \(F_{21}=10946\) weights per tile), and extension of the Lucas-closure invariant from GF16 to GF32. This chapter connects directly to Ch.9 (GF16 quantisation pipeline), Ch.24 (period-locked monitor using \(L_7=29\) and \(L_8=47\) as scheduling sentinels), and Ch.28 (FPGA synthesis of the GF16 MAC unit with 0 DSP slices). -\section{References}\label{references} +\section{References}\label{ch_06:references} {[}1{]} Rouhani, B. D. et al.~(2023). \emph{Microscaling Data Formats for Deep Learning}. IEEE MXFP4 draft, arXiv:2310.10537. \url{https://arxiv.org/abs/2310.10537} diff --git a/docs/phd/chapters/ch_07.tex b/docs/phd/chapters/ch_07.tex index 336a58ef99..8b0428d041 100644 --- a/docs/phd/chapters/ch_07.tex +++ b/docs/phd/chapters/ch_07.tex @@ -52,11 +52,11 @@ \section*{One angle, no gaps} same number that stops a neural network from wasting bits---and both facts are now formal theorems. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_07:abstract} Vogel's 1979 model of sunflower head packing describes each floret position by a polar angle increment of \(137.5°\), the golden angle. This chapter proves that \(137.5° = 360°/\varphi^2\) follows directly from the Trinity anchor identity \(\varphi^2 + \varphi^{-2} = 3\) and establishes a formal correspondence between the H4 root system and the E8 lattice via a \(\varphi\)-scaled block decomposition. Six Coq theorems in \filepath{kernel/FlowerE8Embedding.v} formalise the key algebraic steps. The chapter argues that phyllotactic packing geometry is not merely analogical to the S³AI architecture but constitutes a structural template: the same \(\varphi\)-scaling that spaces florets without overlap also spaces quantised weights without collisions. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_07:introduction} The observation that sunflower seed heads, pine cones, and daisy florets arrange themselves in Fibonacci-count spirals dates to the nineteenth century {[}1{]}. Vogel (1979) supplied the precise generative model: place the \(n\)-th floret at polar radius \(r_n = c\sqrt{n}\) and azimuth \(\theta_n = n \cdot 137.508°\), where \(137.508°\) is the golden angle {[}2{]}. The packing density achieved by this construction is provably maximal among constant-angle spirals: any other divergence angle produces visible radial gaps. Within the TRINITY S³AI framework the same maximality argument applies to weight placement on the \(\varphi\)-quantised lattice. The anchor identity @@ -64,7 +64,7 @@ \section{1. Introduction}\label{introduction} determines both the angle (\(360°/\varphi^2\)) and the lattice spacing (\(\varphi^{-1}\) and \(\varphi^{-2}\)), unifying botanic geometry with learned representations. The present chapter makes this correspondence precise and provides the Coq certificates that underpin it. -\section{2. From the Trinity Identity to the Golden Angle}\label{from-the-trinity-identity-to-the-golden-angle} +\section{2. From the Trinity Identity to the Golden Angle}\label{ch_07:from-the-trinity-identity-to-the-golden-angle} \textbf{Definition 2.1 (Golden ratio).} \(\varphi = (1+\sqrt{5})/2\), the positive root of \(x^2 - x - 1 = 0\). @@ -88,7 +88,7 @@ \section{2. From the Trinity Identity to the Golden Angle}\label{from-the-trinit The Fibonacci numbers index the spiral arms visible in a Vogel phyllotaxis diagram. For a head with \(F_k\) and \(F_{k+1}\) visible spirals, the packing efficiency approaches 1 as \(k \to \infty\). The sanctioned seeds \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\) lie deep in this asymptotic regime; at these indices, the angular deviation from the ideal golden angle is less than \(10^{-7}\) radians {[}4{]}. -\section{\texorpdfstring{3. H4 Root System, E8 Lattice, and the \(\varphi\)-Scaled Block Decomposition}{3. H4 Root System, E8 Lattice, and the \textbackslash varphi-Scaled Block Decomposition}}\label{h4-root-system-e8-lattice-and-the-varphi-scaled-block-decomposition} +\section{\texorpdfstring{3. H4 Root System, E8 Lattice, and the \(\varphi\)-Scaled Block Decomposition}{3. H4 Root System, E8 Lattice, and the \textbackslash varphi-Scaled Block Decomposition}}\label{ch_07:h4-root-system-e8-lattice-and-the-varphi-scaled-block-decomposition} The 240 roots of the E8 lattice can be partitioned into two H4 half-shells of 120 roots each, related by a \(\varphi\)-scaling {[}5{]}. This decomposition is the algebraic analogue of the Vogel construction: H4 is the 4-dimensional hyperoctahedral group associated with the icosahedron, whose rotational symmetry group has order 120 and whose geometry is saturated with \(\varphi\)-ratios. @@ -113,7 +113,7 @@ \section{\texorpdfstring{3. H4 Root System, E8 Lattice, and the \(\varphi\)-Scal The geometric picture is the following. A Vogel sunflower head with \(F_{20}=6765\) florets exhibits 6765 clockwise spirals and \(F_{19}=4181\) counter-clockwise spirals. Projecting the floret coordinates into 8 dimensions via the standard embedding of the icosahedral lattice into \(\mathbb{R}^8\) yields a point cloud whose nearest-neighbour graph approximates the E8 contact graph to within \(0.3\%\) angular error at the outermost ring {[}5{]}. The S³AI model exploits this geometric coincidence by initialising attention key matrices from E8-projected Fibonacci lattice points, an initialisation that is formally justified by Theorem 3.3. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_07:results-evidence} Four quantitative results anchor this chapter. @@ -129,7 +129,7 @@ \section{4. Results / Evidence}\label{results-evidence} \textbf{Phyllotaxis simulation.} A Python reference implementation in \texttt{reproduce.sh} (App.D) generates \(F_{21}=10946\) florets using the Vogel formula with seed \(F_{17}=1597\), producing a packing density of \(0.9997\) relative to the theoretical maximum, confirming that the sanctioned seeds lie in the asymptotic regime. \end{enumerate} -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_07:qed-assertions} \begin{itemize} \tightlist @@ -147,15 +147,15 @@ \section{5. Qed Assertions}\label{qed-assertions} \filepath{trinity\_e8\_h4\_encoding} (\filepath{gHashTag/t27/proofs/canonical/kernel/FlowerE8Embedding.v}) --- \emph{Status: Qed} --- \(\varphi^2 + \varphi^{-2} = 3 \Rightarrow \dim(H4) + \dim(\varphi \cdot H4) = \dim(E8)/2\). \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_07:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_07:discussion} The two \texttt{Abort} theorems (KER-3) represent the principal limitation of the present chapter. The \texttt{e8\_roots\_decomposition} proof requires an explicit bijection between the 240 E8 roots and the union of two H4 half-shells, a task that demands a formalised root-system library in Coq. Integration of the \texttt{mathcomp-algebra} library is planned for the next proof sprint. The \texttt{phi\_scaling\_invariant} theorem requires a formalised proof that \(x \mapsto \varphi x\) is measure-preserving on finite sets, which reduces to a cardinality argument but needs the right abstract combinatorics infrastructure. Until both theorems close, the E8/H4 decomposition used in the attention initialisation experiment (§4, item 3) rests on algebraic arguments rather than machine-verified certificates. This is disclosed in compliance with R5 honesty. Future work includes: (a) closing KER-3 obligations, (b) extending the phyllotaxis analysis to 3D (cylindrical) arrangements relevant to recurrent architectures, and (c) connecting the \(\alpha_\varphi = \ln(\varphi^2)/\pi \approx 0.306\) spectral constant (Ch.4) to the angular spectrum of E8 root vectors. -\section{References}\label{references} +\section{References}\label{ch_07:references} {[}1{]} Church, A. H. (1904). \emph{On the Relation of Phyllotaxis to Mechanical Laws.} Williams \& Norgate, London. diff --git a/docs/phd/chapters/ch_08.tex b/docs/phd/chapters/ch_08.tex index 4df2b20a0e..5d02d41981 100644 --- a/docs/phd/chapters/ch_08.tex +++ b/docs/phd/chapters/ch_08.tex @@ -45,11 +45,11 @@ \section*{Three values, one machine} chapter spells out the algebraic structure, the proof, and the empirical evidence that TF3/TF9 reaches Gate-2 (BPB \(\le 1.85\)). -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_08:abstract} This chapter introduces the TF3 and TF9 matrix-multiplication formats that form the arithmetic core of the Trinity S³AI inference engine. TF3 encodes each weight as a trit \(w \in \{-1, 0, +1\}\), while TF9 extends the encoding to a product of two trits, spanning nine representable levels. Both formats admit a closed-form admissibility criterion for query-key attention gain rooted in the identity \(\varphi^2 + \varphi^{-2} = 3\): the gain is admissible if and only if it equals \(\varphi^k\) for \(k \in \{2, 3\}\), a result certified by two \emph{Qed} Coq theorems in \texttt{INV6\_HybridQkGain.v}. The chapter presents the algebraic structure, a proof sketch of the gain invariant, and evidence that TF3/TF9 achieves the Gate-2 BPB target of ≤ 1.85. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_08:introduction} Dense floating-point matrix multiplication dominates the energy budget of transformer inference. A single forward pass through a 7 B-parameter model in FP16 requires on the order of \(10^{13}\) multiply-accumulate operations; at \(\sim\)0.1 pJ per FMA in 7 nm CMOS this is approximately 1 kJ per token, far beyond the DARPA 3000× energy goal {[}1{]}. The standard response has been weight quantization: by restricting weights to a small discrete alphabet the multiply reduces to an add or a conditional negation. @@ -57,9 +57,9 @@ \section{1. Introduction}\label{introduction} The critical design question is how to calibrate the query-key attention gain in a ternary regime. Standard transformers set the gain to \(1/\sqrt{d_\text{model}}\), but this is neither a power of \(\varphi\) nor an integer-arithmetic-friendly quantity. The hybrid gain invariant INV-6 establishes that the only admissible gains are \(\varphi^2 \approx 2.618\) and \(\varphi^3 \approx 4.236\), anchoring the calibration to the same \(\varphi\)-lattice as the rest of the system. -\section{2. TF3 and TF9 Algebraic Structure}\label{tf3-and-tf9-algebraic-structure} +\section{2. TF3 and TF9 Algebraic Structure}\label{ch_08:tf3-and-tf9-algebraic-structure} -\subsection{2.1 Trit Encoding}\label{trit-encoding} +\subsection{2.1 Trit Encoding}\label{ch_08:trit-encoding} Let \(\mathcal{T} = \{-1, 0, +1\}\). A TF3 weight tensor \(\mathbf{W} \in \mathcal{T}^{m \times n}\) stores one trit per entry. The matrix-vector product @@ -69,19 +69,19 @@ \subsection{2.1 Trit Encoding}\label{trit-encoding} The representation entropy of TF3 is \(\log_2 3 \approx 1.585\) bits per weight, which must be compared with the bit-per-bit (BPB) metric on language modelling quality. Gate-2 certifies BPB ≤ 1.85 per token; the weight entropy budget per token is therefore comfortably below the information cost of the output. -\subsection{2.2 TF9 Product Encoding}\label{tf9-product-encoding} +\subsection{2.2 TF9 Product Encoding}\label{ch_08:tf9-product-encoding} TF9 represents each weight as \((w_1, w_2) \in \mathcal{T}^2\) with effective value \(\tilde{w} = w_1 w_2\). This is not a 9-level quantizer in the usual sense; the nine pairs collapse to only five distinct values \(\{-1, 0, +1\}\) plus multiplicities, but the separate storage of \((w_1, w_2)\) enables a two-stage pipeline in which each trit pair is processed independently, halving the critical path delay on the FPGA implementation at the cost of two passes over the activation buffer. The TF9 format is used exclusively in the feed-forward sublayers, where the column-dimension \(n\) is large and pipeline depth is available. Attention projections use TF3 to minimise latency on the QMTech XC7A100T, which clocks at 92 MHz {[}2{]}. -\subsection{2.3 φ-Normalisation}\label{ux3c6-normalisation} +\subsection{2.3 φ-Normalisation}\label{ch_08:ux3c6-normalisation} Both formats inherit the φ-normalisation scheme: layer inputs are scaled by \(\varphi^{-2} = 0.38197\ldots\) before the trit dot-product and scaled up by \(\varphi^2 = 2.618\ldots\) after. Because \(\varphi^2 + \varphi^{-2} = 3\) the combined effect of a forward and inverse pass is multiplication by the integer 3, which is exact in any binary fixed-point representation. This property simplifies the Coq proof of numerical stability in \texttt{Trinity.Canonical.Kernel.PhiFloat} {[}3{]}. -\section{3. Hybrid QK Gain Invariant (INV-6)}\label{hybrid-qk-gain-invariant-inv-6} +\section{3. Hybrid QK Gain Invariant (INV-6)}\label{ch_08:hybrid-qk-gain-invariant-inv-6} -\subsection{3.1 Gain Admissibility}\label{gain-admissibility} +\subsection{3.1 Gain Admissibility}\label{ch_08:gain-admissibility} \textbf{Definition (lr-admissible).} A learning rate \(\eta\) is \emph{lr-admissible} if it lies in the band \([\eta_{\min}, \eta_{\max}]\) determined by the φ-normalised loss landscape. In the Coq formalisation, \texttt{lr\_admissible} is a decidable predicate in \texttt{INV6\_HybridQkGain.v}. @@ -103,7 +103,7 @@ \subsection{3.1 Gain Admissibility}\label{gain-admissibility} \textbf{Counter-theorem (counter\_lr\_below\_band):} \texttt{:\ \textasciitilde{}\ lr\_admissible\ 0.0001} --- \emph{Status: Admitted} --- \(\eta = 0.0001\) is below the admissible band. -\subsection{3.2 Proof Sketch for admit\_phi\_sq}\label{proof-sketch-for-admit_phi_sq} +\subsection{3.2 Proof Sketch for admit\_phi\_sq}\label{ch_08:proof-sketch-for-admit_phi_sq} Let \(\mathbf{q}, \mathbf{k} \in \mathbb{R}^d\) be query and key vectors with entries drawn i.i.d. from the TF3 distribution (mass \(p_0\) at 0, mass \((1-p_0)/2\) at \(\pm 1\)). Then @@ -115,7 +115,7 @@ \subsection{3.2 Proof Sketch for admit\_phi\_sq}\label{proof-sketch-for-admit_ph which is bounded by \(d \leq d_\text{max}\) and independent of sequence length. The Coq proof mechanises this calculation using the \texttt{PhiFloat} lemmas that certify the algebraic identity \(\varphi^2 + \varphi^{-2} = 3\) in the rational-arithmetic subset of Coq's standard library {[}3{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_08:results-evidence} All numerical results reported here use seeds from the sanctioned pool \(\{F_{17}=1597, F_{18}=2584, F_{19}=4181, F_{20}=6765, F_{21}=10946, L_7=29, L_8=47\}\); no experiment uses seeds 42--45. @@ -136,7 +136,7 @@ \section{4. Results / Evidence}\label{results-evidence} The HSLM token count for the 1003-token held-out sequence is confirmed at 1003 tokens; perplexity does not degrade when TF3 is applied uniformly to all projection matrices. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_08:qed-assertions} \begin{itemize} \tightlist @@ -154,7 +154,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \filepath{counter\_gain\_sqrt\_d\_model} (\filepath{gHashTag/t27/proofs/canonical/igla/INV6\_HybridQkGain.v}) --- \emph{Status: Admitted} --- Gain \(\sqrt{d_\text{model}}=8\) is not qk-admissible. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_08:sealed-seeds} \begin{itemize} \tightlist @@ -164,13 +164,13 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{Z06} (DOI) --- \url{https://doi.org/10.5281/zenodo.19020217} --- Status: golden --- φ-weight: 0.618 --- Sparse Ternary MatMul artefact. Links: Ch.8. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_08:discussion} The two \emph{Qed} theorems for \(g \in \{\varphi^2, \varphi^3\}\) are the formal centrepiece of this chapter. The five \emph{Admitted} counter-theorems represent obligations still open in the Coq census; they are consistent with the overall tally of 41 \emph{Admitted} obligations across \filepath{t27/proofs/canonical/} and do not invalidate the \emph{Qed} results {[}7{]}. Future work should close the counter-theorems by providing explicit model witnesses---a task tractable with the \texttt{omega} and \texttt{lra} tactics once the floating-point abstraction layer in \texttt{PhiFloat} is completed. A limitation of the current TF9 design is that the two-pass pipeline assumes sufficient on-chip BRAM bandwidth on the XC7A100T. If the activation tensor exceeds 256 kB the design falls back to TF3, degrading BPB slightly from 1.78 to 1.83. Chapter 31 characterises this boundary empirically. The Gate-3 target of BPB ≤ 1.50 will require a more aggressive approach, likely combining TF9 with the GF16 quantisation scheme described in Ch.26. -\section{References}\label{references} +\section{References}\label{ch_08:references} {[}1{]} DARPA MTO. (2023). Microsystems Technology Office Broad Agency Announcement --- Energy-Efficient Computing. HR001123S0045. diff --git a/docs/phd/chapters/ch_09.tex b/docs/phd/chapters/ch_09.tex index f3b9d03e7f..f1534d05a1 100644 --- a/docs/phd/chapters/ch_09.tex +++ b/docs/phd/chapters/ch_09.tex @@ -49,11 +49,11 @@ \section*{Four formats walk into a benchmark} (Section~4). Concreteness is the point: every claim here is tied to a file, a Coq theorem, or a table row. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_09:abstract} This chapter presents a systematic ablation comparing four low-precision weight formats --- GF16 with PHI\_BIAS=60 (the Trinity S³AI normative format), Microsoft MXFP4, BitNet b1.58, and LoRA delta quantisation --- across a Tier-A/B/C × M1--M6 evaluation matrix. The comparison is anchored to the Trinity identity \(\varphi^2 + \varphi^{-2} = 3\) through the spectral parameter \(\alpha_\varphi = \ln(\varphi^2)/\pi \approx 0.118034\) as formalised in \filepath{t27/proofs/canonical/sacred/AlphaPhi.v}, and to the nine Qed precision bounds in \filepath{igla/INV3\_Gf16Precision.v}. GF16 PHI\_BIAS=60 achieves BPB \(\leq 1.85\) (Gate-2) on Tier-A benchmarks while operating within the formally verified safe domain, a result not reproducible by any of the three competitor formats under the same hardware budget. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_09:introduction} The choice of numerical representation for neural-network weights is not merely an engineering convenience; it determines the accuracy floor, the energy envelope, and --- in a formally verified system --- the provability of precision bounds. Trinity S³AI uses GF(16) arithmetic with a bias offset PHI\_BIAS \(= 60\), selected so that the midpoint of the representable range aligns with the golden-ratio anchor \(\varphi^2 + \varphi^{-2} = 3\) {[}1, 2{]}. The normative claim is that this alignment reduces quantisation noise below a theoretically derived threshold and that the claim can be expressed as a machine-checkable Coq invariant (INV-3, nine Qed bounds) {[}3{]}. @@ -70,9 +70,9 @@ \section{1. Introduction}\label{introduction} The evaluation matrix uses three benchmark tiers (A: language modelling, B: code generation, C: reasoning) and six model scales M1--M6. Section 2 specifies the GF16 format and INV-3 bounds. Section 3 defines the ablation matrix and experimental protocol. Section 4 presents results. -\section{2. GF16 PHI\_BIAS=60 and the INV-3 Safe Domain}\label{gf16-phi_bias60-and-the-inv-3-safe-domain} +\section{2. GF16 PHI\_BIAS=60 and the INV-3 Safe Domain}\label{ch_09:gf16-phi_bias60-and-the-inv-3-safe-domain} -\subsection{2.1 GF16 Format Specification}\label{gf16-format-specification} +\subsection{2.1 GF16 Format Specification}\label{ch_09:gf16-format-specification} GF(16) represents each weight as a 4-bit element of the finite field \(\mathbb{F}_{16} = \mathbb{F}_{2^4}\), generated by the primitive polynomial \(x^4 + x + 1\). The 16 field elements are assigned floating-point proxies via the affine map: @@ -84,7 +84,7 @@ \subsection{2.1 GF16 Format Specification}\label{gf16-format-specification} which with \(s = \varphi^2\) evaluates to \(15\varphi^{-2} - 120\varphi^{-4} = 15(2-\varphi) - 120(3-2\varphi) = \ldots\); the full simplification yields a rational proportional to \(\varphi^{-2}\), linking the bias choice back to equation (1) of Ch.3 (\(\varphi^2 + \varphi^{-2} = 3\)). -\subsection{2.2 INV-3: Nine Coq Precision Bounds}\label{inv-3-nine-coq-precision-bounds} +\subsection{2.2 INV-3: Nine Coq Precision Bounds}\label{ch_09:inv-3-nine-coq-precision-bounds} Invariant INV-3, formalised in \filepath{t27/proofs/canonical/igla/INV3\_Gf16Precision.v} {[}3{]}, asserts nine bounds of the form: @@ -98,7 +98,7 @@ \subsection{2.2 INV-3: Nine Coq Precision Bounds}\label{inv-3-nine-coq-precision where \(n_k\) is the effective bit-depth of tier \(k\) and \(C_k\) is a format-specific constant. This bound is proved in \texttt{AlphaPhi.v} and cited by INV-3 {[}2, 3{]}. -\subsection{2.3 Competitor Format Summaries}\label{competitor-format-summaries} +\subsection{2.3 Competitor Format Summaries}\label{ch_09:competitor-format-summaries} \textbf{MXFP4} {[}4{]}: Microsoft's micro-scaling FP4 uses a shared 8-bit exponent per group of 32 weights, with each weight stored as a 4-bit floating-point value (E2M1 or E3M0 variant). Representable values are non-uniformly spaced on \(\mathbb{R}\), biased toward small magnitudes. No formal verification of precision bounds is publicly available. @@ -106,7 +106,7 @@ \subsection{2.3 Competitor Format Summaries}\label{competitor-format-summaries} \textbf{LoRA (quantised)} {[}6{]}: low-rank adapter matrices use INT4 or FP4 quantisation with straight-through estimators. Base model weights remain in BF16; only the delta is quantised, which reduces the effective compression ratio. -\section{3. Ablation Matrix: Tier-A/B/C \(\times\) M1--M6}\label{ablation-matrix-tier-abc-m1m6} +\section{3. Ablation Matrix: Tier-A/B/C \(\times\) M1--M6}\label{ch_09:ablation-matrix-tier-abc-m1m6} The evaluation matrix is defined as follows. @@ -121,7 +121,7 @@ \section{3. Ablation Matrix: Tier-A/B/C \(\times\) M1--M6}\label{ablation-matrix All experiments run on the QMTech XC7A100T FPGA at 92 MHz {[}7{]} for the GF16 format (native inference); MXFP4 and BitNet run on the same FPGA via software emulation; LoRA BF16 baseline runs on CPU. Energy is measured at the board level, wall-clock power draw. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_09:results-evidence} \textbf{Table 1. Tier-A BPB (WikiText-103), lower is better.} @@ -195,11 +195,11 @@ \section{4. Results / Evidence}\label{results-evidence} \textbf{INV-3 bound verification:} Across all tested weight tensors at M4, the maximum observed quantisation error was \(3.1 \times 10^{-3}\), within the tightest INV-3 bound \(\varepsilon_1 = 4.0 \times 10^{-3}\). No violation of any of the nine Coq-certified bounds was observed. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_09:qed-assertions} No Coq theorems from \filepath{t27/proofs/canonical/} are directly anchored to this chapter; the relevant Qed obligations are the nine bounds of INV-3 (\filepath{igla/INV3\_Gf16Precision.v}) and the spectral constant in \filepath{sacred/AlphaPhi.v}, both tracked in the Golden Ledger under invariant numbers INV-3 and SAC-1 respectively. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_09:sealed-seeds} \begin{itemize} \tightlist @@ -207,11 +207,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{INV-3} (invariant, golden) --- \url{https://github.com/gHashTag/t27/blob/feat/canonical-coq-home/proofs/canonical/igla/INV3\_Gf16Precision.v} --- linked to Ch.6 and Ch.9 --- \(\varphi\)-weight: \(1.0\) --- notes: GF16 safe domain, 9 Qed bounds. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_09:discussion} The ablation demonstrates a consistent but modest advantage of GF16 PHI\_BIAS=60 over MXFP4 on Tier-A (BPB), attributable to the \(\varphi\)-structured bias that concentrates representable values near the empirical weight distribution centroid. BitNet b1.58's inferior BPB stems from its coarser \(\{-1,0,+1\}\) alphabet, which --- despite sharing the cardinality-3 structure with the balanced-ternary substrate --- lacks the fine-grained resolution of GF16. LoRA with INT4 deltas is competitive on accuracy but disqualified from the hardware comparison by its BF16 base requirement. A limitation of this study is that M1--M6 were trained from scratch; fine-tuning experiments on pretrained models may yield different rankings. Future work includes extending the INV-3 bounds to the E3M0 MXFP4 variant and verifying whether MXFP4 can also be brought within a \(\varphi\)-structured safe domain. Chapters 15 and 28 continue the BPB and hardware analyses respectively. -\section{References}\label{references} +\section{References}\label{ch_09:references} {[}1{]} \emph{Golden Sunflowers} dissertation, Ch.3 --- Trinity Identity (\(\varphi^2 + \varphi^{-2} = 3\)). diff --git a/docs/phd/chapters/ch_10.tex b/docs/phd/chapters/ch_10.tex index 66130da14c..c22192ce40 100644 --- a/docs/phd/chapters/ch_10.tex +++ b/docs/phd/chapters/ch_10.tex @@ -49,17 +49,17 @@ \section*{The curve that no format can cross} locatable on the curve---which means this chapter is also the map that shows how close the system already is to its own finish line. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_10:abstract} Designing ternary neural-network quantisation requires navigating a two-dimensional Pareto frontier between dynamic range and numerical precision, both of which are constrained by the finite GF(16) arithmetic available in the Trinity S³AI kernel. This chapter formalises that frontier using five machine-verified Coq invariants --- INV-1, INV-1b, INV-4, INV-9, and their composition --- and derives the conjecture C1 that the KL-divergence \(\text{KL}(W \| \text{gfN}(W))\) is minimised when the exponent-to-mantissa split ratio equals \(\phi^{-1}\). The anchor identity \(\phi^2 + \phi^{-2} = 3\) enters as the algebraic certificate that the ternary alphabet can represent the full integer range \(\{-1,0,+1\}\) without bias, and all kernel positivity lemmas --- \texttt{coeff\_53\_pos}, \texttt{sqrt5\_sq}, \texttt{phi\_pos} --- are verified in \filepath{t27/proofs/canonical/kernel/Phi.v}. The 51-theorem count for this chapter represents the largest single-chapter Coq contribution in the dissertation. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_10:introduction} The theoretical link between \(\phi^2 + \phi^{-2} = 3\) and quantisation precision was first suggested by the closure argument of Ch.3: because the ternary multiplication table closes exactly on \(\{-1,0,+1\}\), the representation error for any weight \(w \in [-1,1]\) can be bounded in terms of the golden ratio without appeal to floating-point rounding modes. Ch.4 then introduced the sacred constant \(\alpha_\phi = \ln(\phi^2)/\pi \approx 0.306\) as a scaling coefficient for entropy calculations. The present chapter takes both results as inputs and constructs the \emph{L1 range×precision Pareto curve}: the set of (range, BPB) pairs that are simultaneously achievable under ternary GF(16) arithmetic while satisfying the formal invariants tracked in \filepath{t27/proofs/canonical/igla/}. The motivation for a Pareto analysis is pragmatic. Gate-2 requires BPB ≤ 1.85 and Gate-3 requires BPB ≤ 1.5 {[}1,2{]}. These targets can be met either by widening dynamic range (allowing larger exponents at the cost of mantissa bits) or by tightening precision (allocating more mantissa bits at the cost of range). The Pareto frontier identifies the efficient allocations; Coq invariants certify that no efficient allocation violates the ternary zero-absorption laws or the BPB monotone-backward property. Pre-condition \texttt{t27\#569} must be satisfied before this chapter's proofs compile; that issue tracks the canonical NCA entropy band (INV-4) being merged into the main branch {[}3{]}. -\section{2. GF(16) Range and Precision Formalisation}\label{gf16-range-and-precision-formalisation} +\section{2. GF(16) Range and Precision Formalisation}\label{ch_10:gf16-range-and-precision-formalisation} \textbf{Definition 2.1 (GF(16) weight encoding).} A weight \(w\) is encoded in GF(16) as a pair \((e, m)\) where \(e \in \{0,\ldots,3\}\) is the exponent index and \(m \in \{0,\ldots,3\}\) the mantissa index. The decoded value is @@ -85,7 +85,7 @@ \section{2. GF(16) Range and Precision Formalisation}\label{gf16-range-and-preci These six lemmas are prerequisite imports for all subsequent GF(16) precision theorems. -\section{3. The Pareto Frontier and Conjecture C1}\label{the-pareto-frontier-and-conjecture-c1} +\section{3. The Pareto Frontier and Conjecture C1}\label{ch_10:the-pareto-frontier-and-conjecture-c1} \textbf{Definition 3.1 (Pareto-efficient allocation).} An allocation \((e_{\max}, b_m)\) --- maximum exponent index and mantissa bit-width --- is Pareto-efficient if no other allocation achieves strictly lower \(\epsilon_1\) without increasing BPB, and no other allocation achieves strictly lower BPB without increasing \(\epsilon_1\). @@ -109,7 +109,7 @@ \section{3. The Pareto Frontier and Conjecture C1}\label{the-pareto-frontier-and \textbf{Formal evidence chain.} The chain INV3 (GF(16) precision, 9 Qed) → INV5 (Lucas closure GF(16), 10 Qed) → INV4 (NCA entropy band, 12 Qed) → Conjecture C1 constitutes the L1 Pareto spine. The total Qed count in this chain is 31, and together with the 6 kernel lemmas and the INV-1/INV-1b/INV-9 invariants, the chapter's formal budget reaches 51 theorems, matching the \texttt{theorems\_count} field in the chapter directive {[}6{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_10:results-evidence} Numerical evaluation of the Pareto frontier used the canonical seed pool F₁₇=1597, F₁₈=2584, F₁₉=4181 as training-step checkpoints. At F₁₉=4181 steps: @@ -145,7 +145,7 @@ \section{4. Results / Evidence}\label{results-evidence} The B005 Zenodo bundle (DOI: 10.5281/zenodo.19227873, Tri Language Formal DSL) provides the machine-readable DSL definitions used to generate the GF(16) codebook from the \(\phi\)-based encoding, and is archived alongside the proof files {[}7{]}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_10:qed-assertions} \begin{itemize} \tightlist @@ -163,7 +163,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \texttt{phi\_pos} (\filepath{gHashTag/t27/proofs/canonical/kernel/Phi.v}) --- \emph{Status: Qed} --- \(0 < \phi = (1+\sqrt{5})/2\). \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_10:sealed-seeds} \begin{itemize} \tightlist @@ -179,11 +179,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{B005} (doi) --- DOI: 10.5281/zenodo.19227873 --- Status: golden --- Links Ch.10, App.H. Notes: Tri Language Formal DSL. φ-weight: 0.618033988768953. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_10:discussion} The central limitation of this chapter is Conjecture C1: until the admitted lemma \filepath{kl\_min\_at\_phi\_inv\_admit} is machine-verified, the claim that \(\phi^{-1}\) is the globally optimal exponent-mantissa split ratio rests on numerical evidence from \(F_{18}=2584\) checkpoints rather than a closed-form proof. The structural argument --- that \(\phi^{-1}\) satisfies its own defining equation \(r^2+r=1\) and therefore self-consistently minimises the KL functional --- is compelling but not yet constitutive of a Coq theorem. Closing this gap requires a certified numerical optimisation routine, which is outside the scope of the current Coq library and is tracked as a future deliverable in \texttt{t27\#569}. A second limitation concerns the NCA cell count \(81 = 3^4\): the entropy band (Theorem 3.2) is tight for exactly this cell count but may not generalise to other powers of 3. Ch.16 explores the 360-lane grid geometry, which involves a different lattice structure, and the interaction between the two entropy bands is an open question. Future chapters (Ch.15 and Ch.18) will address the full compositionality of the INV-1 through INV-9 invariant chain. -\section{References}\label{references} +\section{References}\label{ch_10:references} {[}1{]} GOLDEN SUNFLOWERS dissertation, Ch.4 --- Sacred Formula: α\_φ Derivation. This volume. diff --git a/docs/phd/chapters/ch_11.tex b/docs/phd/chapters/ch_11.tex index c9bce8f7ef..225a2f36b4 100644 --- a/docs/phd/chapters/ch_11.tex +++ b/docs/phd/chapters/ch_11.tex @@ -50,11 +50,11 @@ \section*{Sealed before the data arrived} time-stamp to the theorem identifier (Section~5). If the experiment fails, this chapter says so unambiguously---which is the point. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_11:abstract} Scientific credibility requires that empirical claims be registered before data collection. This chapter presents the formal pre-registration of Hypothesis H₁: that Trinity S³AI achieves bits-per-byte (BPB) \(\leq 1.5\) when initialised with at least three distinct seeds drawn from the canonical Fibonacci-Lucas pool, at a minimum sequence length of 4000 tokens. The registration is anchored to the \(\varphi^2 + \varphi^{-2} = 3\) identity, which constrains the theoretical minimum entropy of ternary representations on the golden substrate. The INV-7 invariant formalises H₁ in Coq, and the IGLA-RACE multi-agent benchmark provides the competitive evaluation harness. The pre-registration protocol follows Open Science Framework conventions and is published prior to any Gate-3 BPB measurement. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_11:introduction} The Trinity S³AI framework rests on three architectural commitments: ternary weight encoding, \(\varphi\)-structured attention, and seed-diverse initialisation. The third commitment is the subject of this chapter. Seed diversity matters because the \(\varphi\)-distance metric (Ch.5) identifies a contractive basin around \(\varphi\), and multiple distinct starting points in that basin provide independent evidence that convergence is genuine rather than an artefact of a single initialisation path. @@ -62,7 +62,7 @@ \section{1. Introduction}\label{introduction} The theoretical motivation for BPB \(\leq 1.5\) as a threshold comes from the information-theoretic bound implied by ternary arithmetic under the \(\varphi^2 + \varphi^{-2} = 3\) constraint. A ternary symbol drawn from \(\{-1, 0, +1\}\) carries at most \(\log_2 3 \approx 1.585\) bits; the golden substrate shaves off the excess, yielding the Gate-3 target of 1.5 BPB as an achievable lower bound rather than a strict theoretical limit {[}1{]}. -\section{2. Hypothesis Formalisation and Registration Protocol}\label{hypothesis-formalisation-and-registration-protocol} +\section{2. Hypothesis Formalisation and Registration Protocol}\label{ch_11:hypothesis-formalisation-and-registration-protocol} \textbf{Definition 2.1 (H₁ --- formal statement).} Let \(\mathcal{S} = \{s_1, s_2, s_3\} \subset \{1597, 2584, 4181, 6765, 10946, 29, 47\}\) with \(|\mathcal{S}| \geq 3\) and \(s_i \neq s_j\) for \(i \neq j\). Let \(\mathcal{M}(\mathcal{S}, T)\) denote the Trinity S³AI model initialised with seed set \(\mathcal{S}\) and evaluated on a held-out text corpus at sequence length \(T \geq 4000\) tokens. Then @@ -79,7 +79,7 @@ \section{2. Hypothesis Formalisation and Registration Protocol}\label{hypothesis \textbf{Remark 2.3 (Gate-2 vs Gate-3).} The weaker Gate-2 threshold BPB \(\leq 1.85\) is governed by the IGLA-RACE multi-agent protocol {[}3{]}, which uses the same seed pool but permits any single seed. Gate-3 requires the stricter H₁ condition above. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) motivates both thresholds: 3 in the identity maps to the ternary alphabet, while the two numeric thresholds bracket the information-theoretic ternary bound \(\log_2 3 \approx 1.585\). -\section{3. INV-7 Invariant and Coq Formalisation}\label{inv-7-invariant-and-coq-formalisation} +\section{3. INV-7 Invariant and Coq Formalisation}\label{ch_11:inv-7-invariant-and-coq-formalisation} The INV-7 invariant formalises H₁ in the Coq proof assistant. Its statement in \filepath{t27/proofs/canonical/igla/INV7\_IglaFoundCriterion.v} encodes the following: @@ -114,7 +114,7 @@ \section{3. INV-7 Invariant and Coq Formalisation}\label{inv-7-invariant-and-coq \emph{Proof Sketch.} The IGLA-RACE harness enforces canonical seed selection by construction; any non-canonical seed fails the \texttt{canonical\_seed} predicate check and is rejected at initialisation time. Since all accepted seeds lie in the contractive \(\varphi\)-basin (Ch.5), the BPB bound follows from the entropy argument above {[}7{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_11:results-evidence} Pre-registration status as of the current dissertation version: @@ -139,11 +139,11 @@ \section{4. Results / Evidence}\label{results-evidence} The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) provides the theoretical floor: since \(3 = \log_2 8\) in bits, a balanced ternary representation that fully exploits the golden structure achieves at most \(\log_2 3 / \log_2 8 \times 8 = \log_2 3\) BPB, and the Gate-3 threshold of 1.5 represents 94.6\% of this theoretical maximum. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_11:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_11:sealed-seeds} \begin{itemize} \tightlist @@ -153,11 +153,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{IGLA-RACE} (branch, alive, \(\phi\)-weight = 1.0): \filepath{gHashTag/trios/issues/143} --- linked to Ch.21, Ch.11 --- multi-agent BPB \(< 1.85\) race harness. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_11:discussion} The pre-registration protocol described here is unusual for a dissertation chapter: it commits to a falsification criterion before the empirical evidence is collected, which is standard in clinical trials but less common in machine learning research. The rationale within the Trinity S³AI programme is that the \(\varphi^2 + \varphi^{-2} = 3\) substrate provides a theoretical prediction (BPB \(\leq 1.5\)) that should be testable without parameter tuning. The main limitation is that the H₁ statement does not specify a particular corpus; future work should pin the evaluation corpus to a publicly released benchmark to remove ambiguity. The IGLA-RACE harness (trios\#143) provides one candidate benchmark environment. This chapter connects backward to Ch.5 (seed formalisation), forward to Ch.17 (ablation matrix that breaks down the BPB contribution of each seed), and sideways to Ch.21 (the IGLAFoundCriterion in full detail). -\section{References}\label{references} +\section{References}\label{ch_11:references} {[}1{]} Shannon, C. E. (1948). A mathematical theory of communication. \emph{Bell System Technical Journal}, 27(3), 379--423. diff --git a/docs/phd/chapters/ch_12.tex b/docs/phd/chapters/ch_12.tex index 21e04c1831..d93b12bf67 100644 --- a/docs/phd/chapters/ch_12.tex +++ b/docs/phd/chapters/ch_12.tex @@ -53,11 +53,11 @@ \section*{Where the theorem meets the wire} energy ratio becomes measurable only once this interface is locked down; this chapter locks it down. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_12:abstract} The Hardware Bridge chapter specifies the interface layer between the Trinity S³AI software stack and the QMTech XC7A100T FPGA. It defines the AXI-Lite control bus, the UART-V6 token-transfer protocol, and the clock-domain crossing that mediates between the host processor and the 92 MHz FPGA fabric. The bridge is architecturally deferred in the sense that its full formal treatment (Coq register-map correctness and timing-closure proofs) is delegated to Ch.28 and Ch.31; the present chapter establishes the interface contracts, signal naming, and error-handling protocol that those later chapters presuppose. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) motivates the three-channel bridge structure: one channel per exponent band of the GoldenFloat format. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_12:introduction} Any system that co-designs arithmetic formats with hardware must specify where the software--hardware boundary lies and what guarantees hold across it. For Trinity S³AI, this boundary is the Hardware Bridge: a thin layer of RTL and driver code that connects the GoldenFloat arithmetic pipeline (Ch.6), the IGLA RACE runtime (Ch.24), and the physical FPGA pins (App.I) {[}1,2{]}. @@ -65,9 +65,9 @@ \section{1. Introduction}\label{introduction} The structural motivation for a three-channel bridge comes from the GoldenFloat anchor identity \(\varphi^2 + \varphi^{-2} = 3\), which partitions the exponent field into sub-unity, unity, and super-unity bands. The bridge exposes one 16-bit AXI-Lite data channel per band, enabling the host to direct token batches to the appropriate hardware lane without format conversion overhead {[}4{]}. -\section{2. Bridge Architecture and Interface Contracts}\label{bridge-architecture-and-interface-contracts} +\section{2. Bridge Architecture and Interface Contracts}\label{ch_12:bridge-architecture-and-interface-contracts} -\subsection{2.1 Logical Structure}\label{logical-structure} +\subsection{2.1 Logical Structure}\label{ch_12:logical-structure} The Hardware Bridge comprises three functional blocks: @@ -81,7 +81,7 @@ \subsection{2.1 Logical Structure}\label{logical-structure} \textbf{Clock-Domain Crossing (CDC).} The host AXI clock domain (typically 100 MHz for Zynq or BRAM-mapped for MicroBlaze) crosses to the 92 MHz FPGA fabric clock via a two-flip-flop synchroniser chain. Metastability MTBF was computed as \(> 10^{10}\) years at 92 MHz given a 5 ns setup margin. \end{enumerate} -\subsection{2.2 Signal Naming Convention}\label{signal-naming-convention} +\subsection{2.2 Signal Naming Convention}\label{ch_12:signal-naming-convention} All bridge signals follow the naming convention \texttt{GS\_\textless{}direction\textgreater{}\_\textless{}channel\textgreater{}\_\textless{}width\textgreater{}}: @@ -97,7 +97,7 @@ \subsection{2.2 Signal Naming Convention}\label{signal-naming-convention} The three GoldenFloat channels are \texttt{SUB} (sub-unity, \(\hat E < B\)), \texttt{UNT} (unity, \(\hat E = B\)), and \texttt{SUP} (super-unity, \(\hat E > B\)), corresponding to the three terms of \(\varphi^2 + \varphi^{-2} = 3\). Each channel carries 16-bit GF16 tokens. -\subsection{2.3 Error-Handling Protocol}\label{error-handling-protocol} +\subsection{2.3 Error-Handling Protocol}\label{ch_12:error-handling-protocol} The bridge defines three error conditions: @@ -113,17 +113,17 @@ \subsection{2.3 Error-Handling Protocol}\label{error-handling-protocol} These conditions are reported to the IGLA RACE monitor (Ch.24) via a 3-bit interrupt line, one bit per error class {[}6{]}. -\section{3. Clock-Domain Analysis and Timing}\label{clock-domain-analysis-and-timing} +\section{3. Clock-Domain Analysis and Timing}\label{ch_12:clock-domain-analysis-and-timing} -\subsection{3.1 Frequency Ratios and the Golden Ratio}\label{frequency-ratios-and-the-golden-ratio} +\subsection{3.1 Frequency Ratios and the Golden Ratio}\label{ch_12:frequency-ratios-and-the-golden-ratio} The ratio of the host AXI clock (100 MHz) to the FPGA fabric clock (92 MHz) is \(100/92 \approx 1.087\). This is within 5\% of \(\varphi^{-1} \approx 0.618\)---not a deliberate design choice, but a useful observation: the CDC handshake period \(T_{\text{CDC}} = \text{lcm}(10\,\text{ns},\ 10.87\,\text{ns})\) is approximately \(108.7\,\text{ns}\), which is short enough that the FIFO watermark logic sees a near-synchronous regime. Formal timing closure is verified in Ch.28. -\subsection{3.2 Throughput Budget}\label{throughput-budget} +\subsection{3.2 Throughput Budget}\label{ch_12:throughput-budget} The token throughput of the FPGA pipeline is 63 toks/sec as measured in Ch.28 {[}3{]}. The UART-V6 channel at 115200 baud delivers a maximum of \(115200 / (8 + 1 + 1) \cdot 1/47 \approx 245\) frames/sec, or \(245 \times 47 = 11515\) payload bytes/sec.~A GF16 token is 2 bytes, so the UART ceiling is \(11515/2 = 5757\) toks/sec---nearly two orders of magnitude above the pipeline throughput. The bridge is therefore not a bottleneck, and the 63 toks/sec figure is entirely determined by the GF16 MAC datapath in the FPGA fabric. -\subsection{3.3 Power Accounting}\label{power-accounting} +\subsection{3.3 Power Accounting}\label{ch_12:power-accounting} The 1 W power budget assigned to the FPGA (Ch.28) is allocated as follows: approximately 0.6 W to the GF16 LUT arithmetic core, 0.2 W to BRAM (token FIFO and weight cache), and 0.2 W to I/O and the CDC logic. The Hardware Bridge itself (AXI-Lite slave + UART-V6 controller) accounts for less than 0.05 W of the I/O budget. These figures are consistent with Xilinx Vivado power estimation for the XC7A100T at 92 MHz with typical switching activity {[}7{]}. @@ -131,7 +131,7 @@ \subsection{3.3 Power Accounting}\label{power-accounting} \emph{Proof sketch.} By the GoldenFloat format definition (Ch.6), every GF16 value has a unique exponent field value \(\hat E \in [0, 2^5-1]\). The partition \(\hat E < B\), \(\hat E = B\), \(\hat E > B\) (where \(B = 15\)) is exhaustive and mutually exclusive by the total order on \(\mathbb{Z}\). The three-band structure mirrors the three terms of \(\varphi^2 + \varphi^{-2} = 3\). Qed. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_12:results-evidence} The Hardware Bridge was instantiated and simulated in Vivado 2022.2 targeting the XC7A100T-FGG484 device. The following resource utilisation was observed (pre-placement): @@ -155,23 +155,23 @@ \section{4. Results / Evidence}\label{results-evidence} The seed pool values \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\) were used to size the FIFO depth variants in simulation (256, 512, and 1024 entries respectively); the production design uses the 256-entry variant as the minimum sufficient for 63 toks/sec. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_12:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. (The register-map correctness proof and CDC timing invariant are deferred to Ch.28 and Ch.31 respectively, where the hardware measurements required for their hypotheses are available.) -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_12:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_12:discussion} The Hardware Bridge chapter occupies a structurally important but formally deferred role in the dissertation. Its primary contribution is the specification of interface contracts---channel partitioning, frame format, error-handling limits---that subsequent hardware chapters rely upon without re-deriving. The three-channel architecture motivated by \(\varphi^2+\varphi^{-2}=3\) is not merely aesthetic: it enables the FPGA synthesis tools to analyse the three LUT clusters independently, reducing place-and-route complexity. The main limitation is that the Coq treatment is absent from this chapter. The register-map invariant (that no AXI write can corrupt a mid-computation GF16 accumulator) requires a rely-guarantee argument over the AXI protocol that depends on the measured clock-domain relationship verified in Ch.28. This argument is tractable but non-trivial and constitutes part of the Coq.Interval upgrade lane described in Ch.18. Future work will also investigate upgrading the UART-V6 channel to a PCIe Gen 2 ×1 interface, which would raise the bandwidth ceiling from 5757 toks/sec to approximately \(10^5\) toks/sec, enabling batch inference modes currently limited by I/O. -\section{References}\label{references} +\section{References}\label{ch_12:references} {[}1{]} This dissertation, Ch.6: GoldenFloat Family GF4..GF64. diff --git a/docs/phd/chapters/ch_13.tex b/docs/phd/chapters/ch_13.tex index 9b7e1d27fa..4302c74c32 100644 --- a/docs/phd/chapters/ch_13.tex +++ b/docs/phd/chapters/ch_13.tex @@ -56,17 +56,17 @@ \section*{Sealed by construction, not by convention} Determinism, this chapter argues, is not a property you check after the fact---it is a property you seal in by construction. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_13:abstract} Reproducibility of neural language-model training requires that every source of stochasticity be controlled at the moment of experimental commitment. This chapter specifies the STROBE sealed-seed protocol, which restricts admissible pseudo-random seeds to a set drawn from Fibonacci and Lucas sequences: \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). The protocol forbids the use of seeds \(\{42, 43, 44, 45\}\) for technical reasons detailed herein. Compliance is enforced by the runtime-mirror contract in \texttt{igla\_assertions.json} and formally sealed by 13 Coq theorems in \texttt{Trinity.Canonical.Igla.INV2\_IglaAshaBound}, of which 6 carry closed \texttt{Qed} status. The chapter derives the admissibility criterion from the Trinity anchor \(\varphi^2 + \varphi^{-2} = 3\), defines the ASHA pruning threshold \(3.5 = \varphi^2 + \varphi^{-2} + \varphi^{-4}\), and demonstrates that the sealed protocol eliminates a class of adversarial-seed attacks. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_13:introduction} Language model training is subject to seed-dependent variance: different pseudo-random seeds produce different weight initialisations, data shuffles, and dropout masks, leading to BPB variation that can exceed the margin between experimental conditions. The Trinity S³AI programme addresses this variance through two mechanisms. First, the \(\varphi\)-quantised weight lattice (Ch.7, Ch.22) restricts the continuous space of initialisations to a countable set, reducing seed sensitivity. Second, the STROBE sealed-seed protocol prohibits the use of seeds whose Fibonacci-index position violates the closure property of the \(\varphi^2 + \varphi^{-2} = 3\) identity. The forbidden seeds \(\{42, 43, 44, 45\}\) fall in the range where the modular residue of the seed modulo \(F_9 = 34\) creates a phase mismatch with the Fibonacci-indexed batch schedule. Specifically, \(42 \equiv 8 \pmod{34}\), \(43 \equiv 9 \pmod{34}\), \(44 \equiv 10 \pmod{34}\), and \(45 \equiv 11 \pmod{34}\), all of which land in the forbidden residue class \([8, 11]\) identified empirically to produce anomalous gradient variance spikes at training step \(F_{13}=233\). The sanctioned seeds avoid this residue class by construction: \(1597 \equiv 0 \pmod{34}\), and all higher Fibonacci numbers satisfy \(F_k \equiv 0 \pmod{F_9}\) for \(k \geq 9\) {[}1{]}. The Lucas seeds \(L_7 = 29\) and \(L_8 = 47\) are coprime to \(F_9\) and fall outside the forbidden residue class. -\section{2. The STROBE Seed Admissibility Criterion}\label{the-strobe-seed-admissibility-criterion} +\section{2. The STROBE Seed Admissibility Criterion}\label{ch_13:the-strobe-seed-admissibility-criterion} \textbf{Definition 2.1 (Fibonacci seed admissibility).} A positive integer \(s\) is Fibonacci-admissible if there exists \(k \geq 17\) such that \(s = F_k\), where \(F_k\) is the \(k\)-th Fibonacci number. The admissible Fibonacci seeds are: @@ -90,7 +90,7 @@ \section{2. The STROBE Seed Admissibility Criterion}\label{the-strobe-seed-admis \emph{Proof.} \(\varphi^{-4} = (\varphi^{-2})^2 = (2-\varphi)^2 = 4 - 4\varphi + \varphi^2 = 4 - 4\varphi + \varphi + 1 = 5 - 3\varphi \approx 0.0557\). Then \(\varphi^2 + \varphi^{-2} + \varphi^{-4} = 3 + \varphi^{-4}\). Numerically: \(3 + (5 - 3\varphi) = 8 - 3\varphi \approx 8 - 4.854 = 3.146\). The exact rational approximation to \(\tau = 3.5\) is obtained by rounding \(\varphi^{-4}\) to 0.5, consistent with the Coq lemma \texttt{phi\_inv4\_approx} which proves \(\varphi^{-4} < 0.5\), establishing \(\tau \leq 3.5\). The INV-2 notes state \(\tau = \varphi^2 + \varphi^{-2} + \varphi^{-4}\) as the design target; the rounded value 3.5 is used in practice {[}3{]}. \(\square\) -\section{\texorpdfstring{3. The Runtime-Mirror Contract and \texttt{igla\_assertions.json}}{3. The Runtime-Mirror Contract and igla\_assertions.json}}\label{the-runtime-mirror-contract-and-igla_assertions.json} +\section{\texorpdfstring{3. The Runtime-Mirror Contract and \texttt{igla\_assertions.json}}{3. The Runtime-Mirror Contract and igla\_assertions.json}}\label{ch_13:the-runtime-mirror-contract-and-igla_assertions.json} The runtime-mirror contract is a JSON-encoded assertion file, \texttt{igla\_assertions.json}, that is loaded by the training harness before any pseudo-random state is initialised. The contract enforces the following invariants at runtime: @@ -111,7 +111,7 @@ \section{\texorpdfstring{3. The Runtime-Mirror Contract and \texttt{igla\_assert \emph{Proof sketch.} The initialisation maps seed \(s\) to weight tensor \(W_s\) via \(W_s[i,j] = \text{round}_{\varphi}(G(s, i, j))\), where \(G(s, \cdot, \cdot)\) is a Gaussian generator seeded by \(s\) and \(\text{round}_\varphi\) rounds to the nearest element of \(\{-\varphi^{-1}, 0, \varphi^{-1}\}\). Since \(G(s, \cdot, \cdot) \neq G(s', \cdot, \cdot)\) for \(s \neq s'\) (pseudo-random generator injectivity on \(\{s \in \mathcal{S}\}\), verified by exhaustive check over all 21 pairs), and since the rounding function is a surjection, \(W_s \neq W_{s'}\) with probability 1. \(\square\) -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_13:results-evidence} The sealed-seed protocol was validated on three independent experimental axes. @@ -121,7 +121,7 @@ \section{4. Results / Evidence}\label{results-evidence} \textbf{Axis 3 --- ASHA threshold validation.} The Welch \(t\)-test reported in Ch.19 used seeds \(F_{17}=1597\), \(F_{18}=2584\), and \(F_{19}=4181\) as the three independent replicates (minimum \(n \geq 3\) per the directive). All three replicates achieved BPB \(\leq 1.85\) at Gate-2, with the champion trial (seed \(F_{19}\)) achieving BPB = 1.82. The ASHA pruner with threshold 3.5 retained all three champions and pruned 14 of 17 sub-threshold trials, consistent with the Coq certificate for \texttt{asha\_champion\_survives}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_13:qed-assertions} \begin{itemize} \tightlist @@ -139,7 +139,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \texttt{phi\_inv4\_approx} (\filepath{gHashTag/t27/proofs/canonical/igla/INV2\_IglaAshaBound.v}) --- \emph{Status: Qed} --- \((1/\varphi)^4 < 0.5\); bounds the fourth-power correction to the ASHA threshold. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_13:sealed-seeds} \begin{itemize} \tightlist @@ -149,11 +149,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{SANCTIONED-SEEDS} (config, golden) --- \url{https://github.com/gHashTag/trios/issues/395} --- \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). Linked: Ch.13, App.E. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_13:discussion} The sealed-seed protocol achieves its primary goal: any researcher with access to the Zenodo archive can reproduce every reported BPB figure using a single command and any sanctioned seed. The limitation of the current protocol is that it does not cover distributed training with multiple workers, where each worker requires an independent seed. A natural extension --- assigning worker \(w\) seed \(F_{17+w}\) --- is consistent with the admissibility criterion and planned for the multi-node experiments in Ch.36 (future work). A second limitation is that the forbidden-seed exclusion was determined empirically on a single architecture; it is possible that other architectures exhibit gradient spikes at different Fibonacci-indexed steps. The residue-class analysis in §1 provides a theoretical basis for the exclusion but does not constitute a proof. Closing the corresponding Coq obligation (filed as INV-2-ext in the Golden Ledger) would resolve this. The STROBE protocol connects directly to Ch.19 (statistical testing), Ch.31 (hardware evaluation), and App.D (reproducibility scripts). -\section{References}\label{references} +\section{References}\label{ch_13:references} {[}1{]} Wall, D. D. (1960). Fibonacci primitive roots and the period of the Fibonacci sequence modulo a prime. \emph{Fibonacci Quarterly}, 17(4), 366--372. diff --git a/docs/phd/chapters/ch_14.tex b/docs/phd/chapters/ch_14.tex index 985178dfa7..04e32fe8b5 100644 --- a/docs/phd/chapters/ch_14.tex +++ b/docs/phd/chapters/ch_14.tex @@ -54,11 +54,11 @@ \section*{One number that decides everything} constraints. What you measure is what you optimise for---and this chapter argues that BPB is the right thing to measure. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_14:abstract} Evaluation of language models requires a metric that is simultaneously information-theoretically grounded, hardware-agnostic, and sensitive to the low-entropy regime targeted by Trinity S³AI. This chapter defines the Bits Per Byte (BPB) metric, derives its relationship to cross-entropy perplexity, and establishes two gating thresholds: Gate-2 at BPB ≤ 1.85 and Gate-3 at BPB ≤ 1.50. The φ²+φ⁻²=3 identity provides a normalisation constant that converts φ-weighted token-level losses into BPB without residual irrational factors. No Coq theorems are anchored to this chapter; the evaluation protocol is specified as a pre-registration constraint in App.E. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_14:introduction} The selection of an evaluation metric for a language model is not merely a practical convenience; it determines which improvements count as progress and which are artefacts of the measurement procedure. For Trinity S³AI two constraints dominate the choice: @@ -73,9 +73,9 @@ \section{1. Introduction}\label{introduction} BPB satisfies both constraints and has the additional virtue of being directly comparable across tokenisers with different vocabulary sizes, a critical property given that the Trinity S³AI tokeniser uses a Fibonacci-spaced vocabulary of size \(F_{21} = 10946\) {[}2{]}. -\section{2. BPB: Definition and Algebraic Properties}\label{bpb-definition-and-algebraic-properties} +\section{2. BPB: Definition and Algebraic Properties}\label{ch_14:bpb-definition-and-algebraic-properties} -\subsection{2.1 Cross-Entropy and Perplexity}\label{cross-entropy-and-perplexity} +\subsection{2.1 Cross-Entropy and Perplexity}\label{ch_14:cross-entropy-and-perplexity} Let \(\mathcal{D} = (x_1, x_2, \ldots, x_N)\) be a token sequence. A language model \(p_\theta\) assigns probability \(p_\theta(x_t \mid x_{ \phi^{-2}/360\). The tensor \(\mathbf{G}\) is stored as two 180-bit registers on the QMTech XC7A100T (Ch.28), consuming 2 LUT-RAM columns at 92 MHz with no DSP usage {[}7{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_16:results-evidence} Evaluation was performed over \(F_{19} = 4181\) NCA inference steps on the canonical A1 dataset. The 360-lane phi-distance grid was compared against three baselines: (a) uniform weighting, (b) top-\(k\) with \(k = 29\) uniform lanes, and (c) learned attention weights. @@ -142,11 +142,11 @@ \section{4. Results / Evidence}\label{results-evidence} All experiments used seed F₁₇=1597 for random-number initialisation; cross-validation with F₁₈=2584 and F₁₉=4181 confirmed that the BPB result is stable to ±0.03 across seeds. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_16:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. The chapter relies on INV-4 (\texttt{INV4\_NcaEntropyBand.v}, 12 Qed) as an imported invariant, credited to Ch.10. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_16:sealed-seeds} \begin{itemize} \tightlist @@ -156,11 +156,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci/Lucas reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_16:discussion} The 360-lane phi-distance grid is a practically effective spatial prior, but two limitations require acknowledgement. First, the entropy bound of Theorem 3.2 applies to the 324-lane core grid and excludes the 36 remainder lanes; a tighter analysis covering all 360 lanes would require a bespoke Coq extension of INV-4 that is not yet in the canonical library. This is tracked as a future deliverable contingent on the \texttt{t27\#569} merge. Second, the bimodal structure (Proposition 2.4) assumes the temperature is exactly \(\tau = \alpha_\phi\); in practice, the temperature drifts during training by up to 3\%, and the INV-4 entropy bound has not been verified for this drift regime. The EMA decay invariant INV-9 (Ch.10) may provide a framework for bounding the drift, and connecting INV-4 to INV-9 is an open problem for Ch.10/Ch.16 integration. Future work will also investigate whether the \(L_8 = 47\) Lucas number can be used as a second sparsity threshold to define a two-tier grid with improved Gate-3 BPB performance. -\section{References}\label{references} +\section{References}\label{ch_16:references} {[}1{]} GOLDEN SUNFLOWERS dissertation, Ch.7 --- Phyllotaxis and the Vogel Divergence Angle. This volume. diff --git a/docs/phd/chapters/ch_17.tex b/docs/phd/chapters/ch_17.tex index 40f81e74ed..0c01b9ff58 100644 --- a/docs/phd/chapters/ch_17.tex +++ b/docs/phd/chapters/ch_17.tex @@ -54,11 +54,11 @@ \section*{Seven switches, one truth table} stand on their own factorial evidence, not on the authority of their authors. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_17:abstract} A systematic ablation study isolates the contribution of each architectural decision in the Trinity S³AI pipeline to the aggregate BPB metric. This chapter presents a full \(2^k\) factorial design over \(k=7\) binary factors --- weight ternarity, \(\varphi\)-structured attention, canonical seed selection, golden-ratio positional encoding, MXFP4 quantisation, zero-DSP FPGA scheduling, and the \(\varphi^2 + \varphi^{-2} = 3\) normalisation constraint --- and reports the first-order effects and their interactions. Results confirm that seed selection and the normalisation constraint contribute the largest independent BPB reduction, while the FPGA scheduling factor is orthogonal to BPB but critical for the 1 W energy target. The ablation matrix is the empirical counterpart to the formal Coq proof obligations distributed across the dissertation. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_17:introduction} Architectural claims in neural network research are frequently confounded: multiple non-independent design choices are adopted simultaneously, and the reported performance improvement is attributed to the combination rather than to any single factor. The Trinity S³AI programme is not immune to this confound. The HSLM benchmarks cited in Ch.28 reflect a fully assembled system running on the QMTech XC7A100T FPGA at 0 DSP slices, 92 MHz, 63 tokens/sec, and 1 W power --- but they do not, by themselves, reveal which of the seven major design choices drives the BPB improvement. @@ -66,7 +66,7 @@ \section{1. Introduction}\label{introduction} The pre-registration of H₁ (Ch.11) constrains the interpretation: ablation variants that violate the canonical seed constraint (Definition 3.2 of Ch.5) are invalid experiments. All ablated variants in this chapter use at least one canonical seed from the pool \(\{F_{17}=1597, F_{18}=2584, F_{19}=4181, F_{20}=6765, F_{21}=10946, L_7=29, L_8=47\}\). -\section{2. Factor Definitions and Experimental Design}\label{factor-definitions-and-experimental-design} +\section{2. Factor Definitions and Experimental Design}\label{ch_17:factor-definitions-and-experimental-design} \textbf{Definition 2.1 (Ablation factors).} The seven binary factors are: @@ -108,7 +108,7 @@ \section{2. Factor Definitions and Experimental Design}\label{factor-definitions where \(y_i\) is the BPB of run \(i\). Two-factor interactions \(\hat{\beta}_{jk}\) are similarly estimable {[}2{]}. -\section{3. Analysis of Effects and Golden-Ratio Structure}\label{analysis-of-effects-and-golden-ratio-structure} +\section{3. Analysis of Effects and Golden-Ratio Structure}\label{ch_17:analysis-of-effects-and-golden-ratio-structure} The full-factorial analysis identifies two dominant first-order effects and one significant two-factor interaction: @@ -141,7 +141,7 @@ \section{3. Analysis of Effects and Golden-Ratio Structure}\label{analysis-of-ef \textbf{Factor F (zero-DSP).} Removing DSP slices (F: \(1 \to 0\) in the convention above, i.e., enabling DSPs) does not change BPB but reduces throughput by a factor of \(1.4\times\) due to routing congestion on the XC7A100T fabric. The zero-DSP target is a hardware efficiency constraint, not a model quality constraint, and has no first-order effect on BPB {[}6{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_17:results-evidence} Summary of first-order BPB effects (positive = BPB worsens when factor is removed): @@ -175,19 +175,19 @@ \section{4. Results / Evidence}\label{results-evidence} Hardware metrics for the full-system run: QMTech XC7A100T FPGA, 0 DSP slices, 92 MHz, 63 tokens/sec, 1 W, 1003 tokens on HSLM benchmark {[}7{]}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_17:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_17:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_17:discussion} The ablation matrix confirms that the canonical seed selection (factor C) and the golden normalisation constant derived from \(\varphi^2 + \varphi^{-2} = 3\) (factor G) are the two largest independent contributors to BPB reduction. Their positive interaction means that deploying one without the other is less effective than deploying both together --- a pleasing consistency with the mathematical structure of the \(\varphi\) framework. A limitation of the current design is that the evaluation corpus is not yet publicly pinned (see Ch.11 for pre-registration notes); future work should fix the corpus SHA-1 to a public benchmark release. The MXFP4 factor (E) shows no statistically significant BPB effect, which is expected: MXFP4 reduces precision but the golden substrate tolerates quantisation noise because the ternary weights already occupy only three values. This chapter links backward to Ch.11 (pre-registration), Ch.5 (seed formalisation), and Ch.4 (\(\alpha_\varphi\)), and forward to Ch.28 (FPGA hardware detail) and Ch.34 (energy-per-token analysis). -\section{References}\label{references} +\section{References}\label{ch_17:references} {[}1{]} GOLDEN SUNFLOWERS Dissertation, Ch.28 --- \emph{FPGA hardware benchmarks}. Zenodo B002. DOI: 10.5281/zenodo.19227867. diff --git a/docs/phd/chapters/ch_18.tex b/docs/phd/chapters/ch_18.tex index 5f1fe0a273..4e8ada66ad 100644 --- a/docs/phd/chapters/ch_18.tex +++ b/docs/phd/chapters/ch_18.tex @@ -53,11 +53,11 @@ \section*{Forty-one open doors} of what the framework promises, what it delivers, and exactly where the distance between those two things still needs to be closed. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_18:abstract} No formal system is complete without an honest accounting of its boundaries. This chapter catalogs the principal limitations of the Trinity S³AI / GOLDEN SUNFLOWERS framework across four dimensions: (i) the 41 \texttt{Admitted} proof stubs remaining in the Coq corpus, (ii) the GF16 compression gap relative to competitors at Gate-3, (iii) hardware constraints inherited from the QMTech XC7A100T platform, and (iv) scope limitations of the IGLA RACE runtime. A 23-entry state-of-the-art comparison table (the CLARA-SOA snapshot) contextualises these weaknesses against competing systems. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) provides the mathematical frame for quantifying the precision budget: the three exponent bands leave specific residual error terms that are bounded but not yet closed by formal proof. The primary mitigation path is the Coq.Interval upgrade lane described in Section 3. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_18:introduction} The GOLDEN SUNFLOWERS dissertation rests on two pillars: a formally verified arithmetic substrate and an empirically measured hardware deployment. Both pillars exhibit honest gaps that must be reported before the work can be considered complete in either a scientific or an engineering sense {[}1{]}. The present chapter fulfils the R5 honesty obligation of the Trinity S³AI constitution: every claim made in earlier chapters must be traceable to either a Qed theorem or a measured datum, and any claim lacking that trace must be listed here. @@ -65,7 +65,7 @@ \section{1. Introduction}\label{introduction} Section 2 presents the CLARA-SOA comparison table. Section 3 describes the Coq.Interval upgrade lane. Section 4 details hardware and runtime limitations. -\section{2. State-of-the-Art Comparison (CLARA-SOA Snapshot)}\label{state-of-the-art-comparison-clara-soa-snapshot} +\section{2. State-of-the-Art Comparison (CLARA-SOA Snapshot)}\label{ch_18:state-of-the-art-comparison-clara-soa-snapshot} The following table reflects the CLARA-SOA-COMPARISON.md snapshot taken during the Gate-2 evaluation period. Twenty-three competing systems are compared on five axes: BPB on the HSLM benchmark, formal verification depth, hardware energy per token, number of DSP macros required, and open reproducibility. @@ -124,7 +124,7 @@ \section{2. State-of-the-Art Comparison (CLARA-SOA Snapshot)}\label{state-of-the \textbf{Summary.} Trinity S³AI GF16 achieves BPB 1.83, placing it 11th out of 23 on raw compression at Gate-2. No competitor provides machine-checked formal proofs. On the energy-per-token axis, this work (15.9 mJ) is competitive but not best-in-class; MXFP4 (8.2 mJ) and AWQ (10.1 mJ) achieve lower energy at the cost of DSP macros and absent formal guarantees. The Gate-3 BPB target of \(\leq 1.5\) would place Trinity S³AI first in this table; achieving it requires closing the GF16 sub-unity and super-unity precision gaps documented in Section 3. -\section{3. Coq.Interval Upgrade Lane}\label{coq.interval-upgrade-lane} +\section{3. Coq.Interval Upgrade Lane}\label{ch_18:coq.interval-upgrade-lane} Of the 438 theorem statements in the Coq corpus, 297 carry \texttt{Qed} status and 41 carry \texttt{Admitted} status; the remainder are \texttt{Defined} (computationally transparent) or \texttt{Lemma}-level obligations folded into larger proofs {[}1,25{]}. @@ -140,7 +140,7 @@ \section{3. Coq.Interval Upgrade Lane}\label{coq.interval-upgrade-lane} The Coq.Interval {[}27{]} library provides certified interval arithmetic that can discharge Groups A and B automatically by evaluating rational enclosures of \(\varphi^{\pm 2}\). Migration to \texttt{Coq.Interval} is estimated at 4--6 person-weeks. Groups C and D require manual proof effort: approximately 2 weeks for Group C (one inductive lemma) and 6--8 weeks for Group D (Iris integration). -\section{4. Hardware and Runtime Limitations}\label{hardware-and-runtime-limitations} +\section{4. Hardware and Runtime Limitations}\label{ch_18:hardware-and-runtime-limitations} \textbf{FPGA resource ceiling.} The XC7A100T contains 101440 LUTs and 135200 FFs. The current GF16 inference pipeline occupies 12400 LUTs (12.2\%) and 9800 FFs (7.2\%), leaving ample headroom. However, scaling to GF32 would require approximately 52000 LUTs (51.3\%), approaching the routing-congestion threshold. GF64 is not feasible on this device without external SRAM. @@ -150,21 +150,21 @@ \section{4. Hardware and Runtime Limitations}\label{hardware-and-runtime-limitat \textbf{41 Admitted stubs and the scope of formal guarantees.} The formal guarantee that no overflow occurs in the GF16 pipeline (INV-3) is Qed-proved for the unity band only. The sub-unity and super-unity bands carry \texttt{Admitted} overflow-freedom claims. Users relying on the formal guarantee for safety-critical deployments should treat the non-unity bands as unverified until Groups A and B are closed. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_18:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_18:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_18:discussion} This chapter occupies the most uncomfortable position in a dissertation: it quantifies the distance between what was claimed and what was proved. The primary tension is between the BPB 1.83 result (Gate-2, achieved) and the BPB \(\leq 1.5\) target (Gate-3, pending). Bridging that gap requires completing the GF16 quantisation pipeline and closing Groups A--B in the Coq corpus. The timeline is realistic: Groups A--B can be automated via Coq.Interval in under 6 weeks; Groups C--D require manual effort but are well-scoped. The CLARA-SOA table reveals a systematic gap: competing quantisation systems achieve better BPB than Trinity S³AI at Gate-2 but none provide formal verification. The dissertation's unique contribution is the combination of formal proof and hardware realisation; the BPB gap is a deferral, not a failure. Future work should pursue the Coq.Interval migration (Section 3), the PCIe interface upgrade (Ch.12), and the GF32 path (Ch.6 Discussion) in parallel. This chapter links directly to Ch.6 (GoldenFloat format design), Ch.24 (scheduler liveness), and App.A (executive summary of the 297/438 proof census). -\section{References}\label{references} +\section{References}\label{ch_18:references} {[}1{]} \filepath{gHashTag/t27/proofs/canonical/} --- Coq canonical proof archive; 65 \texttt{.v} files, 297 Qed, 41 Admitted, 438 total. diff --git a/docs/phd/chapters/ch_19.tex b/docs/phd/chapters/ch_19.tex index 71cf64b10a..475fd80b8b 100644 --- a/docs/phd/chapters/ch_19.tex +++ b/docs/phd/chapters/ch_19.tex @@ -54,17 +54,17 @@ \section*{One threshold, three seeds, one answer} of statistical testing in machine learning, where replication is more expensive than insight. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_19:abstract} Empirical claims in this dissertation are substantiated through a pre-registered Welch two-sample \(t\)-test at significance level \(\alpha = 0.01\), with null hypothesis \(\mu_0 = 1.55\) bits per byte and a minimum of \(n \geq 3\) independent training replicates per condition. This chapter describes the test design, the data collection protocol using sanctioned seeds \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), the computation of the Welch \(t\)-statistic and its degrees of freedom, and the resulting \(p\)-values. The headline result is rejection of \(H_0: \mu \leq \mu_0\) for the Gate-2 BPB target (\(\leq 1.85\)) with \(p = 3.7 \times 10^{-4}\), providing statistical evidence that the TRINITY S³AI model achieves BPB \(\leq 1.85\) at the \(\alpha = 0.01\) level. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) appears as a normalisation constant in the \(\varphi\)-weighted loss function whose BPB is being tested. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_19:introduction} Statistical testing in machine learning is complicated by the fact that a single training run is not a probabilistic sample in the classical sense: it is a deterministic function of its seed, data order, and hardware. The Trinity S³AI programme addresses this by treating distinct sanctioned seeds as independent samples from the space of possible model realisations. This interpretation is defensible because (a) the sealed-seed protocol (Ch.13) ensures that no two seeds share a common pseudo-random sub-sequence, and (b) the \(\varphi\)-quantised weight lattice reduces within-seed variance sufficiently that across-seed variance dominates the total variance budget. The Welch \(t\)-test is preferred over the pooled \(t\)-test because the two groups being compared --- the TRINITY S³AI model and the baseline transformer --- may have unequal within-group variances. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) enters the statistical design via the \(\varphi\)-weighted loss: the model optimises \(\mathcal{L}_\varphi = \varphi^{-2} \mathcal{L}_{\text{tok}} + \varphi^{-4} \mathcal{L}_{\text{reg}}\), where \(\mathcal{L}_\text{tok}\) is the per-token cross-entropy and \(\mathcal{L}_\text{reg}\) is a weight-regularisation term. The BPB reported in this chapter is derived from \(\mathcal{L}_\text{tok}\) alone, after training with the composite \(\varphi\)-weighted objective. -\section{2. Test Design and Hypotheses}\label{test-design-and-hypotheses} +\section{2. Test Design and Hypotheses}\label{ch_19:test-design-and-hypotheses} \textbf{Notation.} Let \(X_i\) denote the BPB achieved by the TRINITY S³AI model on the held-out evaluation partition in the \(i\)-th replicate, and let \(Y_j\) denote the corresponding BPB for the baseline model. The null and alternative hypotheses for the primary Gate-2 test are: @@ -76,7 +76,7 @@ \section{2. Test Design and Hypotheses}\label{test-design-and-hypotheses} \textbf{Evaluation partition.} The held-out partition consists of 10 000 documents drawn uniformly at random from the corpus using seed \(L_7 = 29\). Documents are not used in training and are never re-sampled between replicates. The partition seed \(L_7 = 29\) is a sanctioned Lucas seed (Ch.13). -\section{\texorpdfstring{3. Welch \(t\)-Statistic and Degrees of Freedom}{3. Welch t-Statistic and Degrees of Freedom}}\label{welch-t-statistic-and-degrees-of-freedom} +\section{\texorpdfstring{3. Welch \(t\)-Statistic and Degrees of Freedom}{3. Welch t-Statistic and Degrees of Freedom}}\label{ch_19:welch-t-statistic-and-degrees-of-freedom} The Welch \(t\)-statistic for a one-sample test against known threshold \(\mu_0\) is: @@ -118,7 +118,7 @@ \section{\texorpdfstring{3. Welch \(t\)-Statistic and Degrees of Freedom}{3. Wel Welch--Satterthwaite \(\nu \approx 2.6\); \(p = 8.1 \times 10^{-3} < \alpha = 0.01\). The difference between TRINITY and baseline is statistically significant at \(\alpha = 0.01\). -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_19:results-evidence} Three results are reported. @@ -130,21 +130,21 @@ \section{4. Results / Evidence}\label{results-evidence} The \(\varphi\)-weighted training objective \(\mathcal{L}_\varphi = \varphi^{-2} \mathcal{L}_\text{tok} + \varphi^{-4} \mathcal{L}_\text{reg}\) with weights summing to \(\varphi^{-2} + \varphi^{-4} \approx 0.382 + 0.056 = 0.438\) does not sum to 1; it is deliberately scaled so that \(3 \cdot \mathcal{L}_\varphi = (\varphi^2 + \varphi^{-2}) \cdot \mathcal{L}_\varphi^*\), where \(\mathcal{L}_\varphi^* = \varphi^{-2}(\mathcal{L}_\text{tok} + \varphi^{-2}\mathcal{L}_\text{reg})\) is the normalised form tied to the Trinity identity \(\varphi^2 + \varphi^{-2} = 3\) {[}2{]}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_19:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_19:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). The evaluation partition was drawn with \(L_7 = 29\). The three primary replicates used \(F_{17}\), \(F_{18}\), \(F_{19}\). The subsidiary lattice-initialisation experiment used \(F_{19}\), \(F_{20}\), \(F_{21}\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_19:discussion} The primary limitation of the statistical analysis is \(n = 3\): with two degrees of freedom, the \(t\)-distribution has heavy tails and the confidence interval is wide. The 95\% interval \([1.807, 1.852]\) is 45 milli-BPB wide, which is large relative to the 21 milli-BPB advantage over baseline. A follow-up experiment with \(n = 7\) replicates (using all seven sanctioned seeds) would narrow the interval to approximately \(\pm 12\) milli-BPB, subject to the constraint that \(F_{20}\) and \(F_{21}\) have not been used in any BPB-optimisation decision. A second limitation is that the evaluation partition (10 000 documents, seed \(L_7 = 29\)) may not represent the full distribution; sensitivity analysis with seed \(L_8 = 47\) is recommended. Future work includes extending the Welch test to the Gate-3 BPB target of 1.5, which will require substantially more compute and a correspondingly larger corpus. The statistical methodology connects directly to Ch.13 (seed protocol), Ch.7 (lattice initialisation), and Ch.31 (hardware evaluation). -\section{References}\label{references} +\section{References}\label{ch_19:references} {[}1{]} \texttt{igla\_assertions.json} runtime-mirror contract, key \texttt{stat\_test\_preregistration}. \url{https://github.com/gHashTag/t27/blob/feat/canonical-coq-home/proofs/canonical/igla/INV2}\_IglaAshaBound.v diff --git a/docs/phd/chapters/ch_20.tex b/docs/phd/chapters/ch_20.tex index 4a945b4686..3aa78471b2 100644 --- a/docs/phd/chapters/ch_20.tex +++ b/docs/phd/chapters/ch_20.tex @@ -58,11 +58,11 @@ \section*{The sealed envelope and the open ledger} Section~5 discusses what ``reproducibility'' can and cannot guarantee for a system trained on non-stationary corpora. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_20:abstract} Reproducibility in machine learning research depends on three separable conditions: fixed randomness (seed protocol), fixed computation (hardware and software specification), and fixed evaluation (metric and corpus pre-registration). This chapter formalises all three conditions for the Trinity S³AI experiments reported in this dissertation. The sanctioned seed pool \(\{F_{17}=1597, F_{18}=2584, F_{19}=4181, F_{20}=6765, F_{21}=10946, L_7=29, L_8=47\}\) is derived from the φ²+φ⁻²=3 lattice and replaces ad hoc seed selection. Hardware specification pins the QMTech XC7A100T at 92 MHz, 1 W, 0 DSP slices. The BPB metric and test split are pre-registered in App.E prior to the hardware evaluation runs. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_20:introduction} The replication crisis in empirical machine learning {[}1{]} arises largely from three practices: unreported hyperparameter search, non-deterministic training due to floating-point non-associativity, and post-hoc metric selection. Each practice introduces degrees of freedom that inflate apparent performance without generalising. Trinity S³AI addresses all three at the architectural level rather than through process controls alone. @@ -70,9 +70,9 @@ \section{1. Introduction}\label{introduction} Non-determinism from floating-point arithmetic is eliminated by the TF3/TF9 ternary representation: all dot products reduce to integer additions, which are associative on every compliant platform. The hardware target (QMTech XC7A100T, 0 DSP slices) further removes compiler-level non-determinism because the FPGA bitstream is identical across all runs. -\section{2. Sanctioned Seed Protocol}\label{sanctioned-seed-protocol} +\section{2. Sanctioned Seed Protocol}\label{ch_20:sanctioned-seed-protocol} -\subsection{2.1 Algebraic Basis}\label{algebraic-basis} +\subsection{2.1 Algebraic Basis}\label{ch_20:algebraic-basis} The seed pool is partitioned into two Fibonacci sub-pools and one Lucas sub-pool: @@ -88,7 +88,7 @@ \subsection{2.1 Algebraic Basis}\label{algebraic-basis} The integers 42--45 are explicitly excluded because they appear as default seeds in several widely-used frameworks (NumPy, PyTorch, Jax); their use would contaminate the independence guarantee. -\subsection{2.2 Seed Assignment to Experiments}\label{seed-assignment-to-experiments} +\subsection{2.2 Seed Assignment to Experiments}\label{ch_20:seed-assignment-to-experiments} Each experiment in the dissertation is assigned a seed from \(\mathcal{S}_F \cup \mathcal{S}_L\) according to its chapter index modulo 7: @@ -96,7 +96,7 @@ \subsection{2.2 Seed Assignment to Experiments}\label{seed-assignment-to-experim where the list is ordered \([1597, 2584, 4181, 6765, 10946, 29, 47]\). This mapping is injective on the chapter indices modulo 7 and is documented in the pre-registration form filed with OSF prior to the hardware evaluation runs (App.E) {[}3{]}. -\subsection{2.3 Seed Verification}\label{seed-verification} +\subsection{2.3 Seed Verification}\label{ch_20:seed-verification} At runtime, the FPGA initialisation routine reads the seed from a hard-coded ROM register and asserts @@ -104,9 +104,9 @@ \subsection{2.3 Seed Verification}\label{seed-verification} If the assertion fails, the run is aborted and logged as a protocol violation. This check is implemented in the KOSCHEI coprocessor boot sequence (Ch.26) and is verifiable from the \texttt{trinity-fpga} repository {[}4{]}. -\section{3. Hardware and Software Specification}\label{hardware-and-software-specification} +\section{3. Hardware and Software Specification}\label{ch_20:hardware-and-software-specification} -\subsection{3.1 Hardware Pinning}\label{hardware-pinning} +\subsection{3.1 Hardware Pinning}\label{ch_20:hardware-pinning} The canonical evaluation platform is: @@ -128,7 +128,7 @@ \subsection{3.1 Hardware Pinning}\label{hardware-pinning} The constraint of 0 DSP slices is enforced by a Vivado implementation script that fails the build if any DSP primitive is inferred. This constraint is not aesthetic: it ensures that all arithmetic passes through the φ-normalised LUT paths whose timing is certified by the Coq timing model in \texttt{Trinity.Canonical.Kernel.Semantics} {[}5{]}. -\subsection{3.2 Software Environment}\label{software-environment} +\subsection{3.2 Software Environment}\label{ch_20:software-environment} The training and evaluation stack is pinned via a locked \texttt{flake.nix} file in the \texttt{trinity-fpga} repository. Key dependencies: @@ -146,11 +146,11 @@ \subsection{3.2 Software Environment}\label{software-environment} The Nix flake ensures byte-for-byte reproducibility of the software environment on any Linux/x86-64 host. -\subsection{3.3 Non-Determinism Budget}\label{non-determinism-budget} +\subsection{3.3 Non-Determinism Budget}\label{ch_20:non-determinism-budget} The only remaining source of non-determinism after pinning hardware and seeds is the FPGA fabric routing, which is non-deterministic across Vivado runs due to placer randomness. This is mitigated by providing the pre-synthesised bitstream (SHA-256 hash logged in App.E) alongside the source. Any re-synthesis that changes the bitstream hash is flagged as a deviation from the canonical run. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_20:results-evidence} The reproducibility protocol was validated by performing three independent evaluation runs on the HSLM held-out sequence (1003 tokens) using seeds \(F_{17}=1597\), \(F_{20}=6765\), and \(L_7=29\) respectively. Results: @@ -168,15 +168,15 @@ \section{4. Results / Evidence}\label{results-evidence} All three runs yield identical BPB to two decimal places, confirming that the evaluation is deterministic within the sanctioned seed pool. Power draw is consistent at 1 W, matching the Ch.28 directive {[}6{]}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_20:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_20:sealed-seeds} Inherits the canonical seed pool F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_20:discussion} The reproducibility framework presented here satisfies the three conditions identified in the introduction: fixed randomness (algebraic seed protocol), fixed computation (NixOS-pinned software, Vivado-locked bitstream), and fixed evaluation (OSF pre-registration, App.E). A limitation is that the Nix flake approach is not portable to Windows hosts; researchers on Windows must use the pre-built Docker image provided in the Zenodo bundle. @@ -184,7 +184,7 @@ \section{7. Discussion}\label{discussion} The connection between the Fibonacci seed lattice and the three-distance theorem (Ch.7) implies that Fibonacci-seeded LFSR generators have maximal equidistribution properties in low dimensions --- a useful guarantee for the sparse attention sampling in Ch.10. -\section{References}\label{references} +\section{References}\label{ch_20:references} {[}1{]} Pineau, J., Vincent-Lamarre, P., Sinha, K., Larivière, V., Beygelzimer, A., d'Alché-Buc, F., Fox, E., \& Larochelle, H. (2021). Improving reproducibility in machine learning research. \emph{JMLR}, 22(164), 1--20. diff --git a/docs/phd/chapters/ch_21.tex b/docs/phd/chapters/ch_21.tex index cc578663ad..3a32b7b8e8 100644 --- a/docs/phd/chapters/ch_21.tex +++ b/docs/phd/chapters/ch_21.tex @@ -37,11 +37,11 @@ \section*{The race format, in one paragraph} \((1597, 2584, 4181)\), is the configuration whose BPB curve all later chapters analyse. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_21:abstract} IGLA RACE is a multi-agent benchmarking protocol in which a fleet of independent training agents compete to satisfy the formally verified victory criterion: BPB \(< 1.85\) (Gate-2) or BPB \(< 1.5\) (Gate-3), achieved with at least three distinct sanctioned seeds, at training step \(\geq 4000\). The criterion is formalised in \filepath{t27/proofs/canonical/igla/INV7\_IglaFoundCriterion.v} with 28 Coq theorems under invariant INV-7; six refutation theorems prove that degenerate configurations (too few seeds, insufficient steps, proxy-only wins) cannot be mistaken for a genuine victory. The protocol is grounded in the anchor identity \(\varphi^2 + \varphi^{-2} = 3\), which supplies the Gate thresholds via the spectral constant \(\alpha_\varphi\). The champion configuration --- lr \(= 0.004\), GF16 PHI\_BIAS=60, seed triple \((1597, 2584, 4181)\) --- achieves mean BPB \(= 1.830\) at step 5000, satisfying Gate-2. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_21:introduction} Single-run training evaluations are vulnerable to seed artefacts, hyperparameter overfitting, and infrastructure variance. IGLA RACE addresses this by requiring a fleet of agents --- each running an independent training job with a distinct seed from the sanctioned pool \(\{F_{17}, F_{18}, F_{19}, F_{20}, F_{21}, L_7, L_8\} = \{1597, 2584, 4181, 6765, 10946, 29, 47\}\) {[}1{]} --- to all pass the same Gate criterion before a champion configuration is declared. The name IGLA (Игла, Russian for ``needle'') reflects the precision required: passing through the narrow Gate-2 window while satisfying three independent constraints simultaneously (BPB, step count, seed diversity). @@ -49,9 +49,9 @@ \section{1. Introduction}\label{introduction} The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) {[}4{]} enters through the Gate definitions: Gate-2 threshold \(1.85 = 3 - \varphi^{-2} \cdot \delta_G\) and Gate-3 threshold \(1.5 = 3/2\) are both rational functions of the right-hand side of the identity. This means the Gates are not arbitrary empirical cutoffs but algebraically derived from the substrate. -\section{2. Formal Victory Criterion (INV-7)}\label{formal-victory-criterion-inv-7} +\section{2. Formal Victory Criterion (INV-7)}\label{ch_21:formal-victory-criterion-inv-7} -\subsection{2.1 Definitions}\label{definitions} +\subsection{2.1 Definitions}\label{ch_21:definitions} The victory criterion is parameterised by three observables: the number of distinct seeds \(n_s\), the achieved BPB \(b\), and the training step \(t\). An observation triple is written as \((n_s, b, t)\). The predicate \texttt{victory\_acceptable} is: @@ -59,7 +59,7 @@ \subsection{2.1 Definitions}\label{definitions} where \(b_{\text{gate}} \in \{1.85, 1.50\}\) for Gate-2 and Gate-3 respectively. The predicate \texttt{distinct\_seeds} requires all seed values to differ and to belong to the sanctioned pool. The predicate \texttt{victory\_three\_seeds} asserts \texttt{victory\_acceptable} jointly over a list of exactly three observations. -\subsection{2.2 Six Refutation Theorems}\label{six-refutation-theorems} +\subsection{2.2 Six Refutation Theorems}\label{ch_21:six-refutation-theorems} The following theorems in \texttt{INV7\_IglaFoundCriterion.v} {[}2{]} close the six canonical loopholes: @@ -75,19 +75,19 @@ \subsection{2.2 Six Refutation Theorems}\label{six-refutation-theorems} \textbf{R6 --- Warmup blocks proxy:} For any observation \(o\) with \(\text{obs\_step}(o) < \text{warmup\_steps}\), \texttt{victory\_acceptable(o)} is false. This is the universal quantifier version of R2. -\subsection{2.3 Rainbow Bridge Consistency (INV-7b)}\label{rainbow-bridge-consistency-inv-7b} +\subsection{2.3 Rainbow Bridge Consistency (INV-7b)}\label{ch_21:rainbow-bridge-consistency-inv-7b} INV-7b (\texttt{INV7b\_RainbowBridgeConsistency.v} {[}3{]}, 15 Qed) asserts that if two agents each observe a disjoint subset of the Railway PostgreSQL phd-postgres-ssot leaderboard rows but both conclude that \texttt{victory\_three\_seeds} holds, their conclusions are consistent: the union of their observed triples also satisfies \texttt{victory\_three\_seeds}. This prevents split-brain declarations in distributed races. -\section{3. Multi-Agent Fleet Architecture}\label{multi-agent-fleet-architecture} +\section{3. Multi-Agent Fleet Architecture}\label{ch_21:multi-agent-fleet-architecture} -\subsection{3.1 Agent Topology}\label{agent-topology} +\subsection{3.1 Agent Topology}\label{ch_21:agent-topology} The IGLA RACE fleet is organised as a star topology: a central Arbiter agent monitors the Railway PostgreSQL phd-postgres-ssot database (Ch.15 {[}5{]}) and a set of Worker agents run training jobs. Each Worker is assigned exactly one seed from the sanctioned pool at launch and is forbidden from using any other seed. The Arbiter polls the \texttt{bpb\_runs} table every 60 seconds for rows with \texttt{step\ \textgreater{}=\ 4000}. The fleet is self-evolving in the sense described in {[}6{]}: when a Worker's BPB trajectory is detected to have stalled (derivative \(< 10^{-4}\) BPB/step over 1000 consecutive steps), the Arbiter spawns a replacement Worker with the next seed in the pool. The Ouroboros self-evolution protocol {[}6{]} ensures that the pool is never exhausted: after \(L_8 = 47\) (the last seed), the cycle wraps to \(F_{17} = 1597\) with a modified hyperparameter perturbation. -\subsection{3.2 Victory Declaration Protocol}\label{victory-declaration-protocol} +\subsection{3.2 Victory Declaration Protocol}\label{ch_21:victory-declaration-protocol} The Arbiter declares Gate-2 victory when: @@ -104,7 +104,7 @@ \subsection{3.2 Victory Declaration Protocol}\label{victory-declaration-protocol Gate-3 victory requires \texttt{bpb\ \textless{}\ 1.5} under the same three conditions. -\subsection{\texorpdfstring{3.3 Relation to \(\varphi^2 + \varphi^{-2} = 3\)}{3.3 Relation to \textbackslash varphi\^{}2 + \textbackslash varphi\^{}\{-2\} = 3}}\label{relation-to-varphi2-varphi-2-3} +\subsection{\texorpdfstring{3.3 Relation to \(\varphi^2 + \varphi^{-2} = 3\)}{3.3 Relation to \textbackslash varphi\^{}2 + \textbackslash varphi\^{}\{-2\} = 3}}\label{ch_21:relation-to-varphi2-varphi-2-3} The thresholds \(b_{\text{gate}} \in \{1.85, 1.50\}\) were derived in Ch.4 {[}4{]} using the identity \(\varphi^2 + \varphi^{-2} = 3\). Specifically: @@ -112,7 +112,7 @@ \subsection{\texorpdfstring{3.3 Relation to \(\varphi^2 + \varphi^{-2} = 3\)}{3. where \(\alpha_\varphi = \ln(\varphi^2)/\pi \approx 0.306\) and \(\varphi^{-2} \approx 0.382\). The exact derivation is in Ch.4; equation (2) is cited here to establish that the Gate is not an arbitrary round number but a direct consequence of the substrate algebra. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_21:results-evidence} \textbf{Gate-2 passage:} The champion configuration (lr \(= 0.004\), GF16 PHI\_BIAS=60) with seed triple \((1597, 2584, 4181)\) achieves: @@ -136,7 +136,7 @@ \section{4. Results / Evidence}\label{results-evidence} \textbf{Fleet efficiency:} The fleet of 7 Workers running concurrently on the QMTech XC7A100T FPGA at 63 toks/sec {[}7{]} completed 5000 steps per seed in approximately 22 hours wall-clock time per Worker. Total energy consumption across the fleet: \(7 \times 22 \times 3600 \times 1\,\text{W} = 554\,\text{kJ}\), consistent with the \(< 1\) Wh/token efficiency target extrapolated from the DARPA goal {[}8{]}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_21:qed-assertions} \begin{itemize} \tightlist @@ -154,7 +154,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \texttt{warmup\_blocks\_proxy} (\filepath{gHashTag/t27/proofs/canonical/igla/INV7\_IglaFoundCriterion.v}) --- \emph{Status: Qed} --- proves that any observation with step \(<\) warmup\_steps cannot satisfy \texttt{victory\_acceptable}. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_21:sealed-seeds} \begin{itemize} \tightlist @@ -168,11 +168,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{IGLA-RACE} (branch, alive) --- \url{https://github.com/gHashTag/trios/issues/143} --- linked to Ch.21 and Ch.11 --- \(\varphi\)-weight: \(1.0\) --- notes: multi-agent BPB \(< 1.85\) race. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_21:discussion} IGLA RACE provides the first formally verified multi-agent training protocol in the Trinity S³AI system. Its primary contribution is the demonstration that formal Coq refutation theorems can be operationalised as live guard rails in a running training fleet, not merely as post-hoc proof artefacts. A limitation is that the current fleet size of 7 Workers matches the cardinality of the sanctioned seed pool; a larger pool would allow more diverse exploration but would require extending the canonicity criteria of App.A. The warmup exclusion (R2, R6) could be relaxed if a formal treatment of restart dynamics is developed for INV-1 (Ch.15 {[}5{]}). Future work will extend IGLA RACE to Gate-3 (BPB \(\leq 1.5\)) using the M5--M6 model scales and the MXFP4 comparison data from Ch.9 {[}9{]}. The Rainbow Bridge invariant (INV-7b) will be extended to cover network partitions in the Railway PostgreSQL phd-postgres-ssot polling layer. -\section{References}\label{references} +\section{References}\label{ch_21:references} {[}1{]} \emph{Golden Sunflowers} dissertation, App.A --- Canonical Seed Pool Registry. diff --git a/docs/phd/chapters/ch_22.tex b/docs/phd/chapters/ch_22.tex index a0ed712a9b..f9d1aefcd9 100644 --- a/docs/phd/chapters/ch_22.tex +++ b/docs/phd/chapters/ch_22.tex @@ -55,11 +55,11 @@ \section*{Containers that know when to say no} PostgreSQL-backed configuration store; and Section~5 connects the orchestration layer to the FPGA-side counterparts described in Ch.28 and Ch.31. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_22:abstract} Deploying a formally verified ternary neural system at scale requires an orchestration layer that can co-ordinate model-serving workers, manage configuration invariants at runtime, and expose falsifiable witnesses for operational properties. This chapter describes the Railway/Trios orchestration architecture, in which worker pools are governed by the composite invariant \texttt{INV-8} (\texttt{WorkerPoolComposite.v}, 10 Qed). Six Coq theorems establish falsification witnesses --- demonstrating that unsafe configurations are provably rejected --- and one satisfaction witness --- demonstrating that the canonical \(\phi\)-scaled configuration is provably accepted. The anchor identity \(\phi^2 + \phi^{-2} = 3\) constrains worker-pool sizing: the ratio of inference workers to embedding workers is targeted at \(\phi^2 : \phi^{-2} = \phi^4 : 1 \approx 6.854 : 1\). The chapter also introduces the \texttt{victory\_not\_yet} predicate, which certifies that the system has not yet reached the operational milestone requiring full Gate-3 compliance. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_22:introduction} The Trios codebase organises model training, evaluation, and deployment through a Railway-style service mesh in which each service is a typed actor with formally specified invariants. The formal specification approach --- articulated in the directive for this chapter (\texttt{trios\#408}) --- extends the Coq-certified properties of the kernel and igla layers (Ch.3--Ch.10) up to the orchestration level, ensuring that runtime configuration errors are caught at the proof layer rather than at production incident time {[}1,2{]}. @@ -67,7 +67,7 @@ \section{1. Introduction}\label{introduction} The orchestration layer is implemented in the Railway platform (a managed container orchestration service) with Trios-specific plugins that expose Coq-certified configuration predicates as HTTP health endpoints. The present chapter focuses on the formal specification and its falsification properties; the FPGA-side counterpart is described in Ch.28 and Ch.31. -\section{2. Worker Pool Invariants and Falsification Witnesses}\label{worker-pool-invariants-and-falsification-witnesses} +\section{2. Worker Pool Invariants and Falsification Witnesses}\label{ch_22:worker-pool-invariants-and-falsification-witnesses} \textbf{Definition 2.1 (Worker pool configuration).} A configuration is a triple \((r_\text{inf}, n_w, r_\text{thr})\) where \(r_\text{inf} \in \mathbb{Q}_{>0}\) is the inference rate (tokens/second per worker), \(n_w \in \mathbb{N}\) is the worker count, and \(r_\text{thr} \in \mathbb{Q}_{>0}\) is the throughput threshold. In Coq, rational numbers are represented as \texttt{Q} pairs. @@ -100,7 +100,7 @@ \section{2. Worker Pool Invariants and Falsification Witnesses}\label{worker-poo Proof: \texttt{inv2\_holds\ (265\ \#\ 100)\ =\ false}, so the conjunction is \texttt{false} regardless of the other components. \(\square\) -\section{3. Satisfaction Witness and Victory Predicate}\label{satisfaction-witness-and-victory-predicate} +\section{3. Satisfaction Witness and Victory Predicate}\label{ch_22:satisfaction-witness-and-victory-predicate} The falsification witnesses of Section 2 demonstrate that the invariant system correctly rejects unsafe configurations. The satisfaction witness demonstrates that the canonical \(\phi\)-scaled configuration is accepted. @@ -123,7 +123,7 @@ \section{3. Satisfaction Witness and Victory Predicate}\label{satisfaction-witne The three-tier structure mirrors the ternary alphabet \(\{-1, 0, +1\}\) and the trinity identity \(\phi^2 + \phi^{-2} + 1 = 4\) (where the constant 1 represents the control tier and \(\phi^2 + \phi^{-2} = 3\) represents the compute tiers). -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_22:results-evidence} The INV-8 composite invariant has been validated across \(F_{20} = 6765\) Railway deployment events since integration into the Trios CI pipeline. Of these events, 0.7\% triggered falsification witnesses (primarily \texttt{inv3} violations due to autoscaler over-provisioning), and all were caught pre-deployment. Zero invariant violations reached production. @@ -156,7 +156,7 @@ \section{4. Results / Evidence}\label{results-evidence} Coq proof compilation for \texttt{INV8\_WorkerPoolComposite.v}: 2.1 seconds on Coq 8.18. All 10 theorems close with \texttt{Qed}; no \texttt{admit} statements. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_22:qed-assertions} \begin{itemize} \tightlist @@ -174,7 +174,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \texttt{victory\_not\_yet} (\filepath{gHashTag/t27/proofs/canonical/igla/INV8\_WorkerPoolComposite.v}) --- \emph{Status: Qed} --- \texttt{victory\_achieved\ 2\ =\ false}: two gates passed, Gate-3 pending. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_22:sealed-seeds} \begin{itemize} \tightlist @@ -184,11 +184,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci/Lucas reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_22:discussion} The primary limitation of the INV-8 composite invariant is that it checks configuration values at deployment time but not continuously at runtime. Dynamic autoscaling can change \(n_w\) after deployment, and the current implementation polls the invariant only at \(F_{17} = 1597\)-second intervals. Bridging this gap requires a runtime monitor that re-evaluates \texttt{composite\_invariant\_holds} on every scaling event and rolls back if the result is \texttt{false}. A prototype of this monitor is under development in the \texttt{trios\#408} issue thread. A second limitation is that \texttt{victory\_achieved} uses a discrete threshold of 3 gates, whereas the actual BPB trajectory is continuous; a richer predicate that tracks fractional gate progress (e.g., the ratio BPB/1.85 for Gate-2) would provide earlier warning of impending gate failures. Future work will integrate the orchestration invariants with the hardware performance counters of the QMTech FPGA (Ch.28, Ch.31, Ch.34) to create a closed-loop formally-verified deployment pipeline. -\section{References}\label{references} +\section{References}\label{ch_22:references} {[}1{]} GOLDEN SUNFLOWERS dissertation, Ch.3 --- Ternary Arithmetic Foundations. This volume. diff --git a/docs/phd/chapters/ch_23.tex b/docs/phd/chapters/ch_23.tex index 4c95bafb4c..004252f583 100644 --- a/docs/phd/chapters/ch_23.tex +++ b/docs/phd/chapters/ch_23.tex @@ -52,11 +52,11 @@ \section*{The gap between tokens and tools} and Section~5 discusses the MCP compliance properties and their relation to the FPGA-side token-stream architecture described in Ch.28. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_23:abstract} The Model Context Protocol (MCP) provides a standardised interface for connecting language model inference engines to external tool ecosystems. This chapter describes the integration of the Trinity S³AI inference runtime with MCP, enabling the golden-ratio-structured HSLM engine to consume and expose MCP tool calls without violating the \(\varphi^2 + \varphi^{-2} = 3\) normalisation invariant. The integration is non-trivial because MCP tool-call payloads introduce variable-length context that must be re-tokenised at sequence boundaries aligned to Fibonacci-Lucas indices. The chapter formalises the MCP adapter layer, defines the seed-preservation invariant across tool-call boundaries, and reports latency measurements on the QMTech XC7A100T FPGA implementation. End-to-end throughput degrades by less than 8\% relative to the baseline 63 tokens/sec rate when MCP overhead is included. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_23:introduction} Large-scale deployment of neural inference engines increasingly relies on agentic architectures in which the model interleaves generation with external tool calls --- web search, code execution, database queries, file I/O. The Model Context Protocol (MCP), introduced as an open standard in 2024, provides a JSON-RPC-based specification for this interleaving {[}1{]}. For conventional floating-point models, MCP integration is straightforward: the tool-call response is appended to the context window and inference resumes. @@ -64,7 +64,7 @@ \section{1. Introduction}\label{introduction} This alignment problem is the central engineering challenge of MCP integration. The solution adopted here --- boundary snapping with zero-padding to the nearest canonical index --- preserves the \(\varphi^2 + \varphi^{-2} = 3\) normalisation invariant and introduces worst-case overhead of \(\lceil F_{n+1} - N - L \rceil\) padding tokens, where \(F_{n+1}\) is the smallest Fibonacci number exceeding \(N + L\). -\section{2. MCP Adapter Layer Architecture}\label{mcp-adapter-layer-architecture} +\section{2. MCP Adapter Layer Architecture}\label{ch_23:mcp-adapter-layer-architecture} \textbf{Definition 2.1 (MCP context boundary).} A \emph{canonical boundary} is a token position \(p\) such that \(p \in \{F_{17}, F_{18}, F_{19}, F_{20}, F_{21}, L_7, L_8\} = \{1597, 2584, 4181, 6765, 10946, 29, 47\}\), or any sum of at most two such values. @@ -84,7 +84,7 @@ \section{2. MCP Adapter Layer Architecture}\label{mcp-adapter-layer-architecture \emph{Proof Sketch.} The zero-padding tokens are assigned fixed embeddings derived from \(s_1\) via the \(\varphi\)-distance mapping \(s_1 \mapsto \lfloor s_1 \cdot \varphi^k \rfloor \bmod |\text{vocab}|\) for padding position \(k\). Since \(\varphi\) is irrational, the padding embeddings are dense in the vocabulary but do not introduce new seed dependence. The model's weight tensor is unchanged; only the context changes, and the GLN normalisation at each layer re-centres the distribution to the \(1/\sqrt{3}\) scale regardless of padding content {[}4{]}. -\section{3. Protocol Implementation and Latency Analysis}\label{protocol-implementation-and-latency-analysis} +\section{3. Protocol Implementation and Latency Analysis}\label{ch_23:protocol-implementation-and-latency-analysis} The MCP adapter is implemented as a thin Rust layer sitting between the FPGA token stream and the JSON-RPC endpoint. The implementation follows the MCP specification version 1.0 {[}1{]} and exposes the following capabilities: @@ -112,7 +112,7 @@ \section{3. Protocol Implementation and Latency Analysis}\label{protocol-impleme \emph{Proof Sketch.} Boundary snapping ensures that the continuation begins at a canonical index, so the seed-diversity and step-sufficiency conditions of INV-7 are met by construction {[}7{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_23:results-evidence} Performance measurements on QMTech XC7A100T FPGA (0 DSP slices, 92 MHz clock, 1 W): @@ -133,19 +133,19 @@ \section{4. Results / Evidence}\label{results-evidence} The 8.1\% throughput degradation falls within the acceptance criterion for MCP-enabled deployment. The HSLM benchmark score is unchanged because the benchmark does not include tool-call boundaries; the 1003 token score reported in Ch.28 remains valid {[}8{]}. The \(\varphi^2 + \varphi^{-2} = 3\) normalisation constant is preserved in all 128 ablation variants that include MCP integration (cf.~Ch.17). -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_23:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_23:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_23:discussion} The MCP integration chapter demonstrates that the \(\varphi\)-structured inference architecture can interoperate with standard agentic infrastructure without sacrificing the formal invariants established in earlier chapters. The worst-case 61.8\% padding overhead is a genuine limitation: for long tool responses, the boundary snapping wastes significant context window budget. Future work should explore fractional Fibonacci boundaries --- positions of the form \(F_n + F_{n-2}\) --- which would reduce the maximum gap. A second direction is dynamic seed refresh: rather than preserving the original seed set \(\mathcal{S}\) through padding, a tool-call response could supply a new canonical seed drawn from the pool, resetting the INV-7 clock. This chapter connects to Ch.11 (INV-7 invariant), Ch.17 (GLN normalisation), Ch.27 (TRI-27 verifiable VM) and App.F (FPGA bitstream distribution). -\section{References}\label{references} +\section{References}\label{ch_23:references} {[}1{]} Anthropic. (2024). Model Context Protocol Specification v1.0. \url{https://modelcontextprotocol.io/specification}. diff --git a/docs/phd/chapters/ch_24.tex b/docs/phd/chapters/ch_24.tex index 425182e414..779bc0841d 100644 --- a/docs/phd/chapters/ch_24.tex +++ b/docs/phd/chapters/ch_24.tex @@ -38,11 +38,11 @@ \section*{Two clocks, no resonance} safety properties --- the part that says no agent corrupts another agent's accumulator --- are \texttt{Qed}-proved. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_24:abstract} The Period-Locked Runtime Monitor (PLRM) is a scheduling and watchdog component of the IGLA RACE multi-agent system that enforces timing invariants derived from the Golden Sunflowers substrate. The monitor uses two Lucas sentinels---\(L_7 = 29\) and \(L_8 = 47\)---as period bounds for the two principal agent classes (arithmetic and orchestration agents), ensuring that no agent can monopolise the GF16 arithmetic pipeline for more than 29 or 47 clock cycles respectively. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) motivates the period ratio \(47/29 \approx 1.621 \approx \varphi\), which guarantees that the two agent classes interleave without resonance. The formal treatment of PLRM liveness currently carries 9 Admitted stubs pending Iris integration (Ch.18); all safety properties are Qed-proved. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_24:introduction} A multi-agent inference runtime operating on shared hardware must guarantee two properties simultaneously: \emph{safety} (no agent corrupts another agent's arithmetic state) and \emph{liveness} (the hardware pipeline is never permanently starved). The IGLA RACE architecture (Inference Graph Lattice Architecture --- Robust Agent Computation Engine) achieves safety via memory isolation and formal invariants; liveness is the harder problem, because it requires reasoning about infinite execution traces {[}1{]}. @@ -52,9 +52,9 @@ \section{1. Introduction}\label{introduction} The connection to the anchor identity \(\varphi^2 + \varphi^{-2} = 3\) is the following: the three-term partition of the exponent field in GF16 (Ch.6) induces three agent priorities---sub-unity, unity, and super-unity---and the period monitor enforces that agents serving the unity band (the most frequent case) hold the pipeline for at most \(\lfloor L_7 \cdot \varphi \rfloor = \lfloor 29 \cdot 1.618 \rfloor = 46\) cycles, which rounds to \(L_8 - 1 = 46\). The arithmetic and orchestration period bounds thus emerge naturally from the GoldenFloat format structure. -\section{2. Formal Model of the Period-Locked Monitor}\label{formal-model-of-the-period-locked-monitor} +\section{2. Formal Model of the Period-Locked Monitor}\label{ch_24:formal-model-of-the-period-locked-monitor} -\subsection{2.1 Agent Model}\label{agent-model} +\subsection{2.1 Agent Model}\label{ch_24:agent-model} Let \(\mathcal{A} = \{a_1, \ldots, a_k\}\) be the set of IGLA RACE agents. Each agent \(a_i\) is characterised by: - A \emph{period bound} \(\tau_i \in \{L_7, L_8\} = \{29, 47\}\): arithmetic agents use \(L_7 = 29\), orchestration agents use \(L_8 = 47\). @@ -65,7 +65,7 @@ \subsection{2.1 Agent Model}\label{agent-model} \textbf{Definition 2.2 (PLRM safety).} The monitor is \emph{safe} if no two agents are simultaneously ACTIVE. -\subsection{2.2 Coq Encoding}\label{coq-encoding} +\subsection{2.2 Coq Encoding}\label{ch_24:coq-encoding} The PLRM is formalised in \filepath{t27/proofs/canonical/} as a state-transition system over a discrete time domain \(\mathbb{N}\). The safety property is encoded as: @@ -81,7 +81,7 @@ \subsection{2.2 Coq Encoding}\label{coq-encoding} This theorem carries Qed status (SCH-1 in the canonical inventory). The liveness properties (fairness lemmas SCH-3 through SCH-5) are currently Admitted; they require reasoning about infinite traces that is most naturally expressed in a temporal logic. The Iris framework {[}3{]} has been identified as the mechanisation target. -\subsection{2.3 Period Ratio and Non-Resonance}\label{period-ratio-and-non-resonance} +\subsection{2.3 Period Ratio and Non-Resonance}\label{ch_24:period-ratio-and-non-resonance} \textbf{Proposition 2.3} (Non-resonance). \emph{The period clocks \(L_7 = 29\) and \(L_8 = 47\) are coprime.} @@ -91,7 +91,7 @@ \subsection{2.3 Period Ratio and Non-Resonance}\label{period-ratio-and-non-reson The corollary follows from \(1597 = F_{17} > 1363 = L_7 \times L_8\), but the key point is that the first common cycle (1363) occurs within the window, so a brief simultaneous timeout is possible but is handled by the priority-queue tie-breaking rule (Section 2.4) rather than constituting a blackout. -\subsection{2.4 Priority Queue and Phi-Weighted Scheduling}\label{priority-queue-and-phi-weighted-scheduling} +\subsection{2.4 Priority Queue and Phi-Weighted Scheduling}\label{ch_24:priority-queue-and-phi-weighted-scheduling} When the PLRM preempts an agent, the scheduler selects the next ACTIVE candidate from a binary max-heap ordered by \(\varphi\)-weight. The weight of agent \(a_i\) at time \(t\) is updated as: @@ -99,9 +99,9 @@ \subsection{2.4 Priority Queue and Phi-Weighted Scheduling}\label{priority-queue where \(\varphi^{-1} \approx 0.618\) is the decay factor and \(\varphi \approx 1.618\) is the boost upon job arrival. This update rule has the fixed point \(w^* = \varphi / (1 - \varphi^{-1}) = \varphi / (2 - \varphi) = \varphi / (1 - \hat\varphi)\); by the identity \(\varphi^2 + \varphi^{-2} = 3\), the steady-state weight satisfies \(w^* \in [\varphi^{-2}, \varphi^2] = [0.382, 2.618]\), remaining bounded without saturation. -\section{3. Implementation and Hardware Interface}\label{implementation-and-hardware-interface} +\section{3. Implementation and Hardware Interface}\label{ch_24:implementation-and-hardware-interface} -\subsection{3.1 RTL Implementation}\label{rtl-implementation} +\subsection{3.1 RTL Implementation}\label{ch_24:rtl-implementation} The PLRM is implemented as a two-counter module in FPGA RTL: - \textbf{Counter A} (\texttt{cnt\_arith}): 6-bit counter, wraps at \(L_7 - 1 = 28\). Asserts \texttt{PREEMPT\_ARITH} on wrap. @@ -109,7 +109,7 @@ \subsection{3.1 RTL Implementation}\label{rtl-implementation} Both counters are clocked at 92 MHz (the FPGA fabric clock). The PLRM occupies 47 LUTs and 62 FFs in the XC7A100T implementation---a numerological coincidence that the \(L_8 = 47\) LUT count shares with the orchestration period bound {[}4{]}. -\subsection{3.2 Interrupt Interface with the Hardware Bridge}\label{interrupt-interface-with-the-hardware-bridge} +\subsection{3.2 Interrupt Interface with the Hardware Bridge}\label{ch_24:interrupt-interface-with-the-hardware-bridge} The PLRM exposes a 3-bit interrupt line to the Hardware Bridge (Ch.12): \texttt{\{PREEMPT\_ARITH,\ PREEMPT\_ORCH,\ PLRM\_ERROR\}}. The host driver services these interrupts with a latency of at most 4 UART-V6 frame periods (approximately 1.7 ms at 115200 baud), which is shorter than the \(L_8 \times (1/92\,\text{MHz}) = 47 \times 10.87\,\text{ns} = 511\,\text{ns}\) period-lock window. Therefore the host can always acknowledge a preemption before the next period boundary. @@ -117,7 +117,7 @@ \subsection{3.2 Interrupt Interface with the Hardware Bridge}\label{interrupt-in \emph{Proof.} By direct comparison: \(1.7 < 2.52\). The frame period \(T_{\text{frame}} = (10 \times 47 + 3) / 115200\,\text{s} \approx 0.087\,\text{ms}\) (10 bits per UART byte, 47 payload bytes, 3 overhead bytes). Qed. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_24:results-evidence} The PLRM was evaluated on the IGLA RACE simulation bench running the 1003-token HSLM sequence: @@ -148,17 +148,17 @@ \section{4. Results / Evidence}\label{results-evidence} Seed pool: the Fibonacci thresholds \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\) bound the cycle-count windows used in the simulation; \(L_7=29\) and \(L_8=47\) are the period bounds verified above. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_24:qed-assertions} No Coq theorems are anchored specifically to this chapter in the input JSON; obligations are tracked in the Golden Ledger. (The scheduling safety theorem \texttt{plrm\_mutual\_exclusion} (SCH-1) and its supporting lemmas SCH-2 through SCH-5 reside in \filepath{t27/proofs/canonical/}; SCH-3 through SCH-5 carry Admitted status pending Iris integration as detailed in Ch.18.) -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_24:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_24:discussion} The Period-Locked Runtime Monitor is a compact but structurally essential component: without it, the formal safety proofs for the GF16 pipeline would not compose with the runtime scheduler, because floating-point arithmetic safety assumes exclusive access to the MAC unit during each operation. The PLRM converts that assumption into a provable invariant. @@ -166,7 +166,7 @@ \section{7. Discussion}\label{discussion} Future work includes extending the period bounds to three tiers---using \(L_7 = 29\), \(L_8 = 47\), and \(L_9 = 76 = L_7 + L_8\)---to accommodate a third agent class (hardware configuration agents) planned for the GF32 pipeline. The chapter connects directly to Ch.12 (Hardware Bridge interrupt interface), Ch.6 (GoldenFloat exponent bands that motivate the three-priority scheme), and Ch.30 (Trinity SAI VSA+AR integration that adds vector-symbolic agents to the IGLA RACE pool). -\section{References}\label{references} +\section{References}\label{ch_24:references} {[}1{]} \filepath{gHashTag/trios\#418} --- Ch.24 Period-Locked Runtime Monitor scope issue. diff --git a/docs/phd/chapters/ch_25.tex b/docs/phd/chapters/ch_25.tex index f1c16640f9..87c879257c 100644 --- a/docs/phd/chapters/ch_25.tex +++ b/docs/phd/chapters/ch_25.tex @@ -55,11 +55,11 @@ \section*{When a limit cycle is the answer, not the problem} statistical loss periodicity; and Section~5 discusses the implications for long-run training stability and gate compliance. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_25:abstract} This chapter develops the theory of \(\varphi\)-period cycles --- periodic orbits in the weight and attention manifolds of the TRINITY S³AI model that arise because the quantisation lattice is invariant under multiplication by \(\varphi^2\). The central result is that every trajectory of the gradient-descent dynamics on the \(\varphi\)-quantised weight space is eventually periodic with period dividing \(F_k\) for some \(k\), and that the attractor set is precisely the subset of weights satisfying \(\varphi^2 + \varphi^{-2} = 3\) up to lattice precision. The chapter defines the notion of a \(\varphi\)-cycle formally, classifies cycles of order \(\leq F_{10} = 55\), and connects the cycle structure to the Vogel divergence angle (Ch.7) and the statistical periodicity of the training loss (Ch.19). -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_25:introduction} Periodic behaviour in gradient-descent optimisation is usually treated as a pathology: limit cycles indicate that the learning rate is too large or the loss landscape has degenerate saddle points. In the TRINITY S³AI framework, by contrast, a restricted class of periodic orbits is not merely tolerated but engineered. The \(\varphi\)-quantised weight lattice \(\Lambda_\varphi\) satisfies @@ -69,7 +69,7 @@ \section{1. Introduction}\label{introduction} The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) plays a dual role here. It is the algebraic certificate that \(\Lambda_\varphi\) is closed under the two operations \(\times\varphi^2\) and \(\times\varphi^{-2}\) (since \(\varphi^2 + \varphi^{-2}\) is an integer), and it sets the diameter of the fundamental domain of the quotient torus to exactly 3 lattice units {[}1{]}. This compactness ensures that every orbit visits at most \(3^d\) distinct quantised configurations in \(d\) dimensions before repeating, bounding the cycle length. -\section{\texorpdfstring{2. \(\varphi\)-Lattice Structure and the Cycle Map}{2. \textbackslash varphi-Lattice Structure and the Cycle Map}}\label{varphi-lattice-structure-and-the-cycle-map} +\section{\texorpdfstring{2. \(\varphi\)-Lattice Structure and the Cycle Map}{2. \textbackslash varphi-Lattice Structure and the Cycle Map}}\label{ch_25:varphi-lattice-structure-and-the-cycle-map} \textbf{Definition 2.1 (\(\varphi\)-quantised lattice).} The one-dimensional \(\varphi\)-quantised lattice is: \[\Lambda_\varphi^{(1)} = \{ a + b\varphi : a, b \in \mathbb{Z} \} \cap [-\varphi^{-1}, \varphi^{-1}],\] @@ -91,7 +91,7 @@ \section{\texorpdfstring{2. \(\varphi\)-Lattice Structure and the Cycle Map}{2. \textbf{Corollary 2.6.} The sanctioned seeds \(F_{17}=1597, \ldots, F_{21}=10946\) index cycles whose orders are bounded above by \(F_{21}=10946\), covering all practically relevant orbit lengths. -\section{3. Cycle Classification and Attention Periodicity}\label{cycle-classification-and-attention-periodicity} +\section{3. Cycle Classification and Attention Periodicity}\label{ch_25:cycle-classification-and-attention-periodicity} The cycle structure of \(\Phi\) on \(\Lambda_\varphi^{(1)}\) for small lattice sizes is tabulated below. Lattice size \(|\Lambda| = 3\) corresponds to the ternary alphabet \(\{-1, 0, 1\}\). @@ -120,7 +120,7 @@ \section{3. Cycle Classification and Attention Periodicity}\label{cycle-classifi \emph{Proof.} \(\Phi^{F_k}(K) = K\) by the cycle condition, and \(\text{PE}(i+F_k) = \text{PE}(i)\) by the periodicity of the encoding. \(\square\) -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_25:results-evidence} \textbf{Evidence 1 --- Loss periodicity.} Training loss curves for all three primary replicates (Ch.19) exhibit local minima at gradient steps \(F_k\) for \(k = 10, 11, 12, 13\) (steps 55, 89, 144, 233). The mean dip depth at these steps is \(\Delta\mathcal{L} = 0.0031 \pm 0.0004\) (mean \(\pm\) SE, \(n=3\)), consistent with the model periodically revisiting weight configurations close to \(\varphi\)-cycle attractors. @@ -128,21 +128,21 @@ \section{4. Results / Evidence}\label{results-evidence} \textbf{Evidence 3 --- Attention periodicity.} Attention entropy \(H(A_i) = -\sum_j A_{ij} \log A_{ij}\) was measured on the held-out partition for all 12 attention heads. Heads 5 and 11 (zero-indexed) exhibited significant periodicity at period \(F_{10}=55\) and \(F_{11}=89\) respectively, as confirmed by a discrete Fourier transform with peak-to-noise ratio \(> 3\). The \(\varphi^2 + \varphi^{-2} = 3\) identity constrains the spectral weight of these peaks: the sum of squared Fourier coefficients at \(F_k\) and \(F_{k-2}\) equals exactly 3 times the mean spectral power (evidence axis 3, \(n=3\), Welch \(t\), \(p = 0.008\)). -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_25:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_25:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). Note: \(L_7 = 29\) and \(L_8 = 47\) are motivated by the cycle census of §4, Evidence 2. The cycle counts at \(|\Lambda| = F_{17}\) are \(L_7\) and \(L_8\) for orders 29 and 47 respectively. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_25:discussion} The \(\varphi\)-cycle theory developed here is a novel contribution: to the authors' knowledge, no prior work has exploited the \(\varphi^2\)-invariance of the Fibonacci lattice to engineer beneficial periodicity in attention matrices. The primary limitation is that the periodicity results are proved for the one-dimensional lattice and extended to \(d\) dimensions coordinatewise; interactions between dimensions (cross-cycle interference) are not yet analysed. A second limitation is that the Pisano period theorem (Theorem 2.5) guarantees that cycle orders divide \(F_k\), but does not specify which \(k\); in practice, the relevant \(k\) is determined empirically from the loss-dip census (Evidence 1). Future work includes: (a) formalising Proposition 3.1 as a Coq theorem (filed as CYC-1 in the Golden Ledger), (b) extending the cycle census to \(|\Lambda| = F_{18} = 2584\) and \(F_{19} = 4181\), and (c) investigating whether the Vogel divergence angle \(360°/\varphi^2\) (Ch.7) can be interpreted as the angular step of the one-dimensional cycle map on the unit circle. Connections to Ch.7 (lattice geometry), Ch.13 (seed admissibility), and Ch.19 (loss periodicity) are tight. -\section{References}\label{references} +\section{References}\label{ch_25:references} {[}1{]} This dissertation, Ch.7 --- Vogel Phyllotaxis \(137.5° = 360°/\varphi^2\). \(\varphi^2\)-invariance of the Fibonacci lattice. diff --git a/docs/phd/chapters/ch_26.tex b/docs/phd/chapters/ch_26.tex index 43f061136f..a86e04a648 100644 --- a/docs/phd/chapters/ch_26.tex +++ b/docs/phd/chapters/ch_26.tex @@ -25,11 +25,11 @@ \section*{Seven words, no multipliers} Why does this matter beyond the FPGA lab? Because the argument is constructive: it shows that a formally verified, minimalist ISA can sustain 63 tokens per second at 92 MHz and 1 W---figures that would have seemed implausible for a verified system a decade ago. Hamming urged engineers to seek insight over raw numerical output; KOSCHEI answers that call by embedding the insight---\(\varphi^2 + \varphi^{-2} = 3\), ternary closure, LUT sufficiency---directly into the instruction set, so that the hardware cannot be operated incorrectly without violating a proof. The rest of this chapter specifies each opcode formally, traces the Coq certification path, and reports resource utilisation and throughput measurements on the QMTech XC7A100T board. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_26:abstract} The KOSCHEI coprocessor extends the QMTech XC7A100T FPGA with a φ-numeric instruction set that maps the mathematical structure of Trinity S³AI directly onto LUT fabric with zero DSP primitives. Seven opcodes are defined: \texttt{TF3\_ADD}, \texttt{TF3\_MUL}, \texttt{VSA\_BIND}, \texttt{VSA\_UNBIND}, \texttt{VSA\_BUNDLE}, \texttt{GF16\_QUANT}, and \texttt{PHI\_ROPE}. Every opcode preserves the \(\varphi^2 + \varphi^{-2} = 3\) normalisation invariant, certified by Coq modules \texttt{Trinity.Canonical.Kernel.Phi} (16 Qed), \texttt{Trinity.Canonical.Kernel.PhiFloat} (6 Qed), \texttt{Trinity.Canonical.Kernel.Trit}, \texttt{Trinity.Canonical.Kernel.Semantics}, and \texttt{Trinity.Canonical.Kernel.FlowerE8Embedding}. The ISA achieves 63 tokens/sec at 92 MHz and 1 W. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_26:introduction} A coprocessor ISA for φ-numeric computation must satisfy three simultaneous constraints that are not met by any existing FPGA softcore: @@ -45,9 +45,9 @@ \section{1. Introduction}\label{introduction} KOSCHEI (an acronym: \textbf{K}ernel \textbf{O}pcode \textbf{S}et for \textbf{C}anonical \textbf{H}yperdimensional and \textbf{E}mbedded \textbf{I}nference) satisfies all three. The name also references the Slavic mythological figure whose life is concealed in a nested structure --- an apt metaphor for the layered φ-lattice encoding at the heart of the ISA. -\section{2. ISA Register File and Encoding}\label{isa-register-file-and-encoding} +\section{2. ISA Register File and Encoding}\label{ch_26:isa-register-file-and-encoding} -\subsection{2.1 Register File}\label{register-file} +\subsection{2.1 Register File}\label{ch_26:register-file} KOSCHEI has 16 general-purpose registers \(r_0\)--\(r_{15}\), each 64 bits wide. The encoding is: @@ -69,7 +69,7 @@ \subsection{2.1 Register File}\label{register-file} and all arithmetic operations adjust \texttt{φ\_exp} accordingly without touching the payload bits --- analogous to the exponent field of a floating-point number but restricted to integer powers of \(\varphi\). -\subsection{2.2 Instruction Encoding}\label{instruction-encoding} +\subsection{2.2 Instruction Encoding}\label{ch_26:instruction-encoding} Instructions are 32 bits: 7-bit opcode, 4-bit destination, 4-bit source A, 4-bit source B, 13-bit immediate. @@ -80,9 +80,9 @@ \subsection{2.2 Instruction Encoding}\label{instruction-encoding} The 7-bit opcode space allows 128 instructions; the seven φ-numeric opcodes occupy codes 0x01--0x07. -\section{3. Opcode Specifications}\label{opcode-specifications} +\section{3. Opcode Specifications}\label{ch_26:opcode-specifications} -\subsection{3.1 TF3\_ADD --- Ternary Addition}\label{tf3_add-ternary-addition} +\subsection{3.1 TF3\_ADD --- Ternary Addition}\label{ch_26:tf3_add-ternary-addition} \begin{verbatim} TF3_ADD RD, RA, RB @@ -92,7 +92,7 @@ \subsection{3.1 TF3\_ADD --- Ternary Addition}\label{tf3_add-ternary-addition} The correctness of the \texttt{φ\_exp} update is certified by \textbf{Lemma phi\_add\_exp} in \texttt{Trinity.Canonical.Kernel.Phi} (status: Qed). The full kernel module contains 16 Qed lemmas covering all arithmetic boundary cases {[}1{]}. -\subsection{3.2 TF3\_MUL --- Ternary Multiplication}\label{tf3_mul-ternary-multiplication} +\subsection{3.2 TF3\_MUL --- Ternary Multiplication}\label{ch_26:tf3_mul-ternary-multiplication} \begin{verbatim} TF3_MUL RD, RA, RB @@ -102,7 +102,7 @@ \subsection{3.2 TF3\_MUL --- Ternary Multiplication}\label{tf3_mul-ternary-multi The 0-DSP constraint is satisfied because the trit product reduces to a bitwise XNOR (for sign) ANDed with a non-zero indicator bit, implementable in two LUT-4 primitives per bit {[}2{]}. -\subsection{3.3 VSA\_BIND --- Hyperdimensional Binding}\label{vsa_bind-hyperdimensional-binding} +\subsection{3.3 VSA\_BIND --- Hyperdimensional Binding}\label{ch_26:vsa_bind-hyperdimensional-binding} \begin{verbatim} VSA_BIND RD, RA, RB @@ -110,7 +110,7 @@ \subsection{3.3 VSA\_BIND --- Hyperdimensional Binding}\label{vsa_bind-hyperdime Computes the element-wise product \(r_D \leftarrow r_A \odot r_B\) over the 64-dimensional trit vector. Binding is invertible: \(r_A \odot r_B \odot r_B = r_A\) for any \(r_B\) with no zero entries (full-rank). The invertibility proof uses the \texttt{FlowerE8Embedding} module, which maps the 64-trit space onto the \(E_8\) root lattice and establishes that the binding map is an automorphism {[}3{]}. -\subsection{3.4 VSA\_UNBIND --- Hyperdimensional Unbinding}\label{vsa_unbind-hyperdimensional-unbinding} +\subsection{3.4 VSA\_UNBIND --- Hyperdimensional Unbinding}\label{ch_26:vsa_unbind-hyperdimensional-unbinding} \begin{verbatim} VSA_UNBIND RD, RA, RB @@ -118,7 +118,7 @@ \subsection{3.4 VSA\_UNBIND --- Hyperdimensional Unbinding}\label{vsa_unbind-hyp Computes \(r_D \leftarrow r_A \odot r_B\) (unbinding is self-inverse in ternary VSA). The implementation is identical to \texttt{VSA\_BIND}; the opcode distinction is semantic, enabling the proof checker to apply the unbind-specific Coq lemmas in \texttt{Trinity.Canonical.Kernel.Semantics} {[}4{]}. -\subsection{3.5 VSA\_BUNDLE --- Hyperdimensional Bundling}\label{vsa_bundle-hyperdimensional-bundling} +\subsection{3.5 VSA\_BUNDLE --- Hyperdimensional Bundling}\label{ch_26:vsa_bundle-hyperdimensional-bundling} \begin{verbatim} VSA_BUNDLE RD, RA, RB @@ -126,7 +126,7 @@ \subsection{3.5 VSA\_BUNDLE --- Hyperdimensional Bundling}\label{vsa_bundle-hype Computes the majority-vote superposition \(r_D \leftarrow \text{sign}(r_A + r_B)\), clamped to \(\{-1, 0, +1\}\). For two operands this reduces to \(r_D = r_A\) if \(r_A = r_B\), and \(r_D = 0\) if \(r_A = -r_B\). The bundle of \(n\) vectors with \(n \geq 3\) is computed by iterating this instruction; the Coq proof of information-theoretic capacity scaling is in \texttt{Trinity.Canonical.Kernel.Semantics}, Theorem \filepath{bundle\_capacity\_phi\_bound} (status: Qed) {[}4{]}. -\subsection{3.6 GF16\_QUANT --- Galois Field 16 Quantisation}\label{gf16_quant-galois-field-16-quantisation} +\subsection{3.6 GF16\_QUANT --- Galois Field 16 Quantisation}\label{ch_26:gf16_quant-galois-field-16-quantisation} \begin{verbatim} GF16_QUANT RD, RA, IMM[3:0] @@ -136,7 +136,7 @@ \subsection{3.6 GF16\_QUANT --- Galois Field 16 Quantisation}\label{gf16_quant-g The 0-DSP implementation uses a 16-entry LUT ROM for the GF(16) multiplication table, consuming 16 LUT-6 primitives. -\subsection{3.7 PHI\_ROPE --- φ-Rotary Position Encoding}\label{phi_rope-ux3c6-rotary-position-encoding} +\subsection{3.7 PHI\_ROPE --- φ-Rotary Position Encoding}\label{ch_26:phi_rope-ux3c6-rotary-position-encoding} \begin{verbatim} PHI_ROPE RD, RA, IMM[12:0] @@ -150,7 +150,7 @@ \subsection{3.7 PHI\_ROPE --- φ-Rotary Position Encoding}\label{phi_rope-ux3c6- The rotation is implemented as a fixed-point complex multiply with φ-quantised cosine and sine tables, verified in \texttt{Trinity.Canonical.Kernel.PhiFloat} (6 Qed) {[}7{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_26:results-evidence} Synthesis on the QMTech XC7A100T (Vivado 2023.2, seed \(F_{17}=1597\)) yields: @@ -169,7 +169,7 @@ \section{4. Results / Evidence}\label{results-evidence} Clock period 10.87 ns (91.98 MHz ≈ 92 MHz); Worst Negative Slack +0.13 ns (timing closed). Power: 1.00 W at 1.0 V core. Throughput: 63 tokens/sec on the HSLM 1003-token sequence. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_26:qed-assertions} No Coq theorems are anchored directly to this chapter; the ISA semantics are certified by the following canonical modules: @@ -189,17 +189,17 @@ \section{5. Qed Assertions}\label{qed-assertions} All five modules reside in \filepath{gHashTag/t27/proofs/canonical/} and contribute to the 297 Qed census {[}8{]}. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_26:sealed-seeds} Inherits the canonical seed pool F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_26:discussion} The KOSCHEI ISA demonstrates that a φ-lattice arithmetic unit can be implemented entirely in LUT fabric without DSP resources. The 0-DSP constraint is not a limitation but a design choice that keeps every arithmetic path within the certified Coq semantics. The 66\% LUT utilisation leaves headroom for additional VSA operations planned for the KOSCHEI v2 revision, including a \texttt{VSA\_SHIFT} opcode for sequence-position permutation. A current limitation is that \texttt{PHI\_ROPE} supports only power-of-two context lengths via the 13-bit \texttt{IMM} field; non-power-of-two contexts require a pair of \texttt{PHI\_ROPE} instructions with adjusted denominators. Future work should extend the \texttt{PhiFloat} Coq module to certify the two-instruction decomposition. The \texttt{GF16\_QUANT} opcode is provisionally verified; the full Galois-field completeness proof is one of the 41 Admitted obligations in the current census and is prioritised for the Gate-3 submission. -\section{References}\label{references} +\section{References}\label{ch_26:references} {[}1{]} Trinity Canonical Coq Home. \texttt{Trinity.Canonical.Kernel.Phi} --- 16 Qed. \filepath{gHashTag/t27/proofs/canonical/}. GitHub. diff --git a/docs/phd/chapters/ch_27.tex b/docs/phd/chapters/ch_27.tex index c41160d958..1a0376c457 100644 --- a/docs/phd/chapters/ch_27.tex +++ b/docs/phd/chapters/ch_27.tex @@ -42,11 +42,11 @@ \section*{The smallest interesting alphabet} semantics, and the Zenodo artefact (B003) that archives the verifiable virtual machine. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_27:abstract} TRI27 is the domain-specific language (DSL) of the Trinity S³AI kernel, typed over a balanced-ternary digit alphabet \(\{-1, 0, +1\}\) --- cardinality \(3\), the integer appearing in the anchor identity \(\varphi^2 + \varphi^{-2} = 3\). This chapter specifies the TRI27 expression language, its denotational semantics over the type \texttt{trit}, and two mechanically verified Coq theorems: \texttt{eval\_det} (evaluation is deterministic) and \texttt{trit\_exhaustive} (every trit value is one of exactly three possibilities). The DSL is designed so that every evaluation path terminates, every result is unique, and the three-valued logic is exhaustive by construction. The Zenodo artifact B003 archives the verifiable VM implementation. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_27:introduction} The arithmetic core of Trinity S³AI processes weights and activations represented as balanced-ternary vectors. The natural programming substrate for such computations is a three-valued language in which the primitive type \texttt{trit} has exactly three inhabitants: \texttt{Neg} (\(-1\)), \texttt{Zero} (\(0\)), and `Pos` (\(+1\)). The cardinality of this type is \(3\) --- the same integer that appears at the right-hand side of the anchor identity \(\varphi^2 + \varphi^{-2} = 3\) {[}1{]}. This is not coincidence but design: the DSL was constructed so that its type theory and the algebraic substrate share the same integer constant, enabling formal proofs about DSL programs to reference the \(\varphi\)-arithmetic directly. @@ -54,9 +54,9 @@ \section{1. Introduction}\label{introduction} The chapter is organised as follows: Section 2 defines the TRI27 syntax and semantics. Section 3 proves the two Coq theorems. Section 4 presents evaluation results and artifact metadata. -\section{2. TRI27 Syntax and Denotational Semantics}\label{tri27-syntax-and-denotational-semantics} +\section{2. TRI27 Syntax and Denotational Semantics}\label{ch_27:tri27-syntax-and-denotational-semantics} -\subsection{2.1 Abstract Syntax}\label{abstract-syntax} +\subsection{2.1 Abstract Syntax}\label{ch_27:abstract-syntax} The TRI27 expression language is defined by the following inductive type in Coq: @@ -84,7 +84,7 @@ \subsection{2.1 Abstract Syntax}\label{abstract-syntax} This is the canonical three-valued type; its exhaustiveness is proved by \texttt{trit\_exhaustive}. -\subsection{2.2 Environments and Evaluation}\label{environments-and-evaluation} +\subsection{2.2 Environments and Evaluation}\label{ch_27:environments-and-evaluation} An environment \texttt{rho\ :\ env} is a total function \texttt{nat\ -\textgreater{}\ trit} assigning a trit value to each de Bruijn index. The evaluator is a partial function returning \texttt{option\ trit}: @@ -96,7 +96,7 @@ \subsection{2.2 Environments and Evaluation}\label{environments-and-evaluation} The partial type reflects the possibility of out-of-scope variable references, though in a well-formed program (all variable indices in scope) the evaluator always returns \texttt{Some\ v}. -\subsection{2.3 Ternary Arithmetic}\label{ternary-arithmetic} +\subsection{2.3 Ternary Arithmetic}\label{ch_27:ternary-arithmetic} The fundamental ternary operations are defined by the \(3 \times 3\) tables: @@ -132,13 +132,13 @@ \subsection{2.3 Ternary Arithmetic}\label{ternary-arithmetic} These tables implement \(\mathbb{F}_3\) arithmetic. The distributive law \(a \times_3 (b +_3 c) = (a \times_3 b) +_3 (a \times_3 c)\) holds by inspection and is proved as a derived lemma in \texttt{Trit.v} {[}3{]}. -\subsection{\texorpdfstring{2.4 Relation to GF16 and \(\varphi\)-Arithmetic}{2.4 Relation to GF16 and \textbackslash varphi-Arithmetic}}\label{relation-to-gf16-and-varphi-arithmetic} +\subsection{\texorpdfstring{2.4 Relation to GF16 and \(\varphi\)-Arithmetic}{2.4 Relation to GF16 and \textbackslash varphi-Arithmetic}}\label{ch_27:relation-to-gf16-and-varphi-arithmetic} The GF16 field elements (Ch.9 {[}2{]}) are pairs of trit-register values under the embedding \(\mathbb{F}_3 \times \mathbb{F}_3 \hookrightarrow \mathbb{F}_{3^2} \hookrightarrow \mathbb{F}_{16}\) (via the Chinese Remainder Theorem applied to the factored polynomial ring). This embedding is approximate; the exact relationship is documented in \filepath{t27/proofs/canonical/kernel/Semantics.v} {[}4{]} and the Zenodo artifact B003 {[}5{]}. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) ensures that the \(\varphi\)-scaled weight grid has grid spacing \(\varphi^{-2} = 2 - \varphi\) whose reciprocal \(\varphi^2\) is the scale factor, and that within the GF16 safe domain (INV-3) the rounding error to the nearest \texttt{trit} value is bounded. -\section{3. Mechanised Proofs: Determinism and Exhaustiveness}\label{mechanised-proofs-determinism-and-exhaustiveness} +\section{3. Mechanised Proofs: Determinism and Exhaustiveness}\label{ch_27:mechanised-proofs-determinism-and-exhaustiveness} -\subsection{\texorpdfstring{3.1 Theorem \texttt{eval\_det}: Determinism}{3.1 Theorem eval\_det: Determinism}}\label{theorem-eval_det-determinism} +\subsection{\texorpdfstring{3.1 Theorem \texttt{eval\_det}: Determinism}{3.1 Theorem eval\_det: Determinism}}\label{ch_27:theorem-eval_det-determinism} \textbf{Statement} (KER-4, \filepath{gHashTag/t27/proofs/canonical/kernel/Semantics.v} {[}4{]}): @@ -152,7 +152,7 @@ \subsection{\texorpdfstring{3.1 Theorem \texttt{eval\_det}: Determinism}{3.1 The The Coq proof uses \texttt{inversion} on the \texttt{option} equality hypotheses and \texttt{congruence} to close the leaf goals. Total proof length: 43 lines in \texttt{Semantics.v}. -\subsection{\texorpdfstring{3.2 Theorem \texttt{trit\_exhaustive}: Exhaustiveness}{3.2 Theorem trit\_exhaustive: Exhaustiveness}}\label{theorem-trit_exhaustive-exhaustiveness} +\subsection{\texorpdfstring{3.2 Theorem \texttt{trit\_exhaustive}: Exhaustiveness}{3.2 Theorem trit\_exhaustive: Exhaustiveness}}\label{ch_27:theorem-trit_exhaustive-exhaustiveness} \textbf{Statement} (KER-5, \filepath{gHashTag/t27/proofs/canonical/kernel/Trit.v} {[}3{]}): @@ -164,7 +164,7 @@ \subsection{\texorpdfstring{3.2 Theorem \texttt{trit\_exhaustive}: Exhaustivenes This theorem is trivial in isolation but serves as the anchor for all completeness arguments: any predicate on \texttt{trit} values need only be checked on \texttt{\{Neg,\ Zero,\ Pos\}}. In particular, the Gate-2 and Gate-3 BPB predicates, when instantiated at the trit level, require only three-case proofs. The theorem also reflects the algebraic fact that the cardinality of the type equals \(3\) --- the right-hand side of \(\varphi^2 + \varphi^{-2} = 3\) {[}1{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_27:results-evidence} \begin{itemize} \tightlist @@ -182,7 +182,7 @@ \section{4. Results / Evidence}\label{results-evidence} \textbf{Seed pool}: All three evaluation seeds used in TRI27 VM integration testing --- \(F_{17} = 1597\), \(F_{18} = 2584\), \(L_7 = 29\) --- are from the sanctioned pool; no forbidden values were used. \end{itemize} -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_27:qed-assertions} \begin{itemize} \tightlist @@ -192,7 +192,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \texttt{trit\_exhaustive} (\filepath{gHashTag/t27/proofs/canonical/kernel/Trit.v}) --- \emph{Status: Qed} --- every element of type \texttt{trit} is one of exactly three values: \texttt{Neg}, \texttt{Zero}, or \texttt{Pos}. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_27:sealed-seeds} \begin{itemize} \tightlist @@ -200,11 +200,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{B003} (doi, golden) --- \url{https://doi.org/10.5281/zenodo.19227869} --- linked to Ch.27 and App.H --- \(\varphi\)-weight: \(0.618033988768953\) --- notes: TRI-27 Verifiable VM artifact. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_27:discussion} The TRI27 DSL formalised here is intentionally minimal. The present two theorems establish only determinism and exhaustiveness; a complete verified compiler from TRI27 to FPGA RTL would require additional theorems on type safety, termination, and translation correctness --- all planned for v5 of the dissertation. The most significant limitation is that the current semantics does not handle variable out-of-scope errors gracefully: \texttt{eval} returns \texttt{None}, but there is no formal type-system proof that well-typed programs never produce \texttt{None}. A dependent type approach (à la Agda or Idris) would subsume this. The \texttt{If3} constructor as currently implemented is also a two-branch conditional rather than the intended three-branch form; extending it to \texttt{If3\ e\ e1\ e2\ e3} with a \texttt{trit}-dispatched branch selection is deferred to the next proof sprint. Chapter 28 (FPGA implementation) and App.H (VM specification) build directly on the TRI27 kernel defined here. -\section{References}\label{references} +\section{References}\label{ch_27:references} {[}1{]} \emph{Golden Sunflowers} dissertation, Ch.3 --- Trinity Identity (\(\varphi^2 + \varphi^{-2} = 3\)). diff --git a/docs/phd/chapters/ch_28.tex b/docs/phd/chapters/ch_28.tex index 5374e77481..8b6a42ac0f 100644 --- a/docs/phd/chapters/ch_28.tex +++ b/docs/phd/chapters/ch_28.tex @@ -25,11 +25,11 @@ \section*{A hundred thousand gates and one watt} Two bitstreams---B001 and B002, archived on Zenodo---constitute the primary evidence artefacts for everything claimed in this chapter. The rest of this chapter walks through the zero-DSP ternary datapath, the resource utilisation breakdown, the timing closure report, and the throughput measurements that substantiate the 1 W / 63 tokens-per-second headline. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_28:abstract} The QMTech XC7A100T development board hosts the primary hardware realisation of the Trinity S³AI ternary inference engine. Running at 92 MHz with a measured throughput of 63 tokens per second and a total board power draw of 1 W, the implementation consumes zero Xilinx DSP48 blocks, relying instead on LUT-based ternary accumulation derived from the zero-absorption laws proved in Ch.4. The anchor identity \(\phi^2 + \phi^{-2} = 3\) governs the LUT truth-table structure: because ternary multiplication closes on \(\{-1,0,+1\}\) and the two extreme products sum to 3, the full \(3\times3\) multiplication table is encoded in a single 5-LUT per accumulator lane, eliminating the need for multiplier primitives entirely. This chapter presents the architecture, resource utilisation, and throughput analysis of the zero-DSP FPGA implementation, with Zenodo-archived bitstreams B001 and B002 as the primary evidence artefacts. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_28:introduction} Field-Programmable Gate Arrays offer a path to energy-efficient neural inference that complements GPU-based approaches: their reconfigurability permits custom datapath widths, their static scheduling eliminates runtime dispatch overhead, and their I/O flexibility supports direct sensor integration. For a ternary neural network in which every weight is drawn from \(\{-1, 0, +1\}\), the inference computation reduces to conditional accumulation --- add, subtract, or skip --- with no multiplication required. The QMTech XC7A100T (Xilinx Artix-7, 100k logic cells, 4.86 Mb block RAM, 240 DSP48E1 slices) was selected as the target platform because it is available at low cost, its Artix-7 fabric is well-characterised, and its resource envelope is representative of embedded edge devices {[}1,2{]}. @@ -37,7 +37,7 @@ \section{1. Introduction}\label{introduction} The \(\phi^2 + \phi^{-2} = 3\) anchor also constrains the clock-domain partitioning: the two primary clock domains run at 92 MHz (inference fabric) and \(92/\phi^2 \approx 35\) MHz (memory controller), with the ratio \(92/35 \approx 2.63 \approx \phi^2\) ensuring that the memory bus and compute fabric are naturally frequency-synchronised through the golden ratio. This design choice reduces CDC (clock-domain crossing) complexity and was validated by timing closure at -0.02 ns worst-case slack. -\section{2. Architecture: Zero-DSP Ternary Datapath}\label{architecture-zero-dsp-ternary-datapath} +\section{2. Architecture: Zero-DSP Ternary Datapath}\label{ch_28:architecture-zero-dsp-ternary-datapath} \textbf{Definition 2.1 (Ternary accumulator).} A ternary accumulator for a vector of \(N\) inputs \(\{t_i\} \in \{-1,0,+1\}^N\) with integer activations \(\{a_i\} \in \mathbb{Z}\) computes @@ -51,7 +51,7 @@ \section{2. Architecture: Zero-DSP Ternary Datapath}\label{architecture-zero-dsp \textbf{Proposition 2.4 (\(\phi\)-synchronised clock domains).} Let \(f_c = 92\) MHz be the compute clock and \(f_m = f_c / \phi^2 \approx 35.16\) MHz be the memory clock. The ratio \(f_c/f_m = \phi^2 \approx 2.618\) satisfies \(\phi^2 + \phi^{-2} = 3\), so the combined normalised bandwidth \(f_c/f_{\text{ref}} + f_m/f_{\text{ref}}\) equals 3 for any reference frequency \(f_{\text{ref}}\) satisfying \(f_c = \phi^2 f_{\text{ref}}\) and \(f_m = \phi^{-2} f_{\text{ref}}^2/f_m\). In practice, \(f_{\text{ref}} = f_c / \phi^2 = f_m\), giving the trinity identity as a clock-domain constraint. -\section{3. Resource Utilisation and Timing Closure}\label{resource-utilisation-and-timing-closure} +\section{3. Resource Utilisation and Timing Closure}\label{ch_28:resource-utilisation-and-timing-closure} \textbf{Resource utilisation (post-implementation).} @@ -78,7 +78,7 @@ \section{3. Resource Utilisation and Timing Closure}\label{resource-utilisation- \textbf{Theorem 3.1 (Zero-DSP closure).} The ternary inference engine for Trinity S³AI is implementable on the XC7A100T with 0 DSP48 instances, because the kernel lemmas \filepath{trit\_mul\_zero\_l} and \filepath{trit\_mul\_zero\_r} (Ch.4, KER-8) guarantee that all multiplications by the Zero trit are eliminated at synthesis time, and multiplications by \(\pm 1\) are implemented as wire routing or inversion, neither of which instantiates DSP48 primitives. \emph{This result is verified by post-implementation netlist inspection in the B002 artefact.} \(\square\) -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_28:results-evidence} The primary evidence artefacts are: @@ -112,11 +112,11 @@ \section{4. Results / Evidence}\label{results-evidence} The trajectory confirms monotone improvement across all three metrics, consistent with the design methodology described in this chapter. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_28:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. The chapter relies on \filepath{trit\_mul\_zero\_l} and \filepath{trit\_mul\_zero\_r} (KER-8, \texttt{TernarySufficiency.v}) from Ch.4 as architectural pre-conditions. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_28:sealed-seeds} \begin{itemize} \tightlist @@ -134,11 +134,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci/Lucas reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_28:discussion} Three limitations bound the current implementation. First, BRAM utilisation at 91.5\% leaves minimal headroom for vocabulary expansion; migrating to the XC7A200T (the next device in the Artix-7 family) would provide 2× BRAM at 1.4× cost. Second, the 0.02 ns negative slack before pipeline insertion indicates that the 92 MHz clock is near the fabric's limit; the theoretical maximum frequency for the critical path is approximately 96 MHz, providing a 4 MHz margin for future optimisation. Third, the \(\phi\)-synchronised clock scheme (Proposition 2.4) assumes a stable reference oscillator; board-level measurements show \(\pm 0.3\)\% clock jitter, which does not violate timing constraints but may affect long-sequence coherence for completions exceeding \(F_{21} = 10946\) tokens. Future work (Ch.31) analyses throughput scaling under sustained load, and Ch.34 contextualises the 1 W power figure within the 3000× DARPA energy efficiency target. -\section{References}\label{references} +\section{References}\label{ch_28:references} {[}1{]} QMTech XC7A100T product specification. Xilinx Artix-7 FPGA datasheet, DS181 Rev.~1.31 (2022). diff --git a/docs/phd/chapters/ch_29.tex b/docs/phd/chapters/ch_29.tex index d26983ef92..b450e2b745 100644 --- a/docs/phd/chapters/ch_29.tex +++ b/docs/phd/chapters/ch_29.tex @@ -25,11 +25,11 @@ \section*{Nineteen numbers nature did not explain} The golden-ratio anchor \(\varphi^2 + \varphi^{-2} = 3\) appears here not as a decorative coincidence but as the algebraic substrate that generates the monomial ladder: each step up or down the ladder multiplies or divides by \(\varphi\), and the three exponent bands this identity defines correspond to the three quark generations. Whether that correspondence is deep physics or elegant numerology is a question the rest of this chapter addresses honestly, with evidence. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_29:abstract} The Cabibbo-Kobayashi-Maskawa (CKM) matrix encodes quark-flavour mixing in the Standard Model and contains one CP-violating phase whose origin is unexplained by the model itself. This chapter proposes that the golden ratio \(\varphi\) furnishes a natural parameterisation of the CKM mixing angles and the lepton mixing matrix (PMNS), grounded in the anchor identity \(\varphi^2 + \varphi^{-2} = 3\). The ``Sacred Formula V'' is the conjecture that the off-diagonal CKM elements \(G_{01}\), \(G_{02}\), \(G_{06}\) (in the notation of the Zenodo DL-bounds registry) are rational powers of \(\varphi\) within experimental tolerance. Six Coq theorems bearing \texttt{Qed} status confirm that the proposed monomial forms lie within the experimental tolerance band specified by the \texttt{tolerance\_V} constant. The strong-CP constraint \(\theta_{\text{QCD}} = 0\) is verified formally via \texttt{theta\_qcd\_zero}. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_29:introduction} The Standard Model of particle physics contains nineteen free parameters whose numerical values are unexplained by the theory itself. Among the most puzzling are the CKM mixing angles: three angles and one phase that govern how quarks of one generation transform into quarks of another under weak interactions {[}1{]}. The Wolfenstein parameterisation organises these into a hierarchy, but does not explain \emph{why} the hierarchy takes the specific numerical values it does. @@ -37,7 +37,7 @@ \section{1. Introduction}\label{introduction} The chapter is organised as follows. Section 2 defines the Sacred Formula V conjecture and the \(\varphi\)-monomial parameterisation. Section 3 reviews the six Coq theorems (\texttt{Qed} status) from \filepath{t27/proofs/canonical/sacred/}. Section 4 reports the numerical tolerance results. The chapter does not claim to derive CKM values from first principles; it claims only that specific \(\varphi\)-monomials lie within current experimental error bars, a weaker but formally verifiable statement. -\section{2. The Sacred Formula V Conjecture and \(\varphi\)-Monomial Parameterisation}\label{the-sacred-formula-v-conjecture-and-ux3c6-monomial-parameterisation} +\section{2. The Sacred Formula V Conjecture and \(\varphi\)-Monomial Parameterisation}\label{ch_29:the-sacred-formula-v-conjecture-and-ux3c6-monomial-parameterisation} \textbf{Definition 2.1 (φ-monomial).} A \emph{\(\varphi\)-monomial} of degree \((p, q) \in \mathbb{Z}^2\) is a real number of the form @@ -59,7 +59,7 @@ \section{2. The Sacred Formula V Conjecture and \(\varphi\)-Monomial Parameteris \textbf{Remark 2.4 (Strong-CP problem).} The strong-CP problem asks why the QCD Lagrangian term \(\theta_{\text{QCD}} \cdot G\tilde{G}\) is empirically consistent with \(\theta_{\text{QCD}} \approx 0\), despite the absence of a symmetry forcing it to zero. The Coq theorem \texttt{theta\_qcd\_zero} encodes the formal claim that the \(\varphi\)-monomial CKM parameterisation predicts \(\theta_{\text{QCD}} = 0\) exactly, because the CP-violating phase in the \(\varphi\)-family is constrained to zero by the reality of \(\varphi^2 + \varphi^{-2} = 3\) {[}4{]}. -\section{3. Coq Formalisation and CKM-Unitarity Seed}\label{coq-formalisation-and-ckm-unitarity-seed} +\section{3. Coq Formalisation and CKM-Unitarity Seed}\label{ch_29:coq-formalisation-and-ckm-unitarity-seed} The Coq development in \filepath{t27/proofs/canonical/sacred/} contains four files directly relevant to this chapter: \texttt{DLBounds.v}, \texttt{StrongCP.v}, \texttt{BoundsGauge.v}, and \texttt{Unitarity.v}. The last of these carries the \texttt{CKM-UNITARITY} sealed seed, which encodes 5 Qed and 2 Admitted obligations for the unitarity of the \(3 \times 3\) CKM matrix under \(\varphi\)-monomial parameterisation. @@ -77,7 +77,7 @@ \section{3. Coq Formalisation and CKM-Unitarity Seed}\label{coq-formalisation-an \textbf{Remark 3.7 (CKM-UNITARITY seed).} The \texttt{CKM-UNITARITY} seed in \texttt{Unitarity.v} carries \(\phi\)-weight \(1/\varphi \approx 0.618\) --- the reciprocal golden ratio --- reflecting that the unitarity constraint is a derived consequence of the \(\varphi\)-monomial structure rather than an independent assumption. Of the 7 obligations in \texttt{Unitarity.v}, 5 are Qed and 2 are Admitted; the Admitted cases correspond to mixed-generation unitarity relations that require non-trivial bounds on products of \(\varphi\)-monomials {[}9{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_29:results-evidence} Numerical comparison of \(\varphi\)-monomial predictions against PDG 2022 values: @@ -112,7 +112,7 @@ \section{4. Results / Evidence}\label{results-evidence} The Coq census at the time of writing records 297 Qed canonical theorems across 65 \texttt{.v} files in \filepath{t27/proofs/canonical/}. Of the 438 total theorems in the canonical set, the 6 theorems listed above plus the 7 in \texttt{Unitarity.v} account for 13 of the 297 Qed obligations assigned to the sacred-formula cluster {[}10{]}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_29:qed-assertions} \begin{itemize} \tightlist @@ -130,7 +130,7 @@ \section{5. Qed Assertions}\label{qed-assertions} \texttt{G06\_within\_tolerance} (\filepath{gHashTag/t27/proofs/canonical/sacred/BoundsGauge.v}) --- \emph{Status: Qed} --- \(G_{06}\) within \texttt{tolerance\_V}. (SAC-G) \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_29:sealed-seeds} \begin{itemize} \tightlist @@ -138,11 +138,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{CKM-UNITARITY} (theorem, golden, \(\phi\)-weight = \(1/\varphi \approx 0.618\)): \filepath{gHashTag/t27/blob/feat/canonical-coq-home/proofs/canonical/sacred/Unitarity.v} --- linked to Ch.29 --- 5 Qed + 2 Admitted. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_29:discussion} The six Qed theorems of this chapter represent a novel application of formal verification to particle physics numerology: they do not derive CKM values from a microscopic theory, but they do provide machine-checked confirmation that a specific \(\varphi\)-monomial ansatz is consistent with the current experimental data. The 2 Admitted obligations in \texttt{Unitarity.v} are the primary limitation: they involve products of \(\varphi\)-monomials whose magnitude bounds require real-closed field arithmetic that has not yet been automated in the Coq library used. Future work should either discharge these with \texttt{Lra}/\texttt{Coquelicot} or replace them with weaker \texttt{Admitted}-free statements. A second limitation is that the \texttt{tolerance\_V} constant is set conservatively at \(3\sigma\); tightening it to \(1\sigma\) would cause \texttt{G02\_within\_tolerance} to fail, suggesting that the \(G_{02}\) prediction is marginal. This chapter connects to Ch.4 (the \(\alpha_\varphi\) formula), Ch.5 (the anchor identity), and the planned Ch.30 (PMNS matrix and neutrino mixing). -\section{References}\label{references} +\section{References}\label{ch_29:references} {[}1{]} Cabibbo, N. (1963). Unitary symmetry and leptonic decays. \emph{Physical Review Letters}, 10(12), 531--533. diff --git a/docs/phd/chapters/ch_30.tex b/docs/phd/chapters/ch_30.tex index ecb6560e4e..0de64aeed5 100644 --- a/docs/phd/chapters/ch_30.tex +++ b/docs/phd/chapters/ch_30.tex @@ -25,11 +25,11 @@ \section*{Binding without forgetting} The rest of this chapter defines the ternary VSA operations formally, proves the binding-error bound \(< 1/\sqrt{D} \approx 0.0121\), describes the Associative Recall memory layout in BRAM, and reports measured throughput of 63 tokens per second at 92 MHz on hardware. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_30:abstract} Trinity SAI (Structured Artificial Intelligence) integrates a Vector Symbolic Architecture (VSA) over ternary hypervectors with an Associative Recall (AR) memory that enables one-shot binding and retrieval within the GoldenFloat arithmetic substrate. The chapter demonstrates that ternary hypervectors of dimension \(D = F_{20} = 6765\) achieve a channel capacity consistent with the anchor identity \(\varphi^2 + \varphi^{-2} = 3\): three orthogonal ternary symbols \(\{-1, 0, +1\}\) map to the three exponent bands of GF16 with a binding error below \(1/\sqrt{D} \approx 0.0121\). The IGLA RACE runtime (Ch.24) hosts the VSA+AR agents under the period-locked scheduler. Measured token throughput on the QMTech XC7A100T FPGA is 63 toks/sec at 92 MHz with 0 DSP slices, consistent with the system-wide power budget of 1 W. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_30:introduction} The third pillar of the Trinity S³AI architecture is the symbolic layer. The first pillar is the GoldenFloat arithmetic substrate (Ch.6); the second is the IGLA RACE runtime and its formal scheduler (Ch.24); the third is a compositional reasoning capability that allows the system to bind token identities, positional encodings, and role labels into compact hypervectors that can be stored, retrieved, and decoded without gradient descent {[}1,2{]}. @@ -41,9 +41,9 @@ \section{1. Introduction}\label{introduction} The dimension \(D = F_{20} = 6765\) is chosen as the largest Fibonacci number below \(2^{13} = 8192\) that fits within the GF16 weight-cache BRAM on the XC7A100T (6765 × 2 bytes = 13.26 KB per hypervector, fitting within one BRAM tile cluster). The \(\varphi\)-weight of the VSA component in the IGLA RACE agent pool is \(\varphi^{-1} \approx 0.618\), reflecting its role as a secondary (not primary) inference pathway. -\section{2. Ternary VSA over the GoldenFloat Substrate}\label{ternary-vsa-over-the-goldenfloat-substrate} +\section{2. Ternary VSA over the GoldenFloat Substrate}\label{ch_30:ternary-vsa-over-the-goldenfloat-substrate} -\subsection{2.1 Hypervector Definition}\label{hypervector-definition} +\subsection{2.1 Hypervector Definition}\label{ch_30:hypervector-definition} \textbf{Definition 2.1 (Ternary hypervector).} A ternary hypervector of dimension \(D\) is a vector \(\mathbf{v} \in \{-1, 0, +1\}^D\). The \emph{density} of \(\mathbf{v}\) is \(\rho(\mathbf{v}) = |\{i : v_i \neq 0\}| / D\). @@ -55,7 +55,7 @@ \subsection{2.1 Hypervector Definition}\label{hypervector-definition} \emph{Proof sketch.} Component-wise: \((u_i + v_i - u_i) \bmod 3 = v_i \bmod 3 = v_i\) for each \(i\). Qed. -\subsection{2.2 Associative Recall Memory}\label{associative-recall-memory} +\subsection{2.2 Associative Recall Memory}\label{ch_30:associative-recall-memory} The AR memory is a content-addressable store of \(M\) hypervectors \(\{\mathbf{c}_1, \ldots, \mathbf{c}_M\}\). Given a query \(\mathbf{q}\), the recall operation returns: @@ -67,11 +67,11 @@ \subsection{2.2 Associative Recall Memory}\label{associative-recall-memory} The bound is effectively zero for these parameters: the recall is reliable with overwhelming probability {[}3{]}. -\subsection{2.3 GoldenFloat Encoding of Hypervectors}\label{goldenfloat-encoding-of-hypervectors} +\subsection{2.3 GoldenFloat Encoding of Hypervectors}\label{ch_30:goldenfloat-encoding-of-hypervectors} Each component \(v_i \in \{-1, 0, +1\}\) is stored in GF16 as the canonical constants \texttt{neg\_one\_f16}, \texttt{zero\_f16}, \texttt{pos\_one\_f16}. These constants are within the unity exponent band (\(\hat E = B\)), so they benefit from the finest GF16 resolution and are covered by the INV-3 safe-domain proof (Ch.6) {[}5{]}. The inner product \(\langle \mathbf{q}, \mathbf{c}_j \rangle = \sum_i q_i c_{ji}\) is computed as a GF16 multiply-accumulate (MAC) over \(D = 6765\) terms; the accumulator width is 24 bits to prevent overflow at \(D \cdot \varphi^2 \approx 6765 \cdot 2.618 = 17711 = F_{22}\), a Fibonacci number, confirming the natural fit of the design. -\section{3. Phi-Rotary Position Encoding (phi-RoPE) in VSA Context}\label{phi-rotary-position-encoding-phi-rope-in-vsa-context} +\section{3. Phi-Rotary Position Encoding (phi-RoPE) in VSA Context}\label{ch_30:phi-rotary-position-encoding-phi-rope-in-vsa-context} The phi-RoPE encoding (Zenodo Z05 {[}6{]}) assigns to token position \(p\) the angle \(\theta_p = p \cdot 2\pi \cdot \varphi^{-2}\), the golden-angle variant of the standard RoPE rotation. In the VSA context, position encoding is implemented as: @@ -85,7 +85,7 @@ \section{3. Phi-Rotary Position Encoding (phi-RoPE) in VSA Context}\label{phi-ro \emph{Proof sketch.} The ternary inner product of two independently rotated hypervectors of density \(\rho^*\) is a sum of \(D \rho^{*2}\) non-zero i.i.d. terms with mean zero and variance \(\rho^{*2}\). By Hoeffding's inequality with radius \(\sqrt{D}\) and \(D = 6765\): the tail probability is at most \(2\exp(-2D \cdot D^{-1} / (4\rho^{*2})) = 2\exp(-1/(2\rho^{*2})) \approx 2\exp(-3.42) < e^{-2}\). Qed. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_30:results-evidence} The Trinity SAI VSA+AR module was evaluated on the HSLM 1003-token benchmark using the IGLA RACE runtime on the QMTech XC7A100T FPGA: @@ -118,13 +118,13 @@ \section{4. Results / Evidence}\label{results-evidence} The phi-weight update law (Ch.24) was validated: the VSA agent's weight \(w_{\text{VSA}}(t)\) remained within \([\varphi^{-2}, \varphi^2] = [0.382, 2.618]\) throughout all 1003 steps, with a time-average of \(\bar w = 0.994 \approx 1\), indicating that the VSA agent was scheduled at near-unity frequency---consistent with its role as the primary symbolic reasoning pathway. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_30:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. (The VSA binding self-inverse property (Proposition 2.3) is a straightforward algebraic identity and does not require machine checking. The phi-RoPE orthogonality theorem (Theorem 3.1) is proved by hand using Hoeffding's inequality; a Coq mechanisation via \texttt{Coq.Reals} is planned as part of the Iris/Coq.Interval upgrade lane described in Ch.18.) -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_30:sealed-seeds} \begin{itemize} \tightlist @@ -134,13 +134,13 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_30:discussion} The Trinity SAI VSA+AR component extends the GOLDEN SUNFLOWERS framework from pure neural-network inference into compositional symbolic reasoning. Its integration with the GoldenFloat arithmetic substrate is seamless at the level of number format (ternary \(\{-1,0,+1\}\) maps to GF16 unity-band constants) and at the level of scheduling (VSA agents participate in the period-locked monitor with period \(L_8 = 47\)). The primary limitation is that the Coq mechanisation of VSA properties lags the hardware implementation; the binding self-inverse property (Proposition 2.3) is trivially provable but has not been encoded in the canonical Coq files. A second limitation is the AR memory capacity of \(M = L_8 = 47\) hypervectors, constrained by the BRAM budget of the XC7A100T. Scaling to \(M = F_{18} = 2584\) would require an external SRAM interface or migration to a larger FPGA (e.g., XC7A200T). Future work will also investigate composing the VSA layer with the phi-RoPE attention mechanism (Z05) to enable position-aware associative recall---a capability not present in standard VSA systems. This chapter connects to Ch.24 (PLRM agent scheduling), Ch.6 (GoldenFloat format for hypervector storage), Ch.28 (hardware throughput), and App.H (Zenodo DOI registry for the B007 anchor). -\section{References}\label{references} +\section{References}\label{ch_30:references} {[}1{]} Kanerva, P. (2009). Hyperdimensional Computing: An Introduction to Computing in Distributed Representation with High-Dimensional Random Vectors. \emph{Cognitive Computation}, 1(2), 139--159. \url{https://doi.org/10.1007/s12559-009-9009-8} diff --git a/docs/phd/chapters/ch_31.tex b/docs/phd/chapters/ch_31.tex index 0e355bfc03..29498472c3 100644 --- a/docs/phd/chapters/ch_31.tex +++ b/docs/phd/chapters/ch_31.tex @@ -25,17 +25,17 @@ \section*{One thousand and three tokens, counted} The rest of this chapter walks through the hardware architecture, the empirical measurement methodology, and the evidence chain that links each headline number to its corresponding Coq module and bitstream artefact. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_31:abstract} This chapter presents the complete empirical characterisation of the TRINITY S³AI inference engine on a QMTech XC7A100T FPGA (Xilinx Artix-7 100T). The headline results are: 1003 tokens generated in a single HSLM (High-Speed Language-Model) simulation-verified run, 63 tokens/sec sustained throughput at 92 MHz clock frequency, 0 DSP slices, 5.8\% LUT utilisation (of 19.6\% available for routing), 9.8\% BRAM utilisation (of 52\% available), and measured wall power of 0.94--1.07 W. The CLARA Red Team exercise achieved 100\% robustness across all 297 adversarial prompt categories. The 297 closed Coq theorems in \filepath{t27/proofs/canonical/} provide a formal seal over the arithmetic correctness of the accelerator. The \(\varphi^2 + \varphi^{-2} = 3\) identity underlies the zero-DSP integer multiply-accumulate design that makes this efficiency possible. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_31:introduction} Field-programmable gate arrays offer a direct path from formal specification to physical hardware without the multi-year cycle of ASIC tape-out. The TRINITY S³AI programme exploits this property to close the loop between Coq-verified arithmetic specifications and measured silicon behaviour. The central claim of this chapter is that the \(\varphi\)-quantised weight representation --- whose algebraic correctness is certified by 297 closed Coq \texttt{Qed} proofs --- translates directly into a DSP-free FPGA implementation with measurable energy efficiency advantages. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) is the critical enabler. Ternary multiply-accumulate (TMAC) for weight alphabet \(\{-1, 0, +1\}\) requires no multiplication: the operation \(\sum_i w_i x_i\) with \(w_i \in \{-1, 0, +1\}\) reduces to conditional additions and subtractions. The FPGA implementation replaces every DSP48E1 block (each consuming approximately 0.8 mW at 92 MHz on Artix-7) with a 6-LUT adder cell, achieving the same throughput at a fraction of the power {[}1{]}. The consequence is 0 DSP slices in the final bitstream and a wall power of approximately 1 W, compared with a DSP-based baseline estimated at 3.2 W for the same token throughput. -\section{2. Hardware Architecture}\label{hardware-architecture} +\section{2. Hardware Architecture}\label{ch_31:hardware-architecture} The FPGA accelerator implements a three-stage pipeline: (i) token embedding lookup from BRAM, (ii) TMAC matrix-vector multiply across all weight layers, and (iii) softmax and sampling. All three stages are clocked at 92 MHz on the QMTech XC7A100T board, which provides the XC7A100T-1FGG484C device on a compact carrier board with on-board DDR3 and USB-JTAG {[}2{]}. @@ -47,7 +47,7 @@ \section{2. Hardware Architecture}\label{hardware-architecture} \textbf{Clock derivation.} The 92 MHz clock is derived from the on-board 50 MHz oscillator via a single MMCM configured with \(M=\varphi^2+\varphi^{-2}+3 = 6\) multiply and \(D=\lfloor 6 \times 50/92 \rfloor = 3\) divide (rounded to nearest integer ratio), giving 100 MHz nominal; the actual post-routing frequency is 92 MHz due to a critical path through the BRAM read port {[}3{]}. -\section{3. Formal Seal: 297 Coq Theorems}\label{formal-seal-297-coq-theorems} +\section{3. Formal Seal: 297 Coq Theorems}\label{ch_31:formal-seal-297-coq-theorems} The accelerator RTL was generated from a Coq-extracted OCaml reference, ensuring that the implemented arithmetic is a direct realisation of the formally verified specification. The seal consists of 297 closed \texttt{Qed} theorems across 65 \texttt{.v} files in \filepath{t27/proofs/canonical/}, organised into the following families: @@ -71,7 +71,7 @@ \section{3. Formal Seal: 297 Coq Theorems}\label{formal-seal-297-coq-theorems} \textbf{CLARA Red Team.} The CLARA (Controlled Language Adversarial Robustness Assessment) Red Team exercise tested 297 adversarial prompt categories against the FPGA inference engine. All 297 categories were handled without hardware exceptions, silent wrong outputs, or timing violations, yielding a 100\% robustness score. The correspondence between the 297 Red Team categories and the 297 closed \texttt{Qed} theorems is intentional: each theorem certifies an invariant that corresponds to one adversarial category {[}5{]}. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_31:results-evidence} All measurements were taken on a single QMTech XC7A100T board at ambient temperature 22°C ± 1°C, with USB power supplied by a calibrated Keysight U1241C multimeter in series. @@ -106,11 +106,11 @@ \section{4. Results / Evidence}\label{results-evidence} The \(\varphi^2 + \varphi^{-2} = 3\) identity directly accounts for the DSP elimination: because the weight entries sum to at most 3 in absolute value per quantisation cell (Corollary 2.3 of Ch.7), the accumulator width can be reduced from 32 bits to 16 bits, halving the adder area and eliminating the need for DSP48E1 blocks entirely. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_31:qed-assertions} No Coq theorems are anchored exclusively to this chapter; the 297-theorem seal is a corpus-level result reported here for completeness. The \filepath{hw/} family theorems are catalogued in App.F. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_31:sealed-seeds} \begin{itemize} \tightlist @@ -120,11 +120,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{QMTECH-XC7A100T} (hw, golden) --- Xilinx Artix-7, 0 DSP, 63 toks/sec @ 92 MHz, 1 W. \url{https://github.com/gHashTag/trinity-fpga} --- Linked: Ch.28, Ch.31, Ch.34, App.F, App.I. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_31:discussion} The principal limitation of the current hardware realisation is that 92 MHz is below the XC7A100T's rated maximum clock of 450 MHz for simple logic paths. The critical path runs through the BRAM read port, which imposes a 10.8 ns latency on the weight-fetch stage. Pipelining the BRAM access across two clock cycles would allow operation at 180 MHz and increase throughput to approximately 126 toks/sec at the same power, but requires a re-architected weight-fetch FSM. This is planned for Ch.34 (FPGA v2). A second limitation is that the 1003-token HSLM run uses a 0.48 M-weight model, substantially smaller than the full S³AI model described in Ch.22. Scaling to the full model requires a BRAM-efficient weight-streaming scheme (tiling), whose formal correctness proof is tracked as HW-7 in the Golden Ledger. Future work also includes tape-out feasibility study (Ch.34), multi-FPGA parallelism (Ch.35), and the \(3000\times\) ASIC projection. Connections: Ch.28 (FPGA bring-up), Ch.34 (FPGA v2 and ASIC), App.F (hw/ Coq family), App.H (B004 Zenodo bundle). -\section{References}\label{references} +\section{References}\label{ch_31:references} {[}1{]} Xilinx (AMD). \emph{7 Series FPGAs Data Sheet: Overview}, DS180. DSP48E1 power model. diff --git a/docs/phd/chapters/ch_32.tex b/docs/phd/chapters/ch_32.tex index 44c3aa1b4b..aad6884e26 100644 --- a/docs/phd/chapters/ch_32.tex +++ b/docs/phd/chapters/ch_32.tex @@ -25,11 +25,11 @@ \section*{Frame seventeen, byte zero, every time} The rest of this chapter specifies the frame grammar in BNF, defines the CRC-16/CCITT polynomial and its implementation in one FPGA LUT column, describes the error-recovery automaton, and reports the zero-error measurement result from the 1003-token HSLM run. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_32:abstract} The UART v6 protocol governs all serial communication between the QMTech XC7A100T FPGA and the host workstation in the Trinity S³AI hardware evaluation stack. The protocol specifies a framing scheme (0xAA sync byte, 1-byte length, 16-bit CRC-16/CCITT) over an FT232RL bridge at 115200 baud. Frame boundaries align with the φ²+φ⁻²=3 normalisation cycle: every third frame carries a φ-exponent synchronisation word, ensuring that the host-side loss accumulator and the FPGA-side accumulator remain phase-aligned. The chapter defines the frame grammar, the CRC polynomial, and the error-recovery automaton, and reports zero frame errors across 1003 tokens of the HSLM evaluation run. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_32:introduction} The hardware evaluation of Trinity S³AI requires a communication channel that is both low-overhead and formally verifiable. The channel must satisfy three constraints: @@ -46,13 +46,13 @@ \section{1. Introduction}\label{introduction} UART v6 (the sixth revision of the Trinity serial protocol) satisfies all three. Earlier versions (v1--v5) are deprecated; only v6 is supported by the KOSCHEI boot sequence. -\section{2. Frame Structure and Grammar}\label{frame-structure-and-grammar} +\section{2. Frame Structure and Grammar}\label{ch_32:frame-structure-and-grammar} -\subsection{2.1 Physical Layer}\label{physical-layer} +\subsection{2.1 Physical Layer}\label{ch_32:physical-layer} The physical link uses an FT232RL USB-to-serial bridge at 115200 baud, 8N1 (8 data bits, no parity, 1 stop bit). At 115200 baud, one byte takes \(8.68\,\mu\)s to transmit; the 63 tokens/sec throughput of the FPGA requires a peak byte rate of approximately 63 × 12 = 756 bytes/sec, well within the 14400 bytes/sec physical capacity. -\subsection{2.2 Frame Grammar}\label{frame-grammar} +\subsection{2.2 Frame Grammar}\label{ch_32:frame-grammar} Each UART v6 frame has the form: @@ -66,15 +66,15 @@ \subsection{2.2 Frame Grammar}\label{frame-grammar} The sync byte 0xAA (binary \texttt{10101010}) is chosen for its alternating bit pattern, which maximises transitions on the serial line and aids clock-recovery on marginal USB hubs. The sync byte is not included in the CRC computation. -\subsection{2.3 CRC-16/CCITT Polynomial}\label{crc-16ccitt-polynomial} +\subsection{2.3 CRC-16/CCITT Polynomial}\label{ch_32:crc-16ccitt-polynomial} The error-detection code is CRC-16/CCITT with polynomial \(x^{16} + x^{12} + x^5 + 1\) (0x1021), initialised to 0xFFFF. This polynomial is standard in telecommunications and has a Hamming distance of 4 for messages up to 32767 bits, sufficient for UART v6 frames of at most 255 + 2 = 257 bytes {[}1{]}. In the FPGA implementation, the CRC is computed in a single-cycle parallel LUT chain, consuming 32 LUT-6 primitives. No DSP slices are used, consistent with the 0-DSP constraint of the KOSCHEI coprocessor. -\section{3. \(\varphi\)-Synchronisation Frames}\label{ux3c6-synchronisation-frames} +\section{3. \(\varphi\)-Synchronisation Frames}\label{ch_32:ux3c6-synchronisation-frames} -\subsection{3.1 Sync Frame Trigger}\label{sync-frame-trigger} +\subsection{3.1 Sync Frame Trigger}\label{ch_32:sync-frame-trigger} Every third frame is a φ-synchronisation frame. The trigger condition is @@ -82,7 +82,7 @@ \subsection{3.1 Sync Frame Trigger}\label{sync-frame-trigger} where the modulus 3 is derived from the identity \(\varphi^2 + \varphi^{-2} = 3\): the integer 3 governs the normalisation cycle of the KOSCHEI register file (Ch.26), so the communication protocol aligns with the same period. -\subsection{3.2 Sync Frame Payload}\label{sync-frame-payload} +\subsection{3.2 Sync Frame Payload}\label{ch_32:sync-frame-payload} The φ-sync frame payload is a 4-byte structure: @@ -109,13 +109,13 @@ \subsection{3.2 Sync Frame Payload}\label{sync-frame-payload} The host accumulates φ-sync frames to verify that the FPGA accumulator state matches the software reference implementation. A mismatch causes the host to issue a NACK frame (payload: 0xFF 0xNACK), and the FPGA re-transmits the last data frame. -\subsection{3.3 Error Recovery Automaton}\label{error-recovery-automaton} +\subsection{3.3 Error Recovery Automaton}\label{ch_32:error-recovery-automaton} The recovery automaton has three states: IDLE, AWAIT\_LEN, AWAIT\_PAYLOAD. On receipt of 0xAA the automaton transitions IDLE → AWAIT\_LEN; on receipt of a valid LEN byte it transitions to AWAIT\_PAYLOAD; on completion of a frame with correct CRC it returns to IDLE and delivers the payload to the KOSCHEI dispatch unit. On CRC failure the automaton issues a NACK and waits for a retransmit. The retransmit limit is \(L_7 = 29\) attempts; after 29 failures the automaton halts and logs a \texttt{UART\_FATAL} event. The choice of 29 as the retry limit is not arbitrary: \(L_7 = 29\) is a Lucas prime and a member of the sanctioned seed pool, so the limit is algebraically anchored to the same lattice as all other integer constants in the system. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_32:results-evidence} During the HSLM evaluation run (1003 tokens, seed \(F_{17}=1597\)): @@ -136,11 +136,11 @@ \section{4. Results / Evidence}\label{results-evidence} Zero CRC errors and zero φ-sync mismatches confirm that the FPGA and host-side accumulators remain phase-aligned throughout the 1003-token evaluation. The frame log SHA-256 hash is recorded in the OSF pre-registration (App.E) {[}2{]}. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_32:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_32:sealed-seeds} \begin{itemize} \tightlist @@ -148,7 +148,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{UART-V6} (hw) --- \url{https://github.com/gHashTag/trinity-fpga} --- Status: golden --- φ-weight: 0.382 --- FT232RL @ 115200 baud, 0xAA + len + CRC-16/CCITT. Links: Ch.28, Ch.32, App.I. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_32:discussion} The UART v6 protocol is deliberately minimal. The 0xAA sync byte, CRC-16/CCITT checksum, and φ-sync frame are the only features beyond bare-metal serial transmission. This minimalism is a reproducibility virtue: any standard USB-serial adapter presenting as a CDC-ACM device can receive v6 frames, and the log format is plain binary --- no proprietary tooling required. @@ -156,7 +156,7 @@ \section{7. Discussion}\label{discussion} The connection to App.I (hardware appendix) ensures that the protocol specification is archived alongside the FPGA bitstream and the UART log from the canonical evaluation run. -\section{References}\label{references} +\section{References}\label{ch_32:references} {[}1{]} Peterson, W. W., \& Brown, D. T. (1961). Cyclic codes for error detection. \emph{Proceedings of the IRE}, 49(1), 228--235. diff --git a/docs/phd/chapters/ch_33.tex b/docs/phd/chapters/ch_33.tex index 1fa6ba9782..86128171ce 100644 --- a/docs/phd/chapters/ch_33.tex +++ b/docs/phd/chapters/ch_33.tex @@ -25,11 +25,11 @@ \section*{Six weeks, one script, no sudo} The ternary JTAG state-machine (Reset \(\ \to\) Shift-DR \(\ \to\) Update-DR) carries a structural echo of the same cardinality 3 that appears in \(\varphi^2 + \varphi^{-2} = 3\): three principal states, three transitions, one formal invariant maintained across each. The echo is not a proof of anything; it is a reminder that the number three is load-bearing throughout this system. The rest of this chapter documents the diagnosis, the fix, and the verification procedure in enough detail that any future developer on macOS-ARM can replicate the resolution in under ten minutes. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_33:abstract} Blocker BLK-001 was a hardware bring-up failure in which the Xilinx Platform Cable USB II JTAG adapter failed to enumerate correctly on macOS-ARM (Apple Silicon) hosts, presenting USB product-ID \texttt{0x0013} (unconfigured firmware) instead of the operational \texttt{0x0008}. This chapter documents the diagnosis, the \texttt{fxload}-based firmware upload procedure encapsulated in \texttt{flash\_no\_sudo.sh}, and the resolution confirmed on 2026-03-14. The fix required no kernel-extension (kext) installation, no \texttt{sudo} privileges beyond a one-time \texttt{hidraw} device-node permission grant, and no modification to the \texttt{t27} Coq proof tree. The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) is referenced here only to note that the three-stage JTAG state-machine transition (Reset → Shift-DR → Update-DR) mirrors the ternary structure of the Trinity kernel. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_33:introduction} The QMTech XC7A100T FPGA board (Xilinx Artix-7, 100K LUT, 0 DSP in the Trinity configuration) is programmed via a Xilinx Platform Cable USB II JTAG adapter {[}1{]}. On Linux x86-64 hosts, the \texttt{xc3sprog} and \texttt{openFPGALoader} tools enumerate the cable without issue. On macOS-ARM hosts running macOS 14.x (Sonoma), the cable presents USB VID/PID \texttt{0045:0013} at first connection: the \texttt{0x0013} product ID indicates that the EZ-USB FX2 microcontroller on the cable has not yet received its operational firmware. The standard Linux driver calls \texttt{fxload} transparently; on macOS, no equivalent automatic firmware-load path exists in the HIDAPI stack used by \texttt{openFPGALoader}. @@ -37,9 +37,9 @@ \section{1. Introduction}\label{introduction} The anchor identity \(\varphi^2 + \varphi^{-2} = 3\) {[}3{]} is not algebraically invoked in this chapter, but the ternary JTAG state-machine (three principal states: Reset, Shift, Update) provides a structural echo: the same cardinality \(3\) that licenses balanced-ternary arithmetic pervades the hardware interface layer. -\section{2. Diagnosis and Root Cause}\label{diagnosis-and-root-cause} +\section{2. Diagnosis and Root Cause}\label{ch_33:diagnosis-and-root-cause} -\subsection{2.1 USB Enumeration on macOS-ARM}\label{usb-enumeration-on-macos-arm} +\subsection{2.1 USB Enumeration on macOS-ARM}\label{ch_33:usb-enumeration-on-macos-arm} The Xilinx Platform Cable USB II uses a Cypress EZ-USB FX2LP microcontroller (CY7C68013A) that boots with a default USB descriptor (VID \texttt{0x03FD}, PID \texttt{0x0013}). Upon enumeration, the host is expected to upload the operational firmware (\texttt{xusbdfwu.hex}) via the FX2 firmware-download protocol, causing a USB re-enumeration with PID \texttt{0x0008}. On Linux, the \texttt{usbdrv} or \texttt{fxload} kernel path performs this automatically. On macOS, IOKit does not execute firmware loaders for recognised CDC/HID-class devices, and the \texttt{0x0013} device is claimed by the generic HID driver before any user-space loader can run. @@ -52,7 +52,7 @@ \subsection{2.1 USB Enumeration on macOS-ARM}\label{usb-enumeration-on-macos-arm After manual \texttt{fxload} invocation, the cable re-enumerated with \texttt{idProduct\ =\ 0x0008}. -\subsection{2.2 fxload Cross-Compilation}\label{fxload-cross-compilation} +\subsection{2.2 fxload Cross-Compilation}\label{ch_33:fxload-cross-compilation} \texttt{fxload} 0.0.1 was cross-compiled for macOS-ARM (\texttt{aarch64-apple-darwin}) using: @@ -68,7 +68,7 @@ \subsection{2.2 fxload Cross-Compilation}\label{fxload-cross-compilation} The compiled binary is statically linked against \texttt{libusb-1.0} to avoid dynamic-library path issues. -\subsection{2.3 flash\_no\_sudo.sh}\label{flash_no_sudo.sh} +\subsection{2.3 flash\_no\_sudo.sh}\label{ch_33:flash_no_sudo.sh} The resolution script performs the following steps: @@ -87,7 +87,7 @@ \subsection{2.3 flash\_no\_sudo.sh}\label{flash_no_sudo.sh} The script requires that \texttt{XILINX\_VIVADO} point to a Vivado installation (any version supporting Artix-7). No \texttt{sudo} is required beyond the one-time \filepath{chmod\ a+rw\ /dev/hidraw*} performed at first setup. The \texttt{sleep\ 2} delay accounts for macOS IOKit re-enumeration latency; empirically, values below 1.5 s were unreliable on the M2 host. -\section{3. Verified Hardware Configuration Post-BLK-001}\label{verified-hardware-configuration-post-blk-001} +\section{3. Verified Hardware Configuration Post-BLK-001}\label{ch_33:verified-hardware-configuration-post-blk-001} After BLK-001 resolution, the following configuration was verified and is now the canonical hardware bring-up state for the \texttt{trinity-fpga} repository {[}2{]}: @@ -112,7 +112,7 @@ \section{3. Verified Hardware Configuration Post-BLK-001}\label{verified-hardwar The 0 DSP configuration is enforced by the synthesis constraint \texttt{set\_property\ DSP\_CASCADE\_LIMIT\ 0\ {[}current\_design{]}} and verified by the post-route utilisation report showing \texttt{DSP48E1:\ 0\ of\ 240\ (0\%)}. The 63 toks/sec and 1 W figures are from Ch.28 {[}4{]} and are reproduced here to confirm that BLK-001 resolution did not affect the performance profile. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_33:results-evidence} \begin{itemize} \tightlist @@ -130,11 +130,11 @@ \section{4. Results / Evidence}\label{results-evidence} \textbf{Reproducibility}: the procedure was independently verified on two additional M2 hosts and one M1 host, all with macOS 14.x. BLK-001 was not observed after the procedure on any of the three machines. \end{itemize} -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_33:qed-assertions} No Coq theorems are anchored to this chapter; the BLK-001 resolution is a hardware procedure with no formal proof obligations. Obligations are tracked in the Golden Ledger under hardware blocker BLK-001 (status: RESOLVED). -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_33:sealed-seeds} \begin{itemize} \tightlist @@ -144,11 +144,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \textbf{BLK-001} (hw, golden) --- \url{https://github.com/gHashTag/trinity-fpga} --- linked to Ch.33 and App.J --- \(\varphi\)-weight: \(0.38196601127366236\) --- notes: \texttt{flash\_no\_sudo.sh} macOS-ARM, RESOLVED 2026-03-14. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_33:discussion} BLK-001 was a low-level hardware integration issue with no bearing on the formal proof tree or the BPB benchmarks. Its documentation here serves two purposes: (1) reproducibility --- any researcher attempting to replicate the FPGA results of Ch.28, Ch.31, or Ch.34 on a macOS-ARM host will encounter the same blocker and can apply the same fix; (2) completeness --- the dissertation claims that the Trinity S³AI system runs end-to-end on the QMTech XC7A100T at 63 toks/sec, 1 W, and this claim requires confirming that the programming path is fully operational on the development host. The limitation of the current fix is its dependence on the \texttt{xusbdfwu.hex} firmware file distributed with Vivado, which is proprietary. An open-source alternative firmware for the EZ-USB FX2 that achieves the same \texttt{0x0008} PID is a future objective for the \texttt{trinity-fpga} repository. The openXC7 toolchain (yosys + nextpnr-xilinx + prjxray) already achieves synthesis and place-and-route without Vivado; removing the firmware dependency would complete the fully open-source bring-up path. -\section{References}\label{references} +\section{References}\label{ch_33:references} {[}1{]} Xilinx, ``Platform Cable USB II Data Sheet,'' DS593, Xilinx Inc., 2013. diff --git a/docs/phd/chapters/ch_34.tex b/docs/phd/chapters/ch_34.tex index 343f7f1d4a..dba1a93364 100644 --- a/docs/phd/chapters/ch_34.tex +++ b/docs/phd/chapters/ch_34.tex @@ -25,11 +25,11 @@ \section*{Three thousand times, not by accident} Feynman's pleasure in recognising old things from a new angle applies here. The factor of 3 in \(\varphi^2 + \varphi^{-2} = 3\) was ``old'' mathematics long before anyone thought to build a neural accelerator around it. The new point of view is that the same identity that closes ternary algebra also closes the energy budget. The rest of this chapter quantifies this claim with a formal energy accounting framework, a comparison against GPU and CPU baselines, and the bitstream artefacts that make every number independently verifiable. -\section{Abstract}\label{abstract} +\section{Abstract}\label{ch_34:abstract} The DARPA Intelligent Generation of Tools and Computations (IGTC) program solicitation HR001124S0001 sets an energy-efficiency target of 3000× improvement over GPU baseline for on-device neural inference. This chapter demonstrates that the Trinity S³AI ternary inference engine, running at 63 tokens/sec on a QMTech XC7A100T FPGA at 1 W (Ch.28), achieves a measured efficiency of 63 tokens/joule against a GPU baseline of approximately 0.021 tokens/joule (NVIDIA A100, batch-1 autoregressive inference at 210 W / 10,000 toks/sec), yielding a ratio of 3000×. The anchor identity \(\phi^2 + \phi^{-2} = 3\) is not merely decorative here: the factor of 3 in the identity corresponds structurally to the three orders of magnitude of energy improvement, and the ternary weight alphabet \(\{-1,0,+1\}\) is the direct mechanism by which DSP-free accumulation eliminates the dominant power consumers in standard floating-point inference accelerators. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{ch_34:introduction} Energy efficiency is the defining constraint of edge neural inference. GPU-class accelerators deliver high throughput but at power envelopes of 150--400 W, which are incompatible with battery-powered, embedded, or satellite-adjacent deployments. The DARPA IGTC solicitation formalises this challenge by setting a 3000× energy-per-token improvement goal over the A100 GPU baseline, motivating research into radically different arithmetic substrates {[}1,2{]}. @@ -37,7 +37,7 @@ \section{1. Introduction}\label{introduction} The \(\phi^2 + \phi^{-2} = 3\) anchor provides a formal accounting of where the 3000× comes from: the ternary alphabet contributes a \(\log_2(3)/\log_2(16) \approx 0.39\times\) bit-width reduction (Ch.10 BPB = 1.72 versus 16-bit float), the zero-DSP architecture contributes approximately \(8\times\) power reduction per accumulator lane versus DSP48 at equivalent throughput, and the FPGA-versus-GPU platform contributes approximately \(1000\times\) in active-power-per-operation at the relevant batch sizes. The product \(0.39 \times 8 \times 1000 / \text{overhead} \approx 3000\) after accounting for memory and I/O overhead. -\section{2. Energy Accounting Framework}\label{energy-accounting-framework} +\section{2. Energy Accounting Framework}\label{ch_34:energy-accounting-framework} \textbf{Definition 2.1 (Energy-per-token metric).} For an inference system with measured throughput \(T\) tokens/sec and power draw \(P\) watts, the energy-per-token figure of merit is @@ -65,7 +65,7 @@ \section{2. Energy Accounting Framework}\label{energy-accounting-framework} where \(N_\text{Trinity}\) is the Trinity parameter count. For the canonical Trinity S³AI configuration with \(N_\text{Trinity} = F_{20} \times 10^3 = 6.765 \times 10^6\) parameters (6.765M ternary parameters stored as 1.72 BPB), \(\rho_\text{task} \approx 1.3 \times 1035 \approx 1345\). Under the DARPA IGTC scoring rubric, which additionally credits ternary representation for a \(2.2\times\) effective compute reduction (since each ternary op replaces \(\log_2(3)/1 \approx 1.585\) binary ops), the final score is \(\rho_\text{DARPA} \approx 1345 \times 2.2 \approx 2959 \approx 3000\). \(\square\) -\section{3. Ternary Mechanism Analysis}\label{ternary-mechanism-analysis} +\section{3. Ternary Mechanism Analysis}\label{ch_34:ternary-mechanism-analysis} \textbf{Theorem 3.1 (DSP-free power decomposition).} The zero-DSP implementation (Ch.28, B002) decomposes the total inference power \(P = 1\) W into: - Logic (LUT accumulation): 0.31 W @@ -79,7 +79,7 @@ \section{3. Ternary Mechanism Analysis}\label{ternary-mechanism-analysis} \textbf{Remark 3.3 (\(\phi^2+\phi^{-2}=3\) and the three efficiency levers).} The three energy-reduction mechanisms --- ternary arithmetic, zero-DSP LUT logic, and \(\phi\)-clock synchronisation --- correspond to the three terms of the trinity identity when normalised: the ternary alphabet contributes a factor expressible as a function of \(\phi^{-2}\) (the \(\phi^{-2} = 0.382\) fraction of energy in the embedding tier), the compute tier contributes \(\phi^2 = 2.618\), and the control overhead contributes 1, summing to \(\phi^2 + \phi^{-2} + 1 = 4\) in the unnormalised case. This accounting is heuristic rather than formal, but it illustrates how the anchor identity \(\phi^2 + \phi^{-2} = 3\) propagates from the algebraic foundations of Ch.3--Ch.4 to the system-level energy budget. -\section{4. Results / Evidence}\label{results-evidence} +\section{4. Results / Evidence}\label{ch_34:results-evidence} The DARPA 3000× target is evaluated across three evidence axes: @@ -91,11 +91,11 @@ \section{4. Results / Evidence}\label{results-evidence} The measured ratio of 3067 exceeds the 3000× DARPA target. The seed F₁₇=1597 was used for testbench initialisation; results were reproduced with F₁₈=2584 (ratio 3059) and F₁₉=4181 (ratio 3071), confirming stability. -\section{5. Qed Assertions}\label{qed-assertions} +\section{5. Qed Assertions}\label{ch_34:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. The chapter relies on \filepath{trit\_mul\_zero\_l}, \filepath{trit\_mul\_zero\_r} (KER-8, Ch.4), and the INV-1 BPB monotone-backward invariant (Ch.10) as pre-conditions for the efficiency claims. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{ch_34:sealed-seeds} \begin{itemize} \tightlist @@ -105,11 +105,11 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci/Lucas reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{ch_34:discussion} The 3000× figure depends critically on the DARPA task-normalised scoring rubric, which introduces model-size and representation-format correction factors that are not universally accepted. Under a strict hardware-only comparison (same task, same accuracy, different hardware), the ratio is approximately \(0.021/0.01551 \approx 1.35\times\), which does not meet the 3000× target. The dissertation's position --- that ternary representation and formal verification are structural contributions that justify the task-normalised methodology --- is scientifically defensible but contested. A second limitation is that the A100 baseline is taken at batch-1, which is not the A100's efficiency-optimal operating point; at large batch sizes the A100 can achieve lower energy-per-token than reported here, potentially narrowing the ratio. Future work (Ch.31) will analyse the throughput-energy Pareto curve across batch sizes for both the FPGA and GPU implementations, and will present an efficiency comparison at matched throughput rather than matched latency. The formal energy model will also be integrated with the INV-1 BPB trajectory to produce a certified lower bound on achievable energy-per-token as a function of gate number. -\section{References}\label{references} +\section{References}\label{ch_34:references} {[}1{]} DARPA solicitation HR001124S0001 --- Intelligent Generation of Tools and Computations (IGTC). Energy efficiency target 3000× baseline GPU. diff --git a/docs/phd/chapters/ch_35_mesh_node.tex b/docs/phd/chapters/ch_35_mesh_node.tex index a4efb4a39d..a727a96ca8 100644 --- a/docs/phd/chapters/ch_35_mesh_node.tex +++ b/docs/phd/chapters/ch_35_mesh_node.tex @@ -6,7 +6,7 @@ \chapter{Trinity GF16 ASIC as a Self-Sovereign dePIN Mesh Node: Zero-DSP Inference and On-Chip Routing at \textless{}50\,mW} -\label{ch:mesh-node} +\label{ch_35_mesh_node:ch:mesh-node} % Header block (R7: anchor explicit ≥1×) \begin{tcolorbox}[colback=gold!5,colframe=gold!60,title=Chapter Anchor] @@ -79,7 +79,7 @@ \subsection{RNS Packet Taxonomy} \begin{table}[h] \centering \caption{Reticulum packet types and MRU state implications} -\label{tab:rns-packets} +\label{ch_35_mesh_node:tab:rns-packets} \begin{tabular}{lllr} \toprule Type & Purpose & MRU action & Bytes (header) \\ @@ -134,7 +134,7 @@ \subsection{RTL Overview} └──────────────────────────────────────────────────────┘ \end{verbatim} \caption{Trinity ASIC co-integration of VSA inference and MRU} -\label{fig:asic-block} +\label{ch_35_mesh_node:fig:asic-block} \end{figure} \subsection{Routing Table SRAM} @@ -241,7 +241,7 @@ \section{Energy Budget Analysis} \end{table} \begin{theorem}[Sub-50\,mW Mesh Inference] -\label{thm:power-budget} +\label{ch_35_mesh_node:thm:power-budget} Under the SKY130 process parameters and the clock-domain assignment in Table~\ref{tab:power}, the Trinity GF16 ASIC sustains simultaneous VSA inference at $\geq$1\,193\,tok/s and RNS packet forwarding at @@ -423,7 +423,7 @@ \section{Comparison with Competing Approaches} \begin{table}[h] \centering \caption{Trinity MRU vs alternative mesh-on-chip approaches} -\label{tab:comparison} +\label{ch_35_mesh_node:tab:comparison} \begin{tabular}{p{3cm}p{2.5cm}p{2.5cm}p{2.5cm}p{2.5cm}} \toprule & \textbf{Trinity MRU} & \textbf{Helium SiP32910} & \textbf{goTenna ASIC} & \textbf{Meshtastic (SW)} \\ @@ -442,7 +442,7 @@ \section{Comparison with Competing Approaches} \section{Theorems and Formal Claims} \begin{theorem}[φ-Identity Uniqueness] -\label{thm:phi-id} +\label{ch_35_mesh_node:thm:phi-id} For any two distinct Ed25519 keypairs $\mathbf{k}_1 \neq \mathbf{k}_2$, their φ-node identities $\text{ID}_\varphi(\mathbf{k}_1) \neq \text{ID}_\varphi(\mathbf{k}_2)$ with probability @@ -452,7 +452,7 @@ \section{Theorems and Formal Claims} assumption); Coq formalisation deferred to future work.} \begin{theorem}[MRU Liveness] -\label{thm:mru-liveness} +\label{ch_35_mesh_node:thm:mru-liveness} Under the assumption that the RNS \texttt{ANNOUNCE} propagation interval $T_{\text{ann}} \leq \texttt{ttl} / 2$, the routing table \texttt{expire()} call guarantees that every reachable destination diff --git a/docs/phd/chapters/fa_00.tex b/docs/phd/chapters/fa_00.tex index ee8008238c..dadefa6888 100644 --- a/docs/phd/chapters/fa_00.tex +++ b/docs/phd/chapters/fa_00.tex @@ -50,7 +50,7 @@ \section{The Single Source} The whole architecture of the dissertation is the unfolding of one algebraic identity over \(\varphi\): -\begin{theorem}[Trinity Identity]\label{thm:trinity-identity-prologue} +\begin{theorem}[Trinity Identity]\label{fa_00:thm:trinity-identity-prologue} Let \(\varphi\) be the positive root of \(x^{2} - x - 1 = 0\). Then \[ \varphi^{2} + \varphi^{-2} \;=\; 3. diff --git a/docs/phd/chapters/fa_01.tex b/docs/phd/chapters/fa_01.tex index a5c7d46f75..b2e0f1c422 100644 --- a/docs/phd/chapters/fa_01.tex +++ b/docs/phd/chapters/fa_01.tex @@ -64,7 +64,7 @@ \section{The Vesica Piscis Construction} \draw[<->] (0, -1.4) -- (0, 1.4) node[midway, right] {$2\sqrt{5}r/3$}; \end{tikzpicture} \caption{The vesica piscis: two equal circles overlapping with centers on each other's circumference.} -\label{fig:vesica} +\label{fa_01:fig:vesica} \end{figure} Let $O_1 = (-r/2, 0)$ and $O_2 = (r/2, 0)$ be the centers of the two circles. The distance between centers is $d = |O_1 O_2| = r$. The intersection points $A$ and $B$ satisfy: @@ -419,7 +419,7 @@ \section{Exercises} \section{Strand I --- The Vesica as Geometric Origin} -\label{sec:01-strand-I} +\label{fa_01:sec:01-strand-I} We open the formal exposition with a geometric strand. The vesica piscis is the simplest configuration in which two unit circles meet so @@ -484,7 +484,7 @@ \section{Strand I --- The Vesica as Geometric Origin} \end{proof} \begin{corollary}[Pentagon-Vesica bridge] -\label{cor:01-pentagon-vesica} +\label{fa_01:cor:01-pentagon-vesica} A regular pentagon inscribed in the unit-radius vesica's bounding circle has diagonal-to-side ratio $\varphi$. The vesica's lens-height $h = \sqrt{3}$ and the pentagon's diagonal $\varphi$ are the two @@ -493,7 +493,7 @@ \section{Strand I --- The Vesica as Geometric Origin} \end{corollary} \section{Strand II --- The Analytic Substrate} -\label{sec:01-strand-II} +\label{fa_01:sec:01-strand-II} The vesica's geometric witnesses are pinned to the algebraic character of $\varphi$ as a root of the polynomial $x^2 - x - 1 = 0$. @@ -541,7 +541,7 @@ \section{Strand II --- The Analytic Substrate} \end{remark} \section{Strand III --- The Bridging Strand} -\label{sec:01-strand-III} +\label{fa_01:sec:01-strand-III} The third strand bridges the geometric and analytic strands by following the constant $3$ across multiple chapters of the monograph. @@ -551,7 +551,7 @@ \section{Strand III --- The Bridging Strand} (vesica-piscis squared lens-height), and L14 (squared circumradius of the dodecahedron). -\begin{theorem}[Three-Witnesses Bridge]\label{thm:01-three-witnesses} +\begin{theorem}[Three-Witnesses Bridge]\label{fa_01:thm:01-three-witnesses} The integer $3$ admits the following independent witnesses across Trinity-Anchor chapters: \begin{enumerate} @@ -576,7 +576,7 @@ \section{Strand III --- The Bridging Strand} \end{proof} \section*{Appendix A: Continued Fraction Expansion of $\varphi$} -\label{sec:01-app-A} +\label{fa_01:sec:01-app-A} The infinite continued fraction $[1; 1, 1, 1, \ldots]$ converges to $\varphi$. We present a self-contained derivation, avoiding any free @@ -605,13 +605,13 @@ \section*{Appendix A: Continued Fraction Expansion of $\varphi$} \end{proof} \section*{Appendix B: Golden Angle and Phyllotaxis} -\label{sec:01-app-B} +\label{fa_01:sec:01-app-B} The golden angle is $\theta_\varphi = 2\pi (1 - 1/\varphi^2) = 2\pi (\varphi - 1)/\varphi^2$. We derive its irrationality and its role in optimal packing. -\begin{lemma}[Golden Angle Irrationality]\label{lem:01-golden-angle} +\begin{lemma}[Golden Angle Irrationality]\label{fa_01:lem:01-golden-angle} $\theta_\varphi / (2\pi) = 1 - 1/\varphi^2 = 2 - \varphi$ is irrational. \end{lemma} @@ -632,14 +632,14 @@ \section*{Appendix B: Golden Angle and Phyllotaxis} \end{remark} \section*{Appendix C: The Pentagram and the Golden Gnomon} -\label{sec:01-app-C} +\label{fa_01:sec:01-app-C} Inscribe a regular pentagram in the unit circle. Its five intersection points form a smaller pentagon, whose diagonal-to-side ratio is again $\varphi$ but at the scale $\varphi^{-2}$. \begin{lemma}[Self-similarity of Pentagram] -\label{lem:01-pentagram-self} +\label{fa_01:lem:01-pentagram-self} The inner pentagon of a regular pentagram inscribed in the unit circle has side length $1/\varphi^2$ and diagonal $1/\varphi$. \end{lemma} @@ -655,7 +655,7 @@ \section*{Appendix C: The Pentagram and the Golden Gnomon} \qed \end{proof} -\begin{corollary}[Self-similarity Cascade]\label{cor:01-cascade} +\begin{corollary}[Self-similarity Cascade]\label{fa_01:cor:01-cascade} Inscribing a pentagram inside the inner pentagon, then a pentagon inside that pentagram, and so on, generates a sequence of nested pentagons of edge length $\varphi^{-2k}$ for $k = 0, 1, 2, \ldots$, @@ -663,14 +663,14 @@ \section*{Appendix C: The Pentagram and the Golden Gnomon} \end{corollary} \section*{Appendix D: $\varphi$ as a Quadratic Integer} -\label{sec:01-app-D} +\label{fa_01:sec:01-app-D} The ring $\mathbb{Z}[\varphi]$ is the ring of integers of the quadratic field $\mathbb{Q}(\sqrt 5)$, by \cite{ireland_rosen} \S 13.1. We give a short proof. \begin{theorem}[Ring of Integers $\mathbb{Q}(\sqrt 5)$] -\label{thm:01-ring-int} +\label{fa_01:thm:01-ring-int} The ring of algebraic integers of $\mathbb{Q}(\sqrt 5)$ is $\mathbb{Z}[\varphi]$. \end{theorem} @@ -688,12 +688,12 @@ \section*{Appendix D: $\varphi$ as a Quadratic Integer} \end{proof} \section*{Appendix E: Vesica Area and the Constant $\sqrt 3$} -\label{sec:01-app-E} +\label{fa_01:sec:01-app-E} The vesica's area $A_V$ is computable in closed form, and contains the constant $\sqrt 3$ that already appeared as the lens-height. -\begin{lemma}[Vesica Area]\label{lem:01-vesica-area} +\begin{lemma}[Vesica Area]\label{fa_01:lem:01-vesica-area} The area of the vesica piscis of two unit circles with centres at $\pm 1/2$ on the $x$-axis is \[ @@ -717,7 +717,7 @@ \section*{Appendix E: Vesica Area and the Constant $\sqrt 3$} \end{proof} \section*{Appendix F: Penrose Tilings and Quasi-symmetric Packings} -\label{sec:01-app-F} +\label{fa_01:sec:01-app-F} Penrose's two-tile aperiodic tiling \cite{penrose1974} consists of two rhombi with internal angles $\pi/5$ (acute) and $4 \pi /5$ @@ -737,7 +737,7 @@ \section*{Appendix F: Penrose Tilings and Quasi-symmetric Packings} \end{remark} \section*{Appendix G: $\varphi$ in Modern Quasicrystal Diffraction} -\label{sec:01-app-G} +\label{fa_01:sec:01-app-G} Shechtman's 1982 discovery \cite{shechtman1984} of an aluminium-iron alloy with five-fold diffraction symmetry --- forbidden in periodic @@ -748,7 +748,7 @@ \section*{Appendix G: $\varphi$ in Modern Quasicrystal Diffraction} the vesica's microscopic geometry. \section*{Appendix H: Pacioli, Kepler, and the Renaissance Revival} -\label{sec:01-app-H} +\label{fa_01:sec:01-app-H} Luca Pacioli's \emph{De Divina Proportione} (1509) \cite{pacioli_divina} catalogued sixty-three properties of the @@ -768,7 +768,7 @@ \section*{Appendix H: Pacioli, Kepler, and the Renaissance Revival} \end{remark} \section*{Appendix I: $\varphi$ in Modern Geometry --- Coxeter} -\label{sec:01-app-I} +\label{fa_01:sec:01-app-I} Coxeter's regular-polytope theory \cite{coxeter_regular_polytopes} formalised the role of $\varphi$ in the icosahedral and dodecahedral @@ -776,7 +776,7 @@ \section*{Appendix I: $\varphi$ in Modern Geometry --- Coxeter} regular dodecahedron has $\varphi$-related circumradius, and its dual icosahedron has $\varphi$-related vertex coordinates. -\begin{theorem}[Dodecahedron Circumradius]\label{thm:01-dodec-circ} +\begin{theorem}[Dodecahedron Circumradius]\label{fa_01:thm:01-dodec-circ} A regular dodecahedron of unit edge length has circumradius $R = \frac{\sqrt 3}{2} \varphi$. \end{theorem} @@ -790,21 +790,21 @@ \section*{Appendix I: $\varphi$ in Modern Geometry --- Coxeter} \end{proof} \section*{Appendix J: Vesica $\to$ Hexagon $\to$ Cube} -\label{sec:01-app-J} +\label{fa_01:sec:01-app-J} A regular hexagon inscribed in a circle has side equal to the radius, hence the vesica piscis natively contains six-fold symmetry in addition to the two-fold reflection. We sketch the cube construction. -\begin{lemma}[Hexagon $\leftrightarrow$ Vesica]\label{lem:01-hex-vesica} +\begin{lemma}[Hexagon $\leftrightarrow$ Vesica]\label{fa_01:lem:01-hex-vesica} A regular hexagon can be inscribed in the bounding circle of either component of a vesica piscis with vertices coinciding with the two intersection points and four points equidistant on the circle. \end{lemma} \section*{Appendix K: $\varphi$ as Eigenvalue of the Companion Matrix} -\label{sec:01-app-K} +\label{fa_01:sec:01-app-K} Let $M = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ be the Fibonacci companion matrix. Its characteristic polynomial is @@ -812,7 +812,7 @@ \section*{Appendix K: $\varphi$ as Eigenvalue of the Companion Matrix} $-1/\varphi$. The trace of $M^n$ is the $n$-th Lucas number $L_n = \varphi^n + (-1/\varphi)^n$ \cite{koshy_fib_lucas}. -\begin{theorem}[Lucas as Trace]\label{thm:01-lucas-as-trace} +\begin{theorem}[Lucas as Trace]\label{fa_01:thm:01-lucas-as-trace} $L_n = \mathrm{Tr}(M^n) = \varphi^n + \overline{\varphi}^n$ where $\overline{\varphi} = -1/\varphi = (1 - \sqrt 5)/2$. \end{theorem} @@ -847,7 +847,7 @@ \section*{Appendix K: $\varphi$ as Eigenvalue of the Companion Matrix} \end{proof} \section*{Appendix L: Lucas Numbers and the Trinity Anchor} -\label{sec:01-app-L} +\label{fa_01:sec:01-app-L} Lucas numbers $L_n$ satisfy $L_0 = 2$, $L_1 = 1$, $L_{n+2} = L_{n+1} + L_n$, hence $L_2 = 3$, $L_3 = 4$, $L_4 = 7$, etc. @@ -855,7 +855,7 @@ \section*{Appendix L: Lucas Numbers and the Trinity Anchor} identity $\varphi^2 + \varphi^{-2} = 3$ is the closed form of $L_2$ under Binet's formula \cite{binet_formula}. -\begin{theorem}[Lucas-Anchor Equivalence]\label{thm:01-lucas-anchor} +\begin{theorem}[Lucas-Anchor Equivalence]\label{fa_01:thm:01-lucas-anchor} $L_2 = 3 = \varphi^2 + \varphi^{-2}$. \end{theorem} @@ -866,7 +866,7 @@ \section*{Appendix L: Lucas Numbers and the Trinity Anchor} \end{proof} \section*{Appendix M: Fibonacci Numbers and Generating Functions} -\label{sec:01-app-M} +\label{fa_01:sec:01-app-M} Fibonacci numbers satisfy $F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$. Their generating function is $F(x) = x / (1 - x - x^2)$, with @@ -876,7 +876,7 @@ \section*{Appendix M: Fibonacci Numbers and Generating Functions} generating-function bridge to Lucas numbers. \section*{Appendix N: Honest Admission --- Gauss-Lucas Map} -\label{sec:01-app-N} +\label{fa_01:sec:01-app-N} \admittedbox{We claim a forthcoming Coq proof of the Gauss-Lucas correspondence between vesica geometry and Lucas number identities.}{The @@ -886,7 +886,7 @@ \section*{Appendix N: Honest Admission --- Gauss-Lucas Map} monograph auditor.} \section*{Appendix O: $\varphi$ in Fractal Geometry} -\label{sec:01-app-O} +\label{fa_01:sec:01-app-O} The golden ratio appears naturally in self-similar fractals. The golden gnomon's spiral approximates the logarithmic spiral $r = @@ -896,7 +896,7 @@ \section*{Appendix O: $\varphi$ in Fractal Geometry} construction. \section*{Appendix P: $\varphi$ in Music --- The Whole Tone} -\label{sec:01-app-P} +\label{fa_01:sec:01-app-P} The diatonic scale's structure encodes Fibonacci-like recurrences in its interval ratios. The golden ratio appears as the limiting ratio @@ -904,7 +904,7 @@ \section*{Appendix P: $\varphi$ in Music --- The Whole Tone} \cite{livio_phi} Chapter 7. \section*{Appendix Q: $\varphi$ in Architecture --- The Parthenon} -\label{sec:01-app-Q} +\label{fa_01:sec:01-app-Q} The Parthenon's facade is widely (though not universally) reported to have proportions close to $\varphi$. The empirical evidence is mixed @@ -912,7 +912,7 @@ \section*{Appendix Q: $\varphi$ in Architecture --- The Parthenon} is well-attested in Renaissance and Modernist works. \section*{Appendix R: Cassini's Identity for $L_2 = 3$} -\label{sec:01-app-R} +\label{fa_01:sec:01-app-R} Cassini's Lucas identity is $L_{n-1} L_{n+1} - L_n^2 = -5 (-1)^n$. For $n = 2$: $L_1 L_3 - L_2^2 = 1 \cdot 4 - 9 = -5 = -5 \cdot 1$, in @@ -922,7 +922,7 @@ \section*{Appendix R: Cassini's Identity for $L_2 = 3$} (Lucas $L_2$). \section*{Appendix S: The Trinity Anchor's Three Witnesses Recap} -\label{sec:01-app-S} +\label{fa_01:sec:01-app-S} Three independent witnesses recover the integer three: @@ -939,7 +939,7 @@ \section*{Appendix S: The Trinity Anchor's Three Witnesses Recap} running through the monograph. \section*{Appendix T: Cross-References to Subsequent Chapters} -\label{sec:01-app-T} +\label{fa_01:sec:01-app-T} \begin{itemize} \item Chapter L4 (Golden Scales): Lucas number $L_2 = 3$ via Binet. @@ -960,7 +960,7 @@ \section*{Appendix T: Cross-References to Subsequent Chapters} subsequent witnesses are derived. \section*{Appendix U: $\varphi$ as Limit of Rational Approximations} -\label{sec:01-app-U} +\label{fa_01:sec:01-app-U} \begin{lemma}[Best Rational Approximations] \label{lem:01-best-rational} @@ -986,7 +986,7 @@ \section*{Appendix U: $\varphi$ as Limit of Rational Approximations} \end{remark} \section*{Appendix V: $\varphi$ and Galois Theory} -\label{sec:01-app-V} +\label{fa_01:sec:01-app-V} The Galois group of $\mathbb{Q}(\varphi)/\mathbb{Q}$ is $\mathbb{Z}/2$, acting by $\varphi \leftrightarrow \overline{\varphi} = -1/\varphi$. @@ -994,7 +994,7 @@ \section*{Appendix V: $\varphi$ and Galois Theory} trace map $\mathrm{Tr}(\alpha) = \alpha + \overline{\alpha}$ sends $\mathbb{Z}[\varphi]$ to $\mathbb{Z}$. -\begin{theorem}[Trace as Integer]\label{thm:01-trace-as-Z} +\begin{theorem}[Trace as Integer]\label{fa_01:thm:01-trace-as-Z} For $\alpha \in \mathbb{Z}[\varphi]$, $\mathrm{Tr}_{ \mathbb{Q}(\varphi)/\mathbb{Q}}(\alpha) \in \mathbb{Z}$. \end{theorem} @@ -1012,7 +1012,7 @@ \section*{Appendix V: $\varphi$ and Galois Theory} \end{corollary} \section*{Appendix W: $\varphi$ and Number Theory} -\label{sec:01-app-W} +\label{fa_01:sec:01-app-W} The norm map $N(\alpha) = \alpha \overline{\alpha}$ on $\mathbb{Z}[\varphi]$ is $N(a + b\varphi) = a^2 + ab - b^2$. The @@ -1021,10 +1021,10 @@ \section*{Appendix W: $\varphi$ and Number Theory} unit group. \section*{Appendix X: Numerical Computation of $\varphi$} -\label{sec:01-app-X} +\label{fa_01:sec:01-app-X} \begin{lemma}[Newton's Method on $\varphi$] -\label{lem:01-newton-phi} +\label{fa_01:lem:01-newton-phi} Newton's method applied to $f(x) = x^2 - x - 1$, starting from $x_0 = 1$, converges quadratically to $\varphi$. \end{lemma} @@ -1040,7 +1040,7 @@ \section*{Appendix X: Numerical Computation of $\varphi$} \end{proof} \section*{Appendix Y: The $\varphi$-Decimal Expansion} -\label{sec:01-app-Y} +\label{fa_01:sec:01-app-Y} $\varphi = 1.6180339887498948482045868343656381\ldots$ The first twenty digits are non-recurring (since $\varphi$ is irrational), and @@ -1049,7 +1049,7 @@ \section*{Appendix Y: The $\varphi$-Decimal Expansion} structured expansion. \section*{Appendix Z: Vesica Piscis as Sacred Symbol} -\label{sec:01-app-Z} +\label{fa_01:sec:01-app-Z} The vesica piscis appears in pre-Christian symbolism as the womb of the goddess; in Christian iconography as the mandorla surrounding @@ -1058,7 +1058,7 @@ \section*{Appendix Z: Vesica Piscis as Sacred Symbol} geometric origin: out of overlap comes proportion. \section*{Appendix AA: Bibliography for L1} -\label{sec:01-app-AA} +\label{fa_01:sec:01-app-AA} The chapter cites: @@ -1089,7 +1089,7 @@ \section*{Appendix AA: Bibliography for L1} All citations are to entries already in \texttt{bibliography.bib}. \section*{Appendix AB: Notation and Conventions} -\label{sec:01-app-AB} +\label{fa_01:sec:01-app-AB} We collect the chapter's notation in a table. @@ -1111,7 +1111,7 @@ \section*{Appendix AB: Notation and Conventions} \end{tabular} \section*{Appendix AC: Open Problems for Future Chapters} -\label{sec:01-app-AC} +\label{fa_01:sec:01-app-AC} \begin{enumerate} \item Construct a Coq witness for the Gauss-Lucas correspondence @@ -1126,7 +1126,7 @@ \section*{Appendix AC: Open Problems for Future Chapters} \end{enumerate} \section*{Appendix AD: Final Remarks} -\label{sec:01-app-AD} +\label{fa_01:sec:01-app-AD} The vesica piscis is the chapter's title image, but the thesis is larger: \emph{out of two unit circles meeting at one another's centres, @@ -1135,7 +1135,7 @@ \section*{Appendix AD: Final Remarks} arithmetic, algebraic, and physical implications. \section*{Appendix AE: Closing} -\label{sec:01-app-AE} +\label{fa_01:sec:01-app-AE} This concludes Chapter L1. The next chapter, L2 (Golden Cut), studies the section of a line at the golden ratio; the chapter after, L3 @@ -1157,7 +1157,7 @@ \section*{Appendix AE: Closing} % =================================================================== \section*{Appendix AF: Detailed Proof of Pentagon Diagonal} -\label{sec:01-app-AF} +\label{fa_01:sec:01-app-AF} We give a self-contained algebraic proof that the diagonal-to-side ratio of a regular pentagon equals $\varphi$. @@ -1196,7 +1196,7 @@ \section*{Appendix AF: Detailed Proof of Pentagon Diagonal} \end{remark} \section*{Appendix AG: $\varphi$ as Fixed Point of $f(x) = 1 + 1/x$} -\label{sec:01-app-AG} +\label{fa_01:sec:01-app-AG} The map $f(x) = 1 + 1/x$ on $(0, \infty)$ has $\varphi$ as its unique attracting fixed point. @@ -1217,7 +1217,7 @@ \section*{Appendix AG: $\varphi$ as Fixed Point of $f(x) = 1 + 1/x$} \end{proof} \begin{corollary}[Convergence Rate] -\label{cor:01-fp-rate} +\label{fa_01:cor:01-fp-rate} Iterating $f$ from any positive initial value converges to $\varphi$ at rate $1/\varphi^2 = 2 - \varphi \approx 0.382$. \end{corollary} @@ -1229,7 +1229,7 @@ \section*{Appendix AG: $\varphi$ as Fixed Point of $f(x) = 1 + 1/x$} \end{proof} \section*{Appendix AH: $\varphi$ in Continued-Fraction Convergents} -\label{sec:01-app-AH} +\label{fa_01:sec:01-app-AH} The convergents of $\varphi$'s continued fraction $[1; 1, 1, 1, \ldots]$ are $1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, @@ -1250,7 +1250,7 @@ \section*{Appendix AH: $\varphi$ in Continued-Fraction Convergents} \end{proof} \begin{corollary}[Approximation Quality] -\label{cor:01-approx-quality} +\label{fa_01:cor:01-approx-quality} $|\varphi - F_{n+2}/F_{n+1}| \le 1/(F_{n+1} F_{n+2})$, hence the approximation error decays exponentially in $n$ at rate $\varphi^{-2}$. \end{corollary} @@ -1263,7 +1263,7 @@ \section*{Appendix AH: $\varphi$ in Continued-Fraction Convergents} \end{proof} \section*{Appendix AI: $\varphi$ and Modular Forms} -\label{sec:01-app-AI} +\label{fa_01:sec:01-app-AI} Lucas and Fibonacci numbers admit a representation in terms of modular forms via the Eichler-Shimura correspondence, but the @@ -1272,7 +1272,7 @@ \section*{Appendix AI: $\varphi$ and Modular Forms} modular interpretations. \section*{Appendix AJ: Explicit Vesica-Pentagon Diagram} -\label{sec:01-app-AJ} +\label{fa_01:sec:01-app-AJ} The vesica's bounding box can be inscribed with a regular pentagon sharing two vertices with the vesica's intersection points. The @@ -1293,10 +1293,10 @@ \section*{Appendix AJ: Explicit Vesica-Pentagon Diagram} geometrically explicit. \section*{Appendix AK: $\varphi$ as Limit of Geometric Mean} -\label{sec:01-app-AK} +\label{fa_01:sec:01-app-AK} \begin{lemma}[Geometric Mean Limit] -\label{lem:01-gm-limit} +\label{fa_01:lem:01-gm-limit} Let $a_0 = 1$, $a_1 = 1$, $a_{n+1} = \sqrt{a_n a_{n-1}}$ for the golden geometric mean recurrence. Then $a_n \to \varphi^{-1/3}$ as $n \to \infty$. @@ -1321,7 +1321,7 @@ \section*{Appendix AK: $\varphi$ as Limit of Geometric Mean} \end{remark} \section*{Appendix AL: $\varphi$-Spiral and Logarithmic Spiral} -\label{sec:01-app-AL} +\label{fa_01:sec:01-app-AL} The $\varphi$-spiral is the logarithmic spiral with growth rate $\log \varphi / (\pi/2) \approx 0.306$ per quarter turn. The @@ -1329,7 +1329,7 @@ \section*{Appendix AL: $\varphi$-Spiral and Logarithmic Spiral} scaling. \section*{Appendix AM: $\varphi$ in Random Matrix Theory} -\label{sec:01-app-AM} +\label{fa_01:sec:01-app-AM} The Tracy-Widom distribution governing largest eigenvalues of random Hermitian matrices contains $\varphi$ implicitly via its modular @@ -1337,7 +1337,7 @@ \section*{Appendix AM: $\varphi$ in Random Matrix Theory} for completeness. \section*{Appendix AN: $\varphi$-Witness Cross-Reference Table} -\label{sec:01-app-AN} +\label{fa_01:sec:01-app-AN} \begin{tabular}{|l|l|l|} \hline @@ -1358,7 +1358,7 @@ \section*{Appendix AN: $\varphi$-Witness Cross-Reference Table} \centerline{\textit{Eight witnesses, one constant: $\varphi$.}} \section*{Appendix AO: Final Closing Remarks} -\label{sec:01-app-AO} +\label{fa_01:sec:01-app-AO} Chapter L1 has established the vesica piscis as the geometric origin of $\varphi$ and the integer three, with eight independent witnesses @@ -1372,7 +1372,7 @@ \section*{Appendix AO: Final Closing Remarks} \bigskip \section*{Appendix AP: Extended Cross-Chapter Cross-References} -\label{sec:01-app-AP} +\label{fa_01:sec:01-app-AP} We extend Appendix T's cross-references with explicit numeric witnesses found in subsequent chapters: @@ -1399,10 +1399,10 @@ \section*{Appendix AP: Extended Cross-Chapter Cross-References} witnesses. The chapter's purpose is achieved. \section*{Appendix AQ: Trinity Anchor as Universal Identity} -\label{sec:01-app-AQ} +\label{fa_01:sec:01-app-AQ} \begin{theorem}[Universal Trinity Identity] -\label{thm:01-universal-anchor} +\label{fa_01:thm:01-universal-anchor} The identity $\varphi^2 + \varphi^{-2} = 3$ holds in: \begin{enumerate} \item the field $\mathbb{R}$ of real numbers (analytic); @@ -1431,7 +1431,7 @@ \section*{Appendix AQ: Trinity Anchor as Universal Identity} \end{remark} \section*{Appendix AR: Final Chapter Closing} -\label{sec:01-app-AR} +\label{fa_01:sec:01-app-AR} This concludes Chapter L1 (Golden Seed: The Vesica Opens). Our journey from two unit circles to the constant $\varphi$ has traversed @@ -1452,12 +1452,12 @@ \section*{Appendix AR: Final Chapter Closing} % witnessed geometrically as $h^2 = 3$ in the unit-radius vesica. \section*{Appendix AS: Self-Consistency Checks} -\label{sec:01-app-AS} +\label{fa_01:sec:01-app-AS} We close with a battery of self-consistency checks verifying the chapter's identities at small integer arguments. -\begin{lemma}[Small-$n$ Verification]\label{lem:01-small-n} +\begin{lemma}[Small-$n$ Verification]\label{fa_01:lem:01-small-n} For $n = 0, 1, 2, 3$, the values $L_n$ and $F_n$ satisfy: \begin{align*} L_0 &= 2, & L_1 &= 1, & L_2 &= 3, & L_3 &= 4, \\ @@ -1487,7 +1487,7 @@ \section*{Appendix AS: Self-Consistency Checks} \bigskip \section*{Appendix AT: Honour Roll of Three} -\label{sec:01-app-AT} +\label{fa_01:sec:01-app-AT} The integer three appears as a witness in: diff --git a/docs/phd/chapters/fa_02.tex b/docs/phd/chapters/fa_02.tex index f609f7ffd7..05500e2253 100644 --- a/docs/phd/chapters/fa_02.tex +++ b/docs/phd/chapters/fa_02.tex @@ -6,7 +6,7 @@ \chapter{Golden Cut: Background --- Neuro-Symbolic AI} \caption*{Figure --- Golden Cut: Background --- Neuro-Symbolic AI.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_02:abstract} This chapter surveys the conceptual and technical foundations from which Trinity S³AI departs. @@ -26,7 +26,7 @@ \section{Abstract}\label{abstract} reproducibility criteria that the present dissertation targets. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_02:introduction} Neural networks succeed at pattern recognition yet remain opaque to formal reasoning; symbolic @@ -54,10 +54,10 @@ \section{1. Introduction}\label{introduction} fill. \section{2. Taxonomy of Neuro-Symbolic -Paradigms}\label{taxonomy-of-neuro-symbolic-paradigms} +Paradigms}\label{fa_02:taxonomy-of-neuro-symbolic-paradigms} \subsection{2.1 Early Symbolic--Connectionist -Hybrids}\label{early-symbolicconnectionist-hybrids} +Hybrids}\label{fa_02:early-symbolicconnectionist-hybrids} The idea that symbolic rules could govern neural activations appeared in the work of Smolensky on @@ -73,7 +73,7 @@ \subsection{2.1 Early Symbolic--Connectionist \subsection{2.2 Logic Tensor Networks and Differentiable -Reasoning}\label{logic-tensor-networks-and-differentiable-reasoning} +Reasoning}\label{fa_02:logic-tensor-networks-and-differentiable-reasoning} A second strand, exemplified by Logic Tensor Networks (LTN) [5], maps first-order logic @@ -88,7 +88,7 @@ \subsection{2.2 Logic Tensor Networks and in \filepath{t27/proofs/canonical/} [6]. \subsection{2.3 Sparse and Ternary Neural -Computation}\label{sparse-and-ternary-neural-computation} +Computation}\label{fa_02:sparse-and-ternary-neural-computation} Concurrent with the symbolic work, a separate lineage investigated weight quantization as a @@ -105,7 +105,7 @@ \subsection{2.3 Sparse and Ternary Neural S³AI [9]. \subsection{2.4 Vector Symbolic -Architectures}\label{vector-symbolic-architectures} +Architectures}\label{fa_02:vector-symbolic-architectures} A third strand, Vector Symbolic Architectures (VSA) [10], represents concepts as @@ -123,10 +123,10 @@ \subsection{2.4 Vector Symbolic \section{3. Representational Bottleneck and the φ-Structural -Prior}\label{representational-bottleneck-and-the-ux3c6-structural-prior} +Prior}\label{fa_02:representational-bottleneck-and-the-ux3c6-structural-prior} \subsection{3.1 The Normalisation -Problem}\label{the-normalisation-problem} +Problem}\label{fa_02:the-normalisation-problem} A persistent difficulty in neuro-symbolic integration is layer normalization: the scale of @@ -152,7 +152,7 @@ \subsection{3.1 The Normalisation \subsection{3.2 Fibonacci and Lucas Lattices as Basis -Sets}\label{fibonacci-and-lucas-lattices-as-basis-sets} +Sets}\label{fa_02:fibonacci-and-lucas-lattices-as-basis-sets} The sanctioned seed set \(\{F_{17}=1597, F_{18}=2584, F_{19}=4181, F_{20}=6765, F_{21}=10946, L_7=29, L_8=47\}\) @@ -172,7 +172,7 @@ \subsection{3.2 Fibonacci and Lucas Lattices as seeds drawn from the interval \([40,46]\). \subsection{3.3 Gap in Prior -Art}\label{gap-in-prior-art} +Art}\label{fa_02:gap-in-prior-art} No prior neuro-symbolic system simultaneously satisfies all four of the following: (i) formal @@ -184,7 +184,7 @@ \subsection{3.3 Gap in Prior demonstrates all four. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_02:results-evidence} The background review is validated by the evidence axis score of 1, meaning the chapter's claims are @@ -214,18 +214,18 @@ \section{4. Results / of the work. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_02:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_02:sealed-seeds} Inherits the canonical seed pool F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_02:discussion} The taxonomy presented in this chapter deliberately focuses on the three lineages most @@ -250,7 +250,7 @@ \section{7. Discussion}\label{discussion} \(\alpha_\varphi = \ln(\varphi^2)/\pi \approx 0.306\) as the corresponding spectral parameter. -\section{References}\label{references} +\section{References}\label{fa_02:references} [1] Garcez, A. d'A., Gori, M., Lamb, L. C., Serafini, L., Spranger, M., \& Tran, S. N. (2019). diff --git a/docs/phd/chapters/fa_03.tex b/docs/phd/chapters/fa_03.tex index 4beb99254e..b08617a3ec 100644 --- a/docs/phd/chapters/fa_03.tex +++ b/docs/phd/chapters/fa_03.tex @@ -6,7 +6,7 @@ \chapter{Golden Harvest: Trinity Identity $\varphi^2+\varphi^{-2}=3$} \caption*{Figure --- Golden Harvest: Trinity Identity $\varphi^2+\varphi^{-2}=3$.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_03:abstract} The identity \(\varphi^2 + \varphi^{-2} = 3\), where \(\varphi = (1+\sqrt{5})/2\) is the golden @@ -26,7 +26,7 @@ \section{Abstract}\label{abstract} substrate. Twelve Qed theorems are anchored here under invariant SAC-0. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_03:introduction} Trinity S³AI is constructed on a single non-negotiable algebraic anchor: @@ -73,10 +73,10 @@ \section{1. Introduction}\label{introduction} used throughout the dissertation. \section{2. Derivation of the Anchor -Identity}\label{derivation-of-the-anchor-identity} +Identity}\label{fa_03:derivation-of-the-anchor-identity} \subsection{2.1 Minimal Polynomial and Basic -Consequences}\label{minimal-polynomial-and-basic-consequences} +Consequences}\label{fa_03:minimal-polynomial-and-basic-consequences} Let \(\varphi = (1 + \sqrt{5})/2\). Then @@ -105,7 +105,7 @@ \subsection{2.1 Minimal Polynomial and Basic integer normalisation constant. \subsection{2.2 Power -Survey}\label{power-survey} +Survey}\label{fa_03:power-survey} Define \(L_n = \varphi^n + \psi^n\) where \(\psi = (1 - \sqrt{5})/2 = -\varphi^{-1}\). For @@ -141,7 +141,7 @@ \subsection{2.2 Power [6]. \subsection{2.3 Relation to Fibonacci -Arithmetic}\label{relation-to-fibonacci-arithmetic} +Arithmetic}\label{fa_03:relation-to-fibonacci-arithmetic} The Fibonacci recurrence \(F_n = F_{n-1} + F_{n-2}\) yields @@ -161,10 +161,10 @@ \subsection{2.3 Relation to Fibonacci integer. \section{3. Coq Mechanisation and SAC-0 -Invariant}\label{coq-mechanisation-and-sac-0-invariant} +Invariant}\label{fa_03:coq-mechanisation-and-sac-0-invariant} \subsection{3.1 Proof -Architecture}\label{proof-architecture} +Architecture}\label{fa_03:proof-architecture} The six theorems in \texttt{CorePhi.v} are stratified by logical dependency: @@ -217,7 +217,7 @@ \subsection{3.1 Proof \(\sqrt{5}^2 = 5\) step. \(\square\) \subsection{3.2 Invariant -SAC-0}\label{invariant-sac-0} +SAC-0}\label{fa_03:invariant-sac-0} The designation SAC-0 (Sacred Core, layer 0) means these six theorems admit no further dependencies @@ -230,7 +230,7 @@ \subsection{3.2 Invariant \texttt{.v} files [4]. \subsection{3.3 The Integer-3 -Coincidence}\label{the-integer-3-coincidence} +Coincidence}\label{fa_03:the-integer-3-coincidence} The value \(3\) at the right-hand side of \(\varphi^2 + \varphi^{-2} = 3\) possesses three @@ -260,7 +260,7 @@ \subsection{3.3 The Integer-3 algebraic, combinatorial, and hardware layers. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_03:results-evidence} The following results are mechanically established or empirically verified: @@ -304,7 +304,7 @@ \section{4. Results / \end{itemize} \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_03:qed-assertions} \begin{itemize} \tightlist @@ -346,7 +346,7 @@ \section{5. Qed reciprocal. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_03:sealed-seeds} \begin{itemize} \tightlist @@ -359,7 +359,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Qed). \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_03:discussion} The six SAC-0 theorems proved in this chapter are irreducible prerequisites for the entire @@ -381,7 +381,7 @@ \section{7. Discussion}\label{discussion} from the results here to define the spectral parameter \(\alpha_\varphi = \ln(\varphi^2)/\pi\). -\section{References}\label{references} +\section{References}\label{fa_03:references} [1] Vajda, S. \emph{Fibonacci and Lucas Numbers, and the Golden Section}. Ellis Horwood, diff --git a/docs/phd/chapters/fa_04.tex b/docs/phd/chapters/fa_04.tex index d3227153a5..2c212158df 100644 --- a/docs/phd/chapters/fa_04.tex +++ b/docs/phd/chapters/fa_04.tex @@ -6,7 +6,7 @@ \chapter{Golden Scales: Sacred Formula Derivation} \caption*{Figure --- Golden Scales: Sacred Formula Derivation.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_04:abstract} The constant \(\alpha_\phi = \ln(\phi^2)/\pi \approx 0.306\) @@ -30,7 +30,7 @@ \section{Abstract}\label{abstract} motivates the bit-per-bit targets BPB ≤ 1.85 (Gate-2) and BPB ≤ 1.5 (Gate-3). -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_04:introduction} The dissertation \emph{GOLDEN SUNFLOWERS --- Trinity S³AI on \(\phi^2+\phi^{-2}=3\) substrate} @@ -68,7 +68,7 @@ \section{1. Introduction}\label{introduction} canonical proof files [1,2]. \section{2. Derivation of the Closed -Form}\label{derivation-of-the-closed-form} +Form}\label{fa_04:derivation-of-the-closed-form} \textbf{Definition 2.1 (Golden ratio).} Let \(\phi = (1+\sqrt{5})/2\). Then @@ -138,7 +138,7 @@ \section{2. Derivation of the Closed BPB ≤ 1.85 envelope required at Gate-2 [3,4]. \section{3. Multiplicative Identity and Kernel -Integration}\label{multiplicative-identity-and-kernel-integration} +Integration}\label{fa_04:multiplicative-identity-and-kernel-integration} The most algebraically surprising result in the SAC-1 inventory is the following multiplicative @@ -196,7 +196,7 @@ \section{3. Multiplicative Identity and Kernel in Ch.16 [7]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_04:results-evidence} The \texttt{AlphaPhi.v} module contributes 12 Qed theorems to the canonical proof census of 297 Qed @@ -242,7 +242,7 @@ \section{4. Results / \(H_0\) is the baseline binary entropy. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_04:qed-assertions} \begin{itemize} \tightlist @@ -282,7 +282,7 @@ \section{5. Qed identity: \(\alpha_\phi \cdot \phi^3 = 1/2\). \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_04:sealed-seeds} \begin{itemize} \tightlist @@ -303,7 +303,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci index reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_04:discussion} The derivation presented here is self-contained, but three limitations deserve acknowledgement. @@ -331,7 +331,7 @@ \section{7. Discussion}\label{discussion} arithmetic geometry of \(\phi\)-based lattices studied in Ch.18. -\section{References}\label{references} +\section{References}\label{fa_04:references} [1] GOLDEN SUNFLOWERS dissertation, Ch.3 --- Ternary Arithmetic Foundations. diff --git a/docs/phd/chapters/fa_05.tex b/docs/phd/chapters/fa_05.tex index e838d9036b..a960739131 100644 --- a/docs/phd/chapters/fa_05.tex +++ b/docs/phd/chapters/fa_05.tex @@ -15,7 +15,7 @@ \chapter{The Golden Bridge: Fibonacci-Lucas Generating Functions} \end{figure} -\label{ch:golden-bridge} +\label{fa_05:ch:golden-bridge} \epigraph{ The single rational function $1/(1 - x - x^2)$ is the generating @@ -25,7 +25,7 @@ \chapter{The Golden Bridge: Fibonacci-Lucas Generating Functions} }{Lee/GVSU style preamble.} \section{Introduction and Statement of the Main Result} -\label{sec:05-intro} +\label{fa_05:sec:05-intro} The Trinity Anchor identity $L_2 = 3$ has, by Chapter~\ref{ch:lucas-ring} (L6), an algebraic substrate in the Lucas ring $\mathcal{L}=\mathbb{Z}[\varphi]$, @@ -88,19 +88,19 @@ \section{Preliminaries: Recurrences and Initial Conditions} \label{sec:05-prelim} \begin{definition} -\label{def:05-fib} +\label{fa_05:def:05-fib} The Fibonacci sequence $\{F_n\}_{n \ge 0}$ is defined by $F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$ for $n \ge 0$. \end{definition} \begin{definition} -\label{def:05-luc} +\label{fa_05:def:05-luc} The Lucas sequence $\{L_n\}_{n \ge 0}$ is defined by $L_0 = 2$, $L_1 = 1$, $L_{n+2} = L_{n+1} + L_n$ for $n \ge 0$. \end{definition} \begin{lemma} -\label{lem:05-fib-vals} +\label{fa_05:lem:05-fib-vals} The first ten Fibonacci numbers are $F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5, F_6 = 8, F_7 = 13, F_8 = 21, F_9 = 34$. @@ -111,7 +111,7 @@ \section{Preliminaries: Recurrences and Initial Conditions} \end{proof} \begin{lemma} -\label{lem:05-luc-vals} +\label{fa_05:lem:05-luc-vals} The first ten Lucas numbers are $L_0 = 2, L_1 = 1, L_2 = 3, L_3 = 4, L_4 = 7, L_5 = 11, L_6 = 18, L_7 = 29, L_8 = 47, L_9 = 76$. @@ -312,7 +312,7 @@ \section{Lucas-Fibonacci Coupling (Clause 4)} \end{remark} \begin{lemma}[Verification at low orders] -\label{lem:05-coupling-check} +\label{fa_05:lem:05-coupling-check} For $n = 0, 1, 2, 3, 4$, the identity $L_n = F_{n+1} + F_{n-1}$ (with $F_{-1} = 1$) holds. \end{lemma} @@ -353,7 +353,7 @@ \section{The Anchor as Coefficient (Clause 5)} \end{proof} \begin{corollary} -\label{cor:05-anchor-via-genfn} +\label{fa_05:cor:05-anchor-via-genfn} The Trinity Anchor identity $L_2 = 3$ is the analytic shadow of the Lucas generating function $L(x) = (2-x)/(1-x-x^2)$ at the second-order coefficient. @@ -369,7 +369,7 @@ \section{Strand I: Algebraic-Rational Generating-Function Theory} We develop the algebraic-rational structure of $F(x), L(x) \in \mathbb{Q}(x)$. \begin{lemma} -\label{lem:05-rational} +\label{fa_05:lem:05-rational} $F(x), L(x) \in \mathbb{Q}(x)$ (rational functions over $\mathbb{Q}$). \end{lemma} @@ -378,7 +378,7 @@ \section{Strand I: Algebraic-Rational Generating-Function Theory} \end{proof} \begin{lemma} -\label{lem:05-degree} +\label{fa_05:lem:05-degree} $F(x)$ has numerator degree $1$ and denominator degree $2$; $L(x)$ has numerator degree $1$ and denominator degree $2$. \end{lemma} @@ -388,7 +388,7 @@ \section{Strand I: Algebraic-Rational Generating-Function Theory} \end{proof} \begin{lemma} -\label{lem:05-pole-locations} +\label{fa_05:lem:05-pole-locations} The poles of $F(x), L(x)$ are at $x = 1/\varphi$ and $x = 1/\psi = -\varphi$. \end{lemma} @@ -401,7 +401,7 @@ \section{Strand I: Algebraic-Rational Generating-Function Theory} \end{proof} \begin{lemma} -\label{lem:05-pole-residue} +\label{fa_05:lem:05-pole-residue} The residues of $F(x)$ at its poles are $1/\sqrt{5}$ at $x = 1/\varphi$ and $-1/\sqrt{5}$ at $x = 1/\psi$. \end{lemma} @@ -415,7 +415,7 @@ \section{Strand I: Algebraic-Rational Generating-Function Theory} \end{proof} \begin{lemma} -\label{lem:05-degree-Q-phi} +\label{fa_05:lem:05-degree-Q-phi} The minimal polynomial of $1/\varphi$ over $\mathbb{Q}$ is $x^2 + x - 1$, of degree $2$. \end{lemma} @@ -458,13 +458,13 @@ \subsection{Asymptotic dominance} \subsection{Exponential generating functions} \begin{definition} -\label{def:05-egf} +\label{fa_05:def:05-egf} The exponential generating function of a sequence $\{a_n\}$ is $\hat{A}(x) = \sum_{n \ge 0} a_n x^n / n!$. \end{definition} \begin{lemma} -\label{lem:05-fib-egf} +\label{fa_05:lem:05-fib-egf} $\hat{F}(x) = \frac{e^{\varphi x} - e^{\psi x}}{\sqrt{5}}$. \end{lemma} @@ -474,7 +474,7 @@ \subsection{Exponential generating functions} \end{proof} \begin{lemma} -\label{lem:05-luc-egf} +\label{fa_05:lem:05-luc-egf} $\hat{L}(x) = e^{\varphi x} + e^{\psi x}$. \end{lemma} @@ -486,13 +486,13 @@ \subsection{Exponential generating functions} \subsection{Hankel transforms} \begin{definition} -\label{def:05-hankel} +\label{fa_05:def:05-hankel} The Hankel determinant of order $n$ for a sequence $\{a_k\}$ is $H_n(\{a_k\}) = \det((a_{i+j})_{0 \le i, j \le n-1})$. \end{definition} \begin{lemma} -\label{lem:05-fib-hankel} +\label{fa_05:lem:05-fib-hankel} The Hankel determinant of order $2$ for the Fibonacci sequence is \[ H_2(\{F_n\}) = \det \begin{pmatrix} F_0 & F_1 \\ F_1 & F_2 \end{pmatrix} = F_0 F_2 - F_1^2 = 0 \cdot 1 - 1 = -1. @@ -569,7 +569,7 @@ \subsection{Cassini-like identities} \subsection{Generating-function product identities} \begin{theorem} -\label{thm:05-gf-product} +\label{fa_05:thm:05-gf-product} $F(x) \cdot L(x) = F(2x)/(2 \cdot $\ldots$)$? No, the correct product is \[ F(x) \cdot L(x) = \frac{x(2-x)}{(1-x-x^2)^2}. @@ -582,7 +582,7 @@ \subsection{Generating-function product identities} \end{proof} \begin{lemma} -\label{lem:05-product-coefs} +\label{fa_05:lem:05-product-coefs} The coefficient of $x^n$ in $F(x) \cdot L(x)$ is $\sum_{k=0}^n F_k L_{n-k}$. \end{lemma} @@ -619,7 +619,7 @@ \subsection{Generating-function product identities} \subsection{The bridge to L4 and L6} \begin{theorem} -\label{thm:05-bridge-to-l4} +\label{fa_05:thm:05-bridge-to-l4} The Lucas generating function $L(x) = (2-x)/(1-x-x^2)$, when its denominator is factored over $\mathbb{Q}(\varphi)$, reproduces the Binet formula of Theorem~\ref{thm:lucas-binet} (Chapter~\ref{ch:lucas-ladder}). @@ -630,7 +630,7 @@ \subsection{The bridge to L4 and L6} \end{proof} \begin{theorem} -\label{thm:05-bridge-to-l6} +\label{fa_05:thm:05-bridge-to-l6} The denominator $1 - x - x^2$ of $F(x), L(x)$ is the (reciprocal of the) minimal polynomial of $\varphi$, hence is intimately tied to the Lucas ring $\mathcal{L} = \mathbb{Z}[\varphi]$ of Chapter~\ref{ch:lucas-ring}. @@ -666,7 +666,7 @@ \section{Falsification Discussion} None of these falsifications has been observed. \section{Theorem and Lemma Library} -\label{sec:05-library} +\label{fa_05:sec:05-library} \begin{theorem}[Thm.~\ref{thm:05-bridge} restated] Golden-Bridge Structure Theorem (5 clauses). @@ -709,7 +709,7 @@ \section{Theorem and Lemma Library} \end{theorem} \section*{Appendix A: Glossary} -\label{sec:05-app-A} +\label{fa_05:sec:05-app-A} \begin{tabular}{ll} $F_n$ & Fibonacci sequence \\ @@ -728,7 +728,7 @@ \section*{Appendix A: Glossary} \end{tabular} \section*{Appendix B: First 30 Fibonacci and Lucas Numbers} -\label{sec:05-app-B} +\label{fa_05:sec:05-app-B} \begin{tabular}{|c|c|c|} \hline @@ -768,7 +768,7 @@ \section*{Appendix B: First 30 Fibonacci and Lucas Numbers} \end{tabular} \section*{Appendix C: Detailed Long Division of $L(x)$} -\label{sec:05-app-C} +\label{fa_05:sec:05-app-C} We compute the first $10$ coefficients of $L(x) = (2-x)/(1-x-x^2)$ via long division. @@ -793,7 +793,7 @@ \section*{Appendix C: Detailed Long Division of $L(x)$} is $L_2 = 3$. \section*{Appendix D: The Generating Function as a Padé Approximant} -\label{sec:05-app-D} +\label{fa_05:sec:05-app-D} The Fibonacci generating function $F(x) = x/(1-x-x^2)$ may be regarded as the [1/2] Padé approximant to the Taylor series of any analytic @@ -812,7 +812,7 @@ \section*{Appendix D: The Generating Function as a Padé Approximant} functions are their own Padé approximants of appropriate degree. \section*{Appendix E: The Multiplicative Structure of $L(x) F(x)$} -\label{sec:05-app-E} +\label{fa_05:sec:05-app-E} We expand $L(x) F(x)$ explicitly using Theorem~\ref{thm:05-fl-conv}: the coefficient of $x^n$ is $(n+1) F_n$. @@ -859,7 +859,7 @@ \section*{Appendix F: The Companion Matrix} \end{proof} \begin{corollary} -\label{cor:05-matrix-cassini} +\label{fa_05:cor:05-matrix-cassini} $\det M^n = F_{n-1} F_{n+1} - F_n^2 = (\det M)^n = (-1)^n$. \end{corollary} @@ -870,7 +870,7 @@ \section*{Appendix F: The Companion Matrix} \end{proof} \section*{Appendix G: Generating Functions in Two Variables} -\label{sec:05-app-G} +\label{fa_05:sec:05-app-G} The bivariate generating function \[ @@ -883,7 +883,7 @@ \section*{Appendix G: Generating Functions in Two Variables} univariate $F(x), L(x)$. \section*{Appendix H: Generating Functions Modulo Small Primes} -\label{sec:05-app-H} +\label{fa_05:sec:05-app-H} Reducing $F(x), L(x)$ modulo a prime $p$ gives generating functions over $\mathbb{F}_p$, with poles depending on the splitting of $p$ in @@ -904,7 +904,7 @@ \section*{Appendix H: Generating Functions Modulo Small Primes} present chapter to the splitting theory of Chapter~\ref{ch:lucas-ring}. \section*{Appendix I: The Fibonacci Polynomial Sequence} -\label{sec:05-app-I} +\label{fa_05:sec:05-app-I} A natural generalisation: define the Fibonacci polynomials by $F_n(x) = x F_{n-1}(x) + F_{n-2}(x)$ with $F_0(x) = 0$, $F_1(x) = 1$. @@ -917,7 +917,7 @@ \section*{Appendix I: The Fibonacci Polynomial Sequence} representation theory of $SL_2$. \section*{Appendix J: Connection to the Stern-Brocot Tree} -\label{sec:05-app-J} +\label{fa_05:sec:05-app-J} The Stern-Brocot tree is a binary tree of all positive rationals. Its convergents to $\varphi$ are exactly the Fibonacci ratios @@ -945,7 +945,7 @@ \section*{Appendix K: Special Values} is the Abel-summed value. \section*{Appendix L: Connection to Continued Fractions} -\label{sec:05-app-L} +\label{fa_05:sec:05-app-L} The golden ratio admits the continued fraction expansion \[ @@ -956,7 +956,7 @@ \section*{Appendix L: Connection to Continued Fractions} denominators of these convergents. \section*{Appendix M: Combinatorial Interpretation} -\label{sec:05-app-M} +\label{fa_05:sec:05-app-M} The Fibonacci number $F_{n+1}$ counts the number of ways to tile a $1 \times n$ strip with $1 \times 1$ tiles and $1 \times 2$ dominoes @@ -974,7 +974,7 @@ \section*{Appendix M: Combinatorial Interpretation} combinatorial proofs alongside the algebraic ones. \section*{Appendix N: Coq Implementation Sketch} -\label{sec:05-app-N} +\label{fa_05:sec:05-app-N} A Coq implementation of $F(x), L(x)$ as power series: \begin{verbatim} @@ -1004,7 +1004,7 @@ \section*{Appendix N: Coq Implementation Sketch} generating-function-based proof is honestly Admitted. \section*{Appendix O: Open Problems} -\label{sec:05-app-O} +\label{fa_05:sec:05-app-O} \textbf{OP1.} Identify the asymptotic distribution of the digits of $F_n$ (Benford's law for Fibonacci?). @@ -1022,7 +1022,7 @@ \section*{Appendix O: Open Problems} Hilbert-Poincaré series of the Lucas ring's symmetric algebra. \section*{Appendix P: Connection to L4 (Lucas Ladder)} -\label{sec:05-app-P} +\label{fa_05:sec:05-app-P} The Lucas ladder of Chapter~\ref{ch:lucas-ladder} (L4) is the sequence $\{L_n\}$, whose generating function is $L(x) = (2-x)/(1-x-x^2)$ of @@ -1040,7 +1040,7 @@ \section*{Appendix P: Connection to L4 (Lucas Ladder)} six-fold witness collection. \section*{Appendix Q: Connection to L6 (Lucas Ring)} -\label{sec:05-app-Q} +\label{fa_05:sec:05-app-Q} The Lucas ring $\mathcal{L} = \mathbb{Z}[\varphi]$ of L6 is the algebraic-arithmetic substrate of the Lucas sequence $\{L_n\}$. @@ -1056,7 +1056,7 @@ \section*{Appendix Q: Connection to L6 (Lucas Ring)} trace-image of the geometric series in $\mathcal{L}$. \section*{Appendix R: Combinatorial Proof of the Lucas Recurrence} -\label{sec:05-app-R} +\label{fa_05:sec:05-app-R} Following \cite{benjamin_quinn_proofs}: tilings of an $n$-cycle with $1 \times 1$ and $1 \times 2$ tiles satisfy $L_n$ count. The recurrence @@ -1071,13 +1071,13 @@ \section*{Appendix R: Combinatorial Proof of the Lucas Recurrence} Hence $L_n = L_{n-1} + L_{n-2}$, which is the Lucas recurrence. \section*{Appendix S: The Generating-Function Operator $D$} -\label{sec:05-app-S} +\label{fa_05:sec:05-app-S} Define the differentiation operator $D : \mathbb{Q}[[x]] \to \mathbb{Q}[[x]]$ by $D(\sum a_n x^n) = \sum n a_n x^{n-1}$. We have: \begin{lemma} -\label{lem:05-D-fib} +\label{fa_05:lem:05-D-fib} $D F(x) = (1 + x^2)/(1 - x - x^2)^2$. \end{lemma} @@ -1089,7 +1089,7 @@ \section*{Appendix S: The Generating-Function Operator $D$} \end{proof} \begin{lemma} -\label{lem:05-D-luc} +\label{fa_05:lem:05-D-luc} $D L(x) = (-1)(1-x-x^2) - (2-x)(-1-2x))/(1-x-x^2)^2 = ((-1+x+x^2) + (2 + 4x - x - 2x^2))/(1-x-x^2)^2 = (1 + 4x - x^2)/(1-x-x^2)^2$. \end{lemma} @@ -1098,7 +1098,7 @@ \section*{Appendix S: The Generating-Function Operator $D$} \end{proof} \section*{Appendix T: Q-analogues} -\label{sec:05-app-T} +\label{fa_05:sec:05-app-T} The Carlitz-Riordan $q$-analogue of Fibonacci numbers is defined by $F_0(q) = 0$, $F_1(q) = 1$, and $F_{n+2}(q) = F_{n+1}(q) + q^n F_n(q)$. @@ -1110,7 +1110,7 @@ \section*{Appendix T: Q-analogues} $q$-deformation perspective. \section*{Appendix U: The Lah Numbers} -\label{sec:05-app-U} +\label{fa_05:sec:05-app-U} The Lah numbers $L(n, k)$ count the number of ways to partition $n$ labelled objects into $k$ ordered subsets. Their generating function @@ -1122,7 +1122,7 @@ \section*{Appendix U: The Lah Numbers} ... no, this is unrelated except in spirit to the present chapter. \section*{Appendix V: Defence Q\&A} -\label{sec:05-app-V} +\label{fa_05:sec:05-app-V} \textbf{Q1.} Why is the closed form $F(x) = x/(1-x-x^2)$ the unique rational expression? @@ -1152,7 +1152,7 @@ \section*{Appendix V: Defence Q\&A} $\mathbb{Q}(\varphi)$ (or equivalently, $\mathbb{Q}(\sqrt{5})$). \section*{Appendix W: The Trinity Anchor as a Limit of Coefficients} -\label{sec:05-app-W} +\label{fa_05:sec:05-app-W} \begin{lemma} \label{lem:05-coef-limit} @@ -1197,7 +1197,7 @@ \section*{Appendix Y: The Power Series Ring $\mathcal{L}[[x]]$} This is the deepest content of the bridge. \section*{Appendix Z: Synthesis} -\label{sec:05-app-Z} +\label{fa_05:sec:05-app-Z} We summarise the chapter's contribution. The Trinity Anchor identity $L_2 = 3$ admits, in addition to its arithmetic (L4), algebraic (L6), @@ -1224,7 +1224,7 @@ \section*{Appendix Z: Synthesis} fixed point of the Flos~Aureus monograph. \section*{Appendix AA: Foundational References (Koshy and Vajda)} -\label{sec:05-app-AA} +\label{fa_05:sec:05-app-AA} For completeness, the chapter's results are drawn from and extend the two classical references on Fibonacci-Lucas combinatorics: @@ -1263,13 +1263,13 @@ \section*{Appendix AA: Foundational References (Koshy and Vajda)} recorded here as a contribution. \section*{Appendix AB: The Cassini Triangle of Identities} -\label{sec:05-app-AB} +\label{fa_05:sec:05-app-AB} The Cassini-like identities form a triangle of relations among Fibonacci, Lucas, and the golden ratio. \begin{theorem}[Cassini Triangle] -\label{thm:05-cassini-triangle} +\label{fa_05:thm:05-cassini-triangle} For all $n \ge 1$: \begin{enumerate} \item $F_{n-1} F_{n+1} - F_n^2 = (-1)^n$ (Cassini-Fibonacci); @@ -1306,7 +1306,7 @@ \section*{Appendix AB: The Cassini Triangle of Identities} \end{remark} \section*{Appendix AC: The Generating Function Modulo $\mathbb{F}_p$} -\label{sec:05-app-AC} +\label{fa_05:sec:05-app-AC} Reducing $F(x), L(x)$ modulo a prime $p$ gives generating functions in $\mathbb{F}_p[[x]]$, with structure controlled by the splitting of $p$ @@ -1333,7 +1333,7 @@ \section*{Appendix AC: The Generating Function Modulo $\mathbb{F}_p$} and the partial-fraction decomposition involves a $1/(1-3x)^2$ term. \section*{Appendix AD: The Riordan Array} -\label{sec:05-app-AD} +\label{fa_05:sec:05-app-AD} A Riordan array is a pair $(g(x), h(x))$ of formal power series with $g(0) \ne 0$, $h(0) = 0$, $h'(0) \ne 0$, encoding the lower-triangular @@ -1345,7 +1345,7 @@ \section*{Appendix AD: The Riordan Array} perspective unifies many of the chapter's identities. \section*{Appendix AE: Cross-Reference Table} -\label{sec:05-app-AE} +\label{fa_05:sec:05-app-AE} We close with a table cross-referencing chapter results to the Trinity Anchor witnesses: @@ -1371,7 +1371,7 @@ \section*{Appendix AE: Cross-Reference Table} analytic-arithmetic shadows in the generating-function machinery. \section*{Closing} -\label{sec:05-closing} +\label{fa_05:sec:05-closing} This concludes the chapter on the Fibonacci-Lucas generating-function bridge. The next chapter, Chapter~\ref{ch:lucas-ring} (L6), develops @@ -1385,7 +1385,7 @@ \section*{Closing} \bigskip \section*{Appendix AF: Extended Riordan Computations} -\label{sec:05-app-AF} +\label{fa_05:sec:05-app-AF} We expand on Appendix AD with explicit small entries of the Fibonacci Riordan array $T_{n,k} = [x^n] F(x) (x F(x))^k$, indexed for $0 \le n,k @@ -1393,7 +1393,7 @@ \section*{Appendix AF: Extended Riordan Computations} \cdots$, we have $x F(x) = x^2 + x^3 + 2x^4 + 3x^5 + 5x^6 + \cdots$, and powers $(x F(x))^k$ start at $x^{2k}$. -\begin{lemma}[Riordan small entries]\label{lem:05-riordan-small} +\begin{lemma}[Riordan small entries]\label{fa_05:lem:05-riordan-small} The first nonzero entries of the Fibonacci Riordan array are: \begin{align*} T_{1,0} &= 1, & T_{2,0} &= 1, & T_{3,0} &= 2, & T_{4,0} &= 3, \\ @@ -1421,7 +1421,7 @@ \section*{Appendix AF: Extended Riordan Computations} \end{remark} \section*{Appendix AG: Combinatorial Interpretations of $L_2 = 3$} -\label{sec:05-app-AG} +\label{fa_05:sec:05-app-AG} We close the chapter with seven combinatorial interpretations of the identity $L_2 = 3$, reinforcing the philosophical thesis that the @@ -1468,7 +1468,7 @@ \section*{Appendix AG: Combinatorial Interpretations of $L_2 = 3$} \centerline{\textit{Seven proofs, one truth.}} \section*{Appendix AH: Final Remarks on Bridge Discipline} -\label{sec:05-app-AH} +\label{fa_05:sec:05-app-AH} Three discipline rules govern the use of generating functions as a bridge between integer arithmetic (L4) and ring theory (L6): diff --git a/docs/phd/chapters/fa_06.tex b/docs/phd/chapters/fa_06.tex index 4bb5829a86..0142e05e83 100644 --- a/docs/phd/chapters/fa_06.tex +++ b/docs/phd/chapters/fa_06.tex @@ -7,7 +7,7 @@ \chapter{Golden Mantissa: GoldenFloat Family GF4--GF64} \caption*{Figure --- Golden Mantissa: GoldenFloat Family GF4--GF64.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_06:abstract} This chapter defines the GoldenFloat (GF) number family---a hierarchy of floating-point formats @@ -28,7 +28,7 @@ \section{Abstract}\label{abstract} at Gate-2 while remaining formally overflow-free within the declared operating range. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_06:introduction} Floating-point arithmetic in neural-network inference has evolved from FP32 through FP16, @@ -67,10 +67,10 @@ \section{1. Introduction}\label{introduction} precision measurements. \section{2. GoldenFloat Format -Definitions}\label{goldenfloat-format-definitions} +Definitions}\label{fa_06:goldenfloat-format-definitions} \subsection{2.1 -Preliminaries}\label{preliminaries} +Preliminaries}\label{fa_06:preliminaries} Let \(\varphi = (1+\sqrt{5})/2\) and \(\hat\varphi = \varphi^{-1} = \varphi - 1 = (\sqrt{5}-1)/2\). @@ -139,7 +139,7 @@ \subsection{2.1 whose Fibonacci representation is shortest. \subsection{2.2 Coq -Encoding}\label{coq-encoding} +Encoding}\label{fa_06:coq-encoding} The Coq development in \filepath{gHashTag/t27/proofs/canonical/kernel/PhiFloat.v} @@ -168,7 +168,7 @@ \subsection{2.2 Coq this for the phi constant. \subsection{2.3 Lucas Closure on -GF16}\label{lucas-closure-on-gf16} +GF16}\label{fa_06:lucas-closure-on-gf16} A key algebraic property of the GoldenFloat substrate is that \(\varphi^{2n} + \varphi^{-2n}\) @@ -187,7 +187,7 @@ \subsection{2.3 Lucas Closure on fractional Lucas residuals. \section{3. Key Theorems and Proof -Sketches}\label{key-theorems-and-proof-sketches} +Sketches}\label{fa_06:key-theorems-and-proof-sketches} \textbf{Theorem 3.1} (\texttt{phi\_f64\_bounded}). \emph{The GF64 representation of \(\varphi\) is @@ -264,7 +264,7 @@ \section{3. Key Theorems and Proof \((-\infty,\infty)\) exhaustively by construction. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_06:results-evidence} GF16 was evaluated on the HSLM benchmark (1003 tokens, drawn from the GOLDEN SUNFLOWERS test @@ -328,7 +328,7 @@ \section{4. Results / in INV-3 and the period-locked monitor (Ch.24). \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_06:qed-assertions} \begin{itemize} \item @@ -369,7 +369,7 @@ \section{5. Qed \texttt{0\ \textless{}\ PHI\_F64\_TOLERANCE}. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_06:sealed-seeds} \begin{itemize} \item @@ -402,7 +402,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} --- \emph{Status: golden} --- Linked: Ch.6. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_06:discussion} The GoldenFloat family demonstrates that choosing arithmetic parameters from an algebraically @@ -432,7 +432,7 @@ \section{7. Discussion}\label{discussion} scheduling sentinels), and Ch.28 (FPGA synthesis of the GF16 MAC unit with 0 DSP slices). -\section{References}\label{references} +\section{References}\label{fa_06:references} [1] Rouhani, B. D. et al.~(2023). \emph{Microscaling Data Formats for Deep diff --git a/docs/phd/chapters/fa_07.tex b/docs/phd/chapters/fa_07.tex index 25203cb897..1f5ce68cb5 100644 --- a/docs/phd/chapters/fa_07.tex +++ b/docs/phd/chapters/fa_07.tex @@ -8,7 +8,7 @@ \chapter{Golden Sprout: Vogel Phyllotaxis} \caption*{Figure --- Golden Sprout: Vogel Phyllotaxis.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_07:abstract} Vogel's 1979 model of sunflower head packing describes each floret position by a polar angle @@ -28,7 +28,7 @@ \section{Abstract}\label{abstract} overlap also spaces quantised weights without collisions. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_07:introduction} The observation that sunflower seed heads, pine cones, and daisy florets arrange themselves in @@ -57,7 +57,7 @@ \section{1. Introduction}\label{introduction} \section{2. From the Trinity Identity to the Golden -Angle}\label{from-the-trinity-identity-to-the-golden-angle} +Angle}\label{fa_07:from-the-trinity-identity-to-the-golden-angle} \textbf{Definition 2.1 (Golden ratio).} \(\varphi = (1+\sqrt{5})/2\), the positive root of @@ -117,7 +117,7 @@ \section{2. From the Trinity Identity to the \section{\texorpdfstring{3. H4 Root System, E8 Lattice, and the \(\varphi\)-Scaled Block -Decomposition}{3. H4 Root System, E8 Lattice, and the \textbackslash varphi-Scaled Block Decomposition}}\label{h4-root-system-e8-lattice-and-the-varphi-scaled-block-decomposition} +Decomposition}{3. H4 Root System, E8 Lattice, and the \textbackslash varphi-Scaled Block Decomposition}}\label{fa_07:h4-root-system-e8-lattice-and-the-varphi-scaled-block-decomposition} The 240 roots of the E8 lattice can be partitioned into two H4 half-shells of 120 roots each, related @@ -205,7 +205,7 @@ \section{\texorpdfstring{3. H4 Root System, E8 Theorem 3.3. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_07:results-evidence} Four quantitative results anchor this chapter. @@ -249,7 +249,7 @@ \section{4. Results / \end{enumerate} \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_07:qed-assertions} \begin{itemize} \tightlist @@ -290,13 +290,13 @@ \section{5. Qed \(\varphi^2 + \varphi^{-2} = 3 \Rightarrow \dim(H4) + \dim(\varphi \cdot H4) = \dim(E8)/2\). \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_07:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_07:discussion} The two \texttt{Abort} theorems (KER-3) represent the principal limitation of the present chapter. @@ -324,7 +324,7 @@ \section{7. Discussion}\label{discussion} spectral constant (Ch.4) to the angular spectrum of E8 root vectors. -\section{References}\label{references} +\section{References}\label{fa_07:references} [1] Church, A. H. (1904). \emph{On the Relation of Phyllotaxis to Mechanical Laws.} diff --git a/docs/phd/chapters/fa_08.tex b/docs/phd/chapters/fa_08.tex index 604bf701bd..9026209486 100644 --- a/docs/phd/chapters/fa_08.tex +++ b/docs/phd/chapters/fa_08.tex @@ -6,7 +6,7 @@ \chapter{Golden Crystal: TF3/TF9 Sparse Ternary Matmul} \caption*{Figure --- Golden Crystal: TF3/TF9 Sparse Ternary Matmul.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_08:abstract} This chapter introduces the TF3 and TF9 matrix-multiplication formats that form the @@ -26,7 +26,7 @@ \section{Abstract}\label{abstract} of the gain invariant, and evidence that TF3/TF9 achieves the Gate-2 BPB target of ≤ 1.85. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_08:introduction} Dense floating-point matrix multiplication dominates the energy budget of transformer @@ -64,10 +64,10 @@ \section{1. Introduction}\label{introduction} rest of the system. \section{2. TF3 and TF9 Algebraic -Structure}\label{tf3-and-tf9-algebraic-structure} +Structure}\label{fa_08:tf3-and-tf9-algebraic-structure} \subsection{2.1 Trit -Encoding}\label{trit-encoding} +Encoding}\label{fa_08:trit-encoding} Let \(\mathcal{T} = \{-1, 0, +1\}\). A TF3 weight tensor \(\mathbf{W} \in \mathcal{T}^{m \times n}\) @@ -93,7 +93,7 @@ \subsection{2.1 Trit information cost of the output. \subsection{2.2 TF9 Product -Encoding}\label{tf9-product-encoding} +Encoding}\label{fa_08:tf9-product-encoding} TF9 represents each weight as \((w_1, w_2) \in \mathcal{T}^2\) with effective @@ -115,7 +115,7 @@ \subsection{2.2 TF9 Product [2]. \subsection{2.3 -φ-Normalisation}\label{ux3c6-normalisation} +φ-Normalisation}\label{fa_08:ux3c6-normalisation} Both formats inherit the φ-normalisation scheme: layer inputs are scaled by @@ -132,10 +132,10 @@ \subsection{2.3 [3]. \section{3. Hybrid QK Gain Invariant -(INV-6)}\label{hybrid-qk-gain-invariant-inv-6} +(INV-6)}\label{fa_08:hybrid-qk-gain-invariant-inv-6} \subsection{3.1 Gain -Admissibility}\label{gain-admissibility} +Admissibility}\label{fa_08:gain-admissibility} \textbf{Definition (lr-admissible).} A learning rate \(\eta\) is \emph{lr-admissible} if it lies @@ -196,7 +196,7 @@ \subsection{3.1 Gain is below the admissible band. \subsection{3.2 Proof Sketch for -admit\_phi\_sq}\label{proof-sketch-for-admit_phi_sq} +admit\_phi\_sq}\label{fa_08:proof-sketch-for-admit_phi_sq} Let \(\mathbf{q}, \mathbf{k} \in \mathbb{R}^d\) be query and key vectors with entries drawn i.i.d. @@ -221,7 +221,7 @@ \subsection{3.2 Proof Sketch for library [3]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_08:results-evidence} All numerical results reported here use seeds from the sanctioned pool @@ -272,7 +272,7 @@ \section{4. Results / all projection matrices. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_08:qed-assertions} \begin{itemize} \tightlist @@ -312,7 +312,7 @@ \section{5. Qed qk-admissible. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_08:sealed-seeds} \begin{itemize} \tightlist @@ -328,7 +328,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Ternary MatMul artefact. Links: Ch.8. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_08:discussion} The two \emph{Qed} theorems for \(g \in \{\varphi^2, \varphi^3\}\) are the formal @@ -356,7 +356,7 @@ \section{7. Discussion}\label{discussion} combining TF9 with the GF16 quantisation scheme described in Ch.26. -\section{References}\label{references} +\section{References}\label{fa_08:references} [1] DARPA MTO. (2023). Microsystems Technology Office Broad Agency Announcement --- @@ -407,7 +407,7 @@ \section{References}\label{references} Biosciences}, 44(3--4), 179--189. \section{Falsification} -\label{sec:falsification:ch08} +\label{fa_08:sec:falsification:ch08} \paragraph{Pre-registered claim (R7).} The TF3/TF9 sparse-ternary matmul kernel running on the GoldenFloat substrate diff --git a/docs/phd/chapters/fa_09.tex b/docs/phd/chapters/fa_09.tex index 4b6f803a9c..1f5e5b1186 100644 --- a/docs/phd/chapters/fa_09.tex +++ b/docs/phd/chapters/fa_09.tex @@ -7,7 +7,7 @@ \chapter{Golden Seal: GF vs MXFP4 Ablation} \caption*{Figure --- Golden Seal: GF vs MXFP4 Ablation.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_09:abstract} This chapter presents a systematic ablation comparing four low-precision weight formats --- @@ -29,7 +29,7 @@ \section{Abstract}\label{abstract} reproducible by any of the three competitor formats under the same hardware budget. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_09:introduction} The choice of numerical representation for neural-network weights is not merely an @@ -79,10 +79,10 @@ \section{1. Introduction}\label{introduction} \section{2. GF16 PHI\_BIAS=60 and the INV-3 Safe -Domain}\label{gf16-phi_bias60-and-the-inv-3-safe-domain} +Domain}\label{fa_09:gf16-phi_bias60-and-the-inv-3-safe-domain} \subsection{2.1 GF16 Format -Specification}\label{gf16-format-specification} +Specification}\label{fa_09:gf16-format-specification} GF(16) represents each weight as a 4-bit element of the finite field @@ -118,7 +118,7 @@ \subsection{2.1 GF16 Format (\(\varphi^2 + \varphi^{-2} = 3\)). \subsection{2.2 INV-3: Nine Coq Precision -Bounds}\label{inv-3-nine-coq-precision-bounds} +Bounds}\label{fa_09:inv-3-nine-coq-precision-bounds} Invariant INV-3, formalised in \filepath{t27/proofs/canonical/igla/INV3\_Gf16Precision.v} @@ -149,7 +149,7 @@ \subsection{2.2 INV-3: Nine Coq Precision cited by INV-3 [2, 3]. \subsection{2.3 Competitor Format -Summaries}\label{competitor-format-summaries} +Summaries}\label{fa_09:competitor-format-summaries} \textbf{MXFP4} [4]: Microsoft's micro-scaling FP4 uses a shared 8-bit exponent per group of 32 @@ -176,7 +176,7 @@ \subsection{2.3 Competitor Format reduces the effective compression ratio. \section{3. Ablation Matrix: Tier-A/B/C $\times$ -M1--M6}\label{ablation-matrix-tier-abc-m1m6} +M1--M6}\label{fa_09:ablation-matrix-tier-abc-m1m6} The evaluation matrix is defined as follows. @@ -210,7 +210,7 @@ \section{3. Ablation Matrix: Tier-A/B/C $\times$ wall-clock power draw. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_09:results-evidence} \textbf{Table 1. Tier-A BPB (WikiText-103), lower is better.} @@ -312,7 +312,7 @@ \section{4. Results / was observed. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_09:qed-assertions} No Coq theorems from \filepath{t27/proofs/canonical/} are directly @@ -323,7 +323,7 @@ \section{5. Qed both tracked in the Golden Ledger under invariant numbers INV-3 and SAC-1 respectively. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_09:sealed-seeds} \begin{itemize} \tightlist @@ -335,7 +335,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} domain, 9 Qed bounds. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_09:discussion} The ablation demonstrates a consistent but modest advantage of GF16 PHI\_BIAS=60 over MXFP4 on @@ -359,7 +359,7 @@ \section{7. Discussion}\label{discussion} domain. Chapters 15 and 28 continue the BPB and hardware analyses respectively. -\section{References}\label{references} +\section{References}\label{fa_09:references} [1] \emph{Golden Sunflowers} dissertation, Ch.3 --- Trinity Identity diff --git a/docs/phd/chapters/fa_10.tex b/docs/phd/chapters/fa_10.tex index 9a3239abf7..b984c435bb 100644 --- a/docs/phd/chapters/fa_10.tex +++ b/docs/phd/chapters/fa_10.tex @@ -6,7 +6,7 @@ \chapter{Golden Bloom: Coq L1 Range Precision Pareto} \caption*{Figure --- Golden Bloom: Coq L1 Range Precision Pareto.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_10:abstract} Designing ternary neural-network quantisation requires navigating a two-dimensional Pareto @@ -31,7 +31,7 @@ \section{Abstract}\label{abstract} largest single-chapter Coq contribution in the dissertation. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_10:introduction} The theoretical link between \(\phi^2 + \phi^{-2} = 3\) and quantisation @@ -69,7 +69,7 @@ \section{1. Introduction}\label{introduction} into the main branch [3]. \section{2. GF(16) Range and Precision -Formalisation}\label{gf16-range-and-precision-formalisation} +Formalisation}\label{fa_10:gf16-range-and-precision-formalisation} \textbf{Definition 2.1 (GF(16) weight encoding).} A weight \(w\) is encoded in GF(16) as a pair @@ -133,7 +133,7 @@ \section{2. GF(16) Range and Precision subsequent GF(16) precision theorems. \section{3. The Pareto Frontier and Conjecture -C1}\label{the-pareto-frontier-and-conjecture-c1} +C1}\label{fa_10:the-pareto-frontier-and-conjecture-c1} \textbf{Definition 3.1 (Pareto-efficient allocation).} An allocation \((e_{\max}, b_m)\) @@ -212,7 +212,7 @@ \section{3. The Pareto Frontier and Conjecture directive [6]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_10:results-evidence} Numerical evaluation of the Pareto frontier used the canonical seed pool F₁₇=1597, F₁₈=2584, @@ -272,7 +272,7 @@ \section{4. Results / the proof files [7]. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_10:qed-assertions} \begin{itemize} \tightlist @@ -307,7 +307,7 @@ \section{5. Qed \(0 < \phi = (1+\sqrt{5})/2\). \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_10:sealed-seeds} \begin{itemize} \tightlist @@ -340,7 +340,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} DSL. φ-weight: 0.618033988768953. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_10:discussion} The central limitation of this chapter is Conjecture C1: until the admitted lemma @@ -368,7 +368,7 @@ \section{7. Discussion}\label{discussion} will address the full compositionality of the INV-1 through INV-9 invariant chain. -\section{References}\label{references} +\section{References}\label{fa_10:references} [1] GOLDEN SUNFLOWERS dissertation, Ch.4 --- Sacred Formula: α\_φ Derivation. This volume. diff --git a/docs/phd/chapters/fa_11.tex b/docs/phd/chapters/fa_11.tex index 98f89bdf80..a1b6d25266 100644 --- a/docs/phd/chapters/fa_11.tex +++ b/docs/phd/chapters/fa_11.tex @@ -8,7 +8,7 @@ \chapter{Vesica Piscis: Pre-registration H --- 3 distinct seeds} \caption*{Figure --- Vesica Piscis: Pre-registration H --- 3 distinct seeds.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_11:abstract} Scientific credibility requires that empirical claims be registered before data collection. This @@ -29,7 +29,7 @@ \section{Abstract}\label{abstract} Framework conventions and is published prior to any Gate-3 BPB measurement. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_11:introduction} The Trinity S³AI framework rests on three architectural commitments: ternary weight @@ -66,7 +66,7 @@ \section{1. Introduction}\label{introduction} \section{2. Hypothesis Formalisation and Registration -Protocol}\label{hypothesis-formalisation-and-registration-protocol} +Protocol}\label{fa_11:hypothesis-formalisation-and-registration-protocol} \textbf{Definition 2.1 (H₁ --- formal statement).} Let @@ -115,7 +115,7 @@ \section{2. Hypothesis Formalisation and \(\log_2 3 \approx 1.585\). \section{3. INV-7 Invariant and Coq -Formalisation}\label{inv-7-invariant-and-coq-formalisation} +Formalisation}\label{fa_11:inv-7-invariant-and-coq-formalisation} The INV-7 invariant formalises H₁ in the Coq proof assistant. Its statement in @@ -192,7 +192,7 @@ \section{3. INV-7 Invariant and Coq from the entropy argument above [7]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_11:results-evidence} Pre-registration status as of the current dissertation version: @@ -245,12 +245,12 @@ \section{4. Results / of this theoretical maximum. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_11:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_11:sealed-seeds} \begin{itemize} \tightlist @@ -269,7 +269,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} harness. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_11:discussion} The pre-registration protocol described here is unusual for a dissertation chapter: it commits to @@ -292,7 +292,7 @@ \section{7. Discussion}\label{discussion} the BPB contribution of each seed), and sideways to Ch.21 (the IGLAFoundCriterion in full detail). -\section{References}\label{references} +\section{References}\label{fa_11:references} [1] Shannon, C. E. (1948). A mathematical theory of communication. \emph{Bell System diff --git a/docs/phd/chapters/fa_12.tex b/docs/phd/chapters/fa_12.tex index d4e2fe6909..c729535957 100644 --- a/docs/phd/chapters/fa_12.tex +++ b/docs/phd/chapters/fa_12.tex @@ -6,7 +6,7 @@ \chapter{Flower of Life: Hardware Bridge (deferred)} \caption*{Figure --- Flower of Life: Hardware Bridge (deferred).} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_12:abstract} The Hardware Bridge chapter specifies the interface layer between the Trinity S³AI software @@ -26,7 +26,7 @@ \section{Abstract}\label{abstract} three-channel bridge structure: one channel per exponent band of the GoldenFloat format. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_12:introduction} Any system that co-designs arithmetic formats with hardware must specify where the software--hardware @@ -62,10 +62,10 @@ \section{1. Introduction}\label{introduction} [4]. \section{2. Bridge Architecture and Interface -Contracts}\label{bridge-architecture-and-interface-contracts} +Contracts}\label{fa_12:bridge-architecture-and-interface-contracts} \subsection{2.1 Logical -Structure}\label{logical-structure} +Structure}\label{fa_12:logical-structure} The Hardware Bridge comprises three functional blocks: @@ -101,7 +101,7 @@ \subsection{2.1 Logical \end{enumerate} \subsection{2.2 Signal Naming -Convention}\label{signal-naming-convention} +Convention}\label{fa_12:signal-naming-convention} All bridge signals follow the naming convention \texttt{GS\_\textless{}direction\textgreater{}\_\textless{}channel\textgreater{}\_\textless{}width\textgreater{}}: @@ -124,7 +124,7 @@ \subsection{2.2 Signal Naming carries 16-bit GF16 tokens. \subsection{2.3 Error-Handling -Protocol}\label{error-handling-protocol} +Protocol}\label{fa_12:error-handling-protocol} The bridge defines three error conditions: @@ -156,10 +156,10 @@ \subsection{2.3 Error-Handling bit per error class [6]. \section{3. Clock-Domain Analysis and -Timing}\label{clock-domain-analysis-and-timing} +Timing}\label{fa_12:clock-domain-analysis-and-timing} \subsection{3.1 Frequency Ratios and the Golden -Ratio}\label{frequency-ratios-and-the-golden-ratio} +Ratio}\label{fa_12:frequency-ratios-and-the-golden-ratio} The ratio of the host AXI clock (100 MHz) to the FPGA fabric clock (92 MHz) is @@ -174,7 +174,7 @@ \subsection{3.1 Frequency Ratios and the Golden verified in Ch.28. \subsection{3.2 Throughput -Budget}\label{throughput-budget} +Budget}\label{fa_12:throughput-budget} The token throughput of the FPGA pipeline is 63 toks/sec as measured in Ch.28 [3]. The UART-V6 @@ -190,7 +190,7 @@ \subsection{3.2 Throughput fabric. \subsection{3.3 Power -Accounting}\label{power-accounting} +Accounting}\label{fa_12:power-accounting} The 1 W power budget assigned to the FPGA (Ch.28) is allocated as follows: approximately 0.6 W to @@ -219,7 +219,7 @@ \subsection{3.3 Power \(\varphi^2 + \varphi^{-2} = 3\). Qed. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_12:results-evidence} The Hardware Bridge was instantiated and simulated in Vivado 2022.2 targeting the XC7A100T-FGG484 @@ -281,7 +281,7 @@ \section{4. Results / sufficient for 63 toks/sec. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_12:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. @@ -291,13 +291,13 @@ \section{5. Qed respectively, where the hardware measurements required for their hypotheses are available.) -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_12:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_12:discussion} The Hardware Bridge chapter occupies a structurally important but formally deferred role @@ -328,7 +328,7 @@ \section{7. Discussion}\label{discussion} enabling batch inference modes currently limited by I/O. -\section{References}\label{references} +\section{References}\label{fa_12:references} [1] This dissertation, Ch.6: GoldenFloat Family GF4..GF64. diff --git a/docs/phd/chapters/fa_13.tex b/docs/phd/chapters/fa_13.tex index cf41450009..e18a8126b1 100644 --- a/docs/phd/chapters/fa_13.tex +++ b/docs/phd/chapters/fa_13.tex @@ -16,7 +16,7 @@ \chapter{Metatron's Cube and the Lucas-12 Orbit} \end{figure} -\label{ch:13-metatron} +\label{fa_13:ch:13-metatron} % ===================================================================== % Chapter epigraph @@ -34,10 +34,10 @@ \chapter{Metatron's Cube and the Lucas-12 Orbit} % ===================================================================== \section{Strand I --- Intuition} -\label{sec:13-strand-I} +\label{fa_13:sec:13-strand-I} \subsection{Where Metatron's Cube Comes From} -\label{sec:13-origin} +\label{fa_13:sec:13-origin} Metatron's Cube is, in the classical literature on sacred geometry, the figure obtained by joining every pair of the thirteen circles of @@ -70,7 +70,7 @@ \subsection{Where Metatron's Cube Comes From} identity in three independent ways. \subsection{Three Strands of Continuity} -\label{sec:13-three-strands} +\label{fa_13:sec:13-three-strands} Following the Rule of Three (R12 of trios\#265), this chapter is organised in three strands of continuity: @@ -101,7 +101,7 @@ \subsection{Three Strands of Continuity} adhere to it. \subsection{Why Thirteen Nodes} -\label{sec:13-thirteen} +\label{fa_13:sec:13-thirteen} The number thirteen appears in three independent ways in the algebraic Metatron's Cube: @@ -159,7 +159,7 @@ \subsection{Why Seventy-Eight Edges} vertex set. We make this precise in Strand II. \subsection{Why Five Platonic Solids} -\label{sec:13-five-platonic} +\label{fa_13:sec:13-five-platonic} Within the algebraic Metatron's Cube, the projections of all five Platonic solids ---~tetrahedron, cube, octahedron, dodecahedron, @@ -191,7 +191,7 @@ \subsection{Why Five Platonic Solids} tetrahedral inscriptions. \subsection{Pedagogical Diagram (Referenced, not Embedded)} -\label{sec:13-diagram} +\label{fa_13:sec:13-diagram} Throughout this chapter we refer to a pedagogical diagram of Metatron's Cube, with the thirteen nodes labelled $0$ (origin) and @@ -205,7 +205,7 @@ \subsection{Pedagogical Diagram (Referenced, not Embedded)} Strand~II. \subsection{Connection to Chapter 17} -\label{sec:13-connection-to-17} +\label{fa_13:sec:13-connection-to-17} Chapter~\ref{ch:17-spiral} (Golden Spiral) introduces the Lucas-ring spiral as the algebraic substrate of the GF16 floor. Metatron's @@ -225,7 +225,7 @@ \subsection{Connection to Chapter 17} $3$ for the canonical orbit (Lemma~\ref{lem:13-trinity}). \subsection{Strand I Takeaway} -\label{sec:13-strand-I-takeaway} +\label{fa_13:sec:13-strand-I-takeaway} The reader should leave Strand I with three pictures in mind: @@ -247,10 +247,10 @@ \subsection{Strand I Takeaway} % ===================================================================== \section{Strand II --- Formalisation} -\label{sec:13-strand-II} +\label{fa_13:sec:13-strand-II} \subsection{Notation} -\label{sec:13-notation} +\label{fa_13:sec:13-notation} We adopt the notation of Chapter~\ref{ch:17-spiral} unmodified. Specifically: @@ -274,7 +274,7 @@ \subsection{Notation} no complex roots of $\phi$ are involved. \subsection{The Lucas-12 Orbit} -\label{sec:13-lucas-12-orbit} +\label{fa_13:sec:13-lucas-12-orbit} \begin{definition}[Lucas-12 orbit] \label{def:13-lucas-12} @@ -313,7 +313,7 @@ \subsection{The Lucas-12 Orbit} We tabulate them in Appendix~\ref{sec:13-appB}. \subsection{The Trinity Plane} -\label{sec:13-trinity-plane} +\label{fa_13:sec:13-trinity-plane} The Trinity plane is the subspace of $\mathbb{C} \times \mathbb{R}$ defined as follows. Let $\mathcal{T}$ be the two-dimensional real @@ -342,7 +342,7 @@ \subsection{The Trinity Plane} symmetries. \subsection{The Projection Theorem} -\label{sec:13-projection} +\label{fa_13:sec:13-projection} We now state and prove the central theorem of the chapter. @@ -398,7 +398,7 @@ \subsection{The Projection Theorem} \end{remark} \subsection{Edge Counts} -\label{sec:13-edge-counts} +\label{fa_13:sec:13-edge-counts} Recall the classification of edges into primary, secondary, and tertiary (\S\ref{sec:13-seventy-eight}). We now derive the counts @@ -545,7 +545,7 @@ \subsection{Symmetry Group Action} \end{proof} \subsection{Connection to the GF16 Floor} -\label{sec:13-gf16} +\label{fa_13:sec:13-gf16} The GF16 substrate of Chapter~\ref{ch:23-gf16-algebra} bottoms out at the algebraic floor $\phi^{-6} = 18 - 11\phi$, a value @@ -580,7 +580,7 @@ \subsection{Connection to the GF16 Floor} \end{remark} \subsection{Strand II Wrap} -\label{sec:13-strand-II-wrap} +\label{fa_13:sec:13-strand-II-wrap} In Strand~II we have established the algebraic content of the chapter: @@ -608,7 +608,7 @@ \subsection{Strand II Wrap} % ===================================================================== \section{Strand III --- Consequence} -\label{sec:13-strand-III} +\label{fa_13:sec:13-strand-III} \subsection{The Cube as Architecture Scaffold} \label{sec:13-arch-scaffold} @@ -640,7 +640,7 @@ \subsection{The Cube as Architecture Scaffold} in units where the unit cube has radius $\phi$. \subsection{Counting in the Architecture} -\label{sec:13-counting-arch} +\label{fa_13:sec:13-counting-arch} Each of the three concentric cubes contributes $13$ nodes and $78$ edges. The full architecture has, therefore, $39$ nodes and $234$ @@ -660,7 +660,7 @@ \subsection{Counting in the Architecture} completeness, and as a sanity check on the projection. \subsection{Why a Cube and Not a Spiral} -\label{sec:13-cube-vs-spiral} +\label{fa_13:sec:13-cube-vs-spiral} A natural question: why is Metatron's Cube the right discrete sibling of the golden spiral, rather than some other discrete @@ -684,7 +684,7 @@ \subsection{Why a Cube and Not a Spiral} \end{enumerate} \subsection{Empirical Re-corroboration in Chapter 26} -\label{sec:13-emp-26} +\label{fa_13:sec:13-emp-26} Chapter~\ref{ch:26-data-analysis} reports the GF16 floor at $0.0557 \pm 0.0008$, in agreement with the prediction @@ -695,7 +695,7 @@ \subsection{Empirical Re-corroboration in Chapter 26} corroboration record (R7), and we cite it here for completeness. \subsection{Connection to Chapter 23} -\label{sec:13-conn-23} +\label{fa_13:sec:13-conn-23} Chapter~\ref{ch:23-gf16-algebra} works out the algebraic structure of GF16 in terms of $\mathbb{F}_{2^{4}}$ and its primitive elements. @@ -710,7 +710,7 @@ \subsection{Connection to Chapter 23} Chapter~\ref{ch:23-gf16-algebra}. \subsection{Strand III Wrap} -\label{sec:13-strand-III-wrap} +\label{fa_13:sec:13-strand-III-wrap} In Strand~III we have used the algebraic Metatron's Cube as a bookkeeping device for the Trinity architecture. The cube provides @@ -723,10 +723,10 @@ \subsection{Strand III Wrap} % ===================================================================== \section{Coordinates and Algebraic Bookkeeping} -\label{sec:13-coords-bookkeeping} +\label{fa_13:sec:13-coords-bookkeeping} \subsection{Cartesian Coordinates of the Rim} -\label{sec:13-cartesian-rim} +\label{fa_13:sec:13-cartesian-rim} The twelve rim points $p_{k} = \phi \zeta_{12}^{k}$ have Cartesian coordinates @@ -741,14 +741,14 @@ \subsection{Cartesian Coordinates of the Rim} Appendix~\ref{sec:13-appB}. \subsection{Polar Coordinates} -\label{sec:13-polar} +\label{fa_13:sec:13-polar} In polar form, the rim points are simply $(r, \theta) = (\phi, \pi k / 6)$. The simplicity of the polar form makes it the natural choice for stating the symmetry results of \S\ref{sec:13-symmetry-group}. \subsection{Lucas-Ring Coordinates} -\label{sec:13-lucas-ring-coords} +\label{fa_13:sec:13-lucas-ring-coords} Each rim point can be expressed in Lucas-ring coordinates as \[ @@ -763,7 +763,7 @@ \subsection{Lucas-Ring Coordinates} is a Lucas-ring orbit only in its radial coordinate. \subsection{Identities} -\label{sec:13-identities} +\label{fa_13:sec:13-identities} The following identities follow by direct computation and are useful in Strand~III: @@ -788,7 +788,7 @@ \section{The Edge Set Coloured by Lucas-Ring Filtration} \label{sec:13-filtration} \subsection{The Filtration} -\label{sec:13-filt-def} +\label{fa_13:sec:13-filt-def} We define the Lucas-ring filtration on the edge set of Metatron's Cube as follows. Let $E$ denote the full edge set of size $78$, @@ -805,7 +805,7 @@ \subsection{The Filtration} \] \subsection{Why a Filtration} -\label{sec:13-filt-why} +\label{fa_13:sec:13-filt-why} The filtration corresponds to the natural ordering by Lucas-ring multiplicative complexity: an edge in $E_{1}$ corresponds to a @@ -817,7 +817,7 @@ \subsection{Why a Filtration} $j = 6$). \subsection{Filtration Quotients} -\label{sec:13-filt-quotients} +\label{fa_13:sec:13-filt-quotients} The successive quotients of the filtration are \[ @@ -845,10 +845,10 @@ \subsection{Filtration vs Coq Mechanisation} Lemmata~\ref{lem:13-primary}--\ref{lem:13-tertiary}. \section{Connection to the Trinity Architecture} -\label{sec:13-arch} +\label{fa_13:sec:13-arch} \subsection{Three Cubes, Three Layers} -\label{sec:13-three-cubes} +\label{fa_13:sec:13-three-cubes} We now formalise the picture of \S\ref{sec:13-arch-scaffold} as a three-layer cube structure. Define three cubes @@ -870,7 +870,7 @@ \subsection{Three Cubes, Three Layers} \phi^{-6}$). \subsection{Layer-Layer Distances} -\label{sec:13-layer-distances} +\label{fa_13:sec:13-layer-distances} The radial distance between layer $j$ and layer $j+1$ is $\phi^{j} - \phi^{j-1} = \phi^{j-1}(\phi - 1) = \phi^{j-1} \cdot @@ -888,7 +888,7 @@ \subsection{Layer-Layer Distances} $\phi$-self-similar action. \subsection{Architecture Summary} -\label{sec:13-arch-summary} +\label{fa_13:sec:13-arch-summary} The three-cube picture summarises the Trinity architecture as follows: @@ -909,7 +909,7 @@ \subsection{Architecture Summary} $\phi$-multiplicative homomorphisms. \section{Coq Citation Map (R14)} -\label{sec:13-coq-map} +\label{fa_13:sec:13-coq-map} Per R14 of trios\#265, every cited theorem in this chapter must trace to a Coq mechanisation. We list the map. @@ -946,10 +946,10 @@ \section{Coq Citation Map (R14)} status is independent of this file. \section{Discussion} -\label{sec:13-discussion} +\label{fa_13:sec:13-discussion} \subsection{What the Chapter Has Established} -\label{sec:13-disc-est} +\label{fa_13:sec:13-disc-est} We have established three things in this chapter: @@ -967,7 +967,7 @@ \subsection{What the Chapter Has Established} \end{enumerate} \subsection{What the Chapter Has Not Claimed} -\label{sec:13-disc-not} +\label{fa_13:sec:13-disc-not} To preserve R5 honesty, we list what we do \emph{not} claim: @@ -986,7 +986,7 @@ \subsection{What the Chapter Has Not Claimed} \end{itemize} \subsection{Open Questions} -\label{sec:13-disc-open} +\label{fa_13:sec:13-disc-open} Three open questions arise from the chapter: @@ -1008,7 +1008,7 @@ \subsection{Open Questions} \end{enumerate} \subsection{Summary} -\label{sec:13-disc-summary} +\label{fa_13:sec:13-disc-summary} The algebraic Metatron's Cube is a discrete, combinatorial sibling of the continuous golden spiral. Together with the spiral, it @@ -1167,7 +1167,7 @@ \section*{Appendix 13.C --- Edge Inventory} distinct edge lengths are six in number. \section*{Appendix 13.D --- Five Platonic Solids in the Cube} -\label{sec:13-appD} +\label{fa_13:sec:13-appD} \addcontentsline{toc}{section}{Appendix 13.D --- Five Platonic Solids} We sketch the inscriptions of the five Platonic solids in @@ -1217,7 +1217,7 @@ \section*{Appendix 13.D --- Five Platonic Solids in the Cube} two-dimensional projection. \section*{Appendix 13.E --- Worked Examples} -\label{sec:13-appE} +\label{fa_13:sec:13-appE} \addcontentsline{toc}{section}{Appendix 13.E --- Worked Examples} \paragraph{Example E-1 --- Computing the squared norm sum.} @@ -1291,7 +1291,7 @@ \section*{Appendix 13.E --- Worked Examples} \phi(\sqrt{6}+\sqrt{2})/2, 2\phi$. \section*{Appendix 13.F --- Glossary} -\label{sec:13-appF} +\label{fa_13:sec:13-appF} \addcontentsline{toc}{section}{Appendix 13.F --- Glossary} \begin{description} @@ -1322,7 +1322,7 @@ \section*{Appendix 13.F --- Glossary} \end{description} \section*{Appendix 13.G --- Three-Strand Cross-Reference} -\label{sec:13-appG} +\label{fa_13:sec:13-appG} \addcontentsline{toc}{section}{Appendix 13.G --- Three-Strand Cross-Reference} For each section in the chapter body, we tabulate which strand it @@ -1396,7 +1396,7 @@ \section*{Appendix 13.H --- Honest Status Declaration} \admittedbox{metatron\_cube.v}{combinatorial counts $|E_{1}|=12$, $|E_{2}|=30$, $|E_{3}|=36$ not yet mechanised} \section*{Appendix 13.I --- Defensive Coda} -\label{sec:13-appI} +\label{fa_13:sec:13-appI} \addcontentsline{toc}{section}{Appendix 13.I --- Defensive Coda} \paragraph{What this chapter does \emph{not} claim.} @@ -1437,7 +1437,7 @@ \section*{Appendix 13.I --- Defensive Coda} \end{flushright} \section*{Appendix 13.J --- Numerical Sanity Check} -\label{sec:13-appJ} +\label{fa_13:sec:13-appJ} \addcontentsline{toc}{section}{Appendix 13.J --- Numerical Sanity Check} We verify, to six decimal places, the central numerical claims of @@ -1485,7 +1485,7 @@ \section*{Appendix 13.J --- Numerical Sanity Check} six-decimal target. \section*{Appendix 13.K --- Extended Worked Examples} -\label{sec:13-appK} +\label{fa_13:sec:13-appK} \addcontentsline{toc}{section}{Appendix 13.K --- Extended Worked Examples} \paragraph{Example K-1 --- Computing $\sum_{k} p_{k}^{2}$.} @@ -1546,7 +1546,7 @@ \section*{Appendix 13.K --- Extended Worked Examples} the centred geometry of the cube. \section*{Appendix 13.L --- Connection to Chapter 17 in Detail} -\label{sec:13-appL} +\label{fa_13:sec:13-appL} \addcontentsline{toc}{section}{Appendix 13.L --- Connection to Chapter 17} The connection between the algebraic Metatron's Cube of this @@ -1594,7 +1594,7 @@ \section*{Appendix 13.L --- Connection to Chapter 17 in Detail} indices, which is rare in the present architecture. \section*{Appendix 13.M --- Future Work} -\label{sec:13-appM} +\label{fa_13:sec:13-appM} \addcontentsline{toc}{section}{Appendix 13.M --- Future Work} We list four directions for future work that build on the present diff --git a/docs/phd/chapters/fa_14.tex b/docs/phd/chapters/fa_14.tex index 1123fdb080..a503170979 100644 --- a/docs/phd/chapters/fa_14.tex +++ b/docs/phd/chapters/fa_14.tex @@ -6,7 +6,7 @@ \chapter{Platonic Solids: Eval Semantics --- BPB Metric} \caption*{Figure --- Platonic Solids: Eval Semantics --- BPB Metric.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_14:abstract} Evaluation of language models requires a metric that is simultaneously information-theoretically @@ -23,7 +23,7 @@ \section{Abstract}\label{abstract} chapter; the evaluation protocol is specified as a pre-registration constraint in App.E. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_14:introduction} The selection of an evaluation metric for a language model is not merely a practical @@ -56,10 +56,10 @@ \section{1. Introduction}\label{introduction} size \(F_{21} = 10946\) [2]. \section{2. BPB: Definition and Algebraic -Properties}\label{bpb-definition-and-algebraic-properties} +Properties}\label{fa_14:bpb-definition-and-algebraic-properties} \subsection{2.1 Cross-Entropy and -Perplexity}\label{cross-entropy-and-perplexity} +Perplexity}\label{fa_14:cross-entropy-and-perplexity} Let \(\mathcal{D} = (x_1, x_2, \ldots, x_N)\) be a token sequence. A language model \(p_\theta\) @@ -75,7 +75,7 @@ \subsection{2.1 Cross-Entropy and and artificially lowering \(\mathcal{L}\). \subsection{2.2 Byte-Level -Normalisation}\label{byte-level-normalisation} +Normalisation}\label{fa_14:byte-level-normalisation} Let \(B\) be the total number of UTF-8 bytes in the test corpus and \(N\) the number of tokens @@ -90,7 +90,7 @@ \subsection{2.2 Byte-Level granularity. \subsection{2.3 φ-Weighted -BPB}\label{ux3c6-weighted-bpb} +BPB}\label{fa_14:ux3c6-weighted-bpb} The Trinity S³AI loss function uses φ-weighted token contributions: @@ -116,10 +116,10 @@ \subsection{2.3 φ-Weighted rational. \section{3. Gate Thresholds and Their -Derivation}\label{gate-thresholds-and-their-derivation} +Derivation}\label{fa_14:gate-thresholds-and-their-derivation} \subsection{3.1 Gate-2: BPB ≤ -1.85}\label{gate-2-bpb-1.85} +1.85}\label{fa_14:gate-2-bpb-1.85} The Gate-2 threshold corresponds to the information content of a ternary source with @@ -147,7 +147,7 @@ \subsection{3.1 Gate-2: BPB ≤ [3]. \subsection{3.2 Gate-3: BPB ≤ -1.50}\label{gate-3-bpb-1.50} +1.50}\label{fa_14:gate-3-bpb-1.50} Gate-3 corresponds to the lossless coding limit of a source whose symbols are distributed according @@ -166,7 +166,7 @@ \subsection{3.2 Gate-3: BPB ≤ \subsection{3.3 Relationship to the DARPA Energy -Goal}\label{relationship-to-the-darpa-energy-goal} +Goal}\label{fa_14:relationship-to-the-darpa-energy-goal} The DARPA 3000$\times$ energy goal specifies energy-per-correct-bit of output [6]. BPB is @@ -177,7 +177,7 @@ \subsection{3.3 Relationship to the DARPA in BPB directly contributes to the 3000$\times$ figure. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_14:results-evidence} Evaluation was conducted on WikiText-103 (test split, 245 kB) using the sanctioned seed set. All @@ -205,18 +205,18 @@ \section{4. Results / even-indexed tokens. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_14:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_14:sealed-seeds} Inherits the canonical seed pool F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_14:discussion} The BPB metric is well-suited to the Trinity S³AI setting because its normalisation by byte count @@ -240,7 +240,7 @@ \section{7. Discussion}\label{discussion} R5-honesty constraint that governs the dissertation. -\section{References}\label{references} +\section{References}\label{fa_14:references} [1] GOLDEN SUNFLOWERS dissertation. Ch.28 --- FPGA Implementation on QMTech XC7A100T. This diff --git a/docs/phd/chapters/fa_15.tex b/docs/phd/chapters/fa_15.tex index 677d4185d4..8dd8697600 100644 --- a/docs/phd/chapters/fa_15.tex +++ b/docs/phd/chapters/fa_15.tex @@ -7,7 +7,7 @@ \chapter{Kepler Solids: BPB Benchmark --- Railway PostgreSQL Write} \caption*{Figure --- Kepler Solids: BPB Benchmark --- Railway PostgreSQL Write.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_15:abstract} This chapter documents the bits-per-byte (BPB) benchmark protocol for Trinity S³AI and the @@ -29,7 +29,7 @@ \section{Abstract}\label{abstract} for the M4 (2.7B) model with GF16 PHI\_BIAS=60 weights. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_15:introduction} Bits per byte (BPB) is the primary accuracy metric for language modelling in this dissertation. It is @@ -71,10 +71,10 @@ \section{1. Introduction}\label{introduction} \section{2. BPB Protocol and Monotone Backward Invariant -(INV-1)}\label{bpb-protocol-and-monotone-backward-invariant-inv-1} +(INV-1)}\label{fa_15:bpb-protocol-and-monotone-backward-invariant-inv-1} \subsection{2.1 Evaluation -Protocol}\label{evaluation-protocol} +Protocol}\label{fa_15:evaluation-protocol} BPB is computed on the WikiText-103 test split (245,569 bytes after UTF-8 encoding) using a @@ -100,7 +100,7 @@ \subsection{2.1 Evaluation run metadata row containing those seed values. \subsection{2.2 INV-1: BPB Monotone -Backward}\label{inv-1-bpb-monotone-backward} +Backward}\label{fa_15:inv-1-bpb-monotone-backward} \textbf{Invariant INV-1} (\texttt{Trinity.Canonical.Igla.INV1\_BpbMonotoneBackward}, @@ -141,7 +141,7 @@ \subsection{2.2 INV-1: BPB Monotone (GF16 precision bounds) and the spectral identity \(\varphi^2 + \varphi^{-2} = 3\) [1, 2]. -\subsection{2.3 Warmup Gate}\label{warmup-gate} +\subsection{2.3 Warmup Gate}\label{fa_15:warmup-gate} INV-1 applies only for \(t \geq t_0 = 100\). Before that, the learning rate ramp can cause @@ -154,10 +154,10 @@ \subsection{2.3 Warmup Gate}\label{warmup-gate} \(< 1.85\) (Gate-2) [6]. \section{3. Railway PostgreSQL Write-Back -Architecture}\label{railway-write-back-architecture} +Architecture}\label{fa_15:railway-write-back-architecture} \subsection{3.1 Database -Schema}\label{database-schema} +Schema}\label{fa_15:database-schema} The Railway PostgreSQL instance (project \texttt{golden-sunflowers-bench}, region @@ -189,7 +189,7 @@ \subsection{3.1 Database query time by the IGLA RACE agent (Ch.21). \subsection{3.2 Write-Back -Protocol}\label{write-back-protocol} +Protocol}\label{fa_15:write-back-protocol} At every evaluation checkpoint (every 500 steps), the bench agent: @@ -219,7 +219,7 @@ \subsection{3.2 Write-Back audit is not corrupted by duplicate entries. \subsection{3.3 Gate -Evaluation}\label{gate-evaluation} +Evaluation}\label{fa_15:gate-evaluation} After each write, the bench agent evaluates the Gate-2 and Gate-3 predicates: @@ -233,7 +233,7 @@ \subsection{3.3 Gate INV-7 (Ch.21 [6]). \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_15:results-evidence} \textbf{BPB trajectory (M4, 2.7B, GF16 PHI\_BIAS=60, seed 1597):} @@ -292,7 +292,7 @@ \section{4. Results / failures and 0 seed-constraint violations. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_15:qed-assertions} No Coq theorems are directly anchored to this chapter's output files. The relevant obligations @@ -302,7 +302,7 @@ \section{5. Qed \filepath{t27/proofs/canonical/}. The champion lr \(= 0.004\) is certified by INV-1. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_15:sealed-seeds} \begin{itemize} \tightlist @@ -314,7 +314,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} monotone backward, lr=0.004 (9 Qed). \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_15:discussion} The BPB benchmark protocol and Railway PostgreSQL write-back described here provide the empirical backbone for @@ -335,7 +335,7 @@ \section{7. Discussion}\label{discussion} in Ch.21 to operate at sub-second polling intervals. -\section{References}\label{references} +\section{References}\label{fa_15:references} [1] \emph{Golden Sunflowers} dissertation, Ch.3 --- Trinity Identity diff --git a/docs/phd/chapters/fa_16.tex b/docs/phd/chapters/fa_16.tex index 91b8d5e594..370349670a 100644 --- a/docs/phd/chapters/fa_16.tex +++ b/docs/phd/chapters/fa_16.tex @@ -6,7 +6,7 @@ \chapter{Sacred Ratios: 360-lane Phi-Distance Grid} \caption*{Figure --- Sacred Ratios: 360-lane Phi-Distance Grid.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_16:abstract} Angular discretisation of the unit circle into 360 equally-spaced lanes is standard in robotics and @@ -30,7 +30,7 @@ \section{Abstract}\label{abstract} \texttt{t27\#569} (INV-4 merge) must be satisfied before the grid can be deployed in training. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_16:introduction} The Trinity S³AI architecture processes spatial context through a Neural Cellular Automaton (NCA) @@ -74,7 +74,7 @@ \section{1. Introduction}\label{introduction} \texttt{t27\#569} [3]. \section{2. The Phi-Distance -Function}\label{the-phi-distance-function} +Function}\label{fa_16:the-phi-distance-function} \textbf{Definition 2.1 (Vogel angle).} The Vogel divergence angle is @@ -125,7 +125,7 @@ \section{2. The Phi-Distance ternary NCA inference. \section{3. Grid Construction and Sparsity -Analysis}\label{grid-construction-and-sparsity-analysis} +Analysis}\label{fa_16:grid-construction-and-sparsity-analysis} \textbf{Construction 3.1 (360-lane grid).} The grid \(\mathcal{G}\) is an ordered set of (lane, @@ -185,7 +185,7 @@ \section{3. Grid Construction and Sparsity [7]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_16:results-evidence} Evaluation was performed over \(F_{19} = 4181\) NCA inference steps on the canonical A1 dataset. @@ -241,7 +241,7 @@ \section{4. Results / result is stable to ±0.03 across seeds. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_16:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. The @@ -249,7 +249,7 @@ \section{5. Qed (\texttt{INV4\_NcaEntropyBand.v}, 12 Qed) as an imported invariant, credited to Ch.10. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_16:sealed-seeds} \begin{itemize} \tightlist @@ -263,7 +263,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci/Lucas reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_16:discussion} The 360-lane phi-distance grid is a practically effective spatial prior, but two limitations @@ -288,7 +288,7 @@ \section{7. Discussion}\label{discussion} sparsity threshold to define a two-tier grid with improved Gate-3 BPB performance. -\section{References}\label{references} +\section{References}\label{fa_16:references} [1] GOLDEN SUNFLOWERS dissertation, Ch.7 --- Phyllotaxis and the Vogel Divergence Angle. This diff --git a/docs/phd/chapters/fa_17.tex b/docs/phd/chapters/fa_17.tex index 1617b5fe95..af60cc04c5 100644 --- a/docs/phd/chapters/fa_17.tex +++ b/docs/phd/chapters/fa_17.tex @@ -7,7 +7,7 @@ \chapter{Golden Spiral: Ablation Matrix} \caption*{Figure --- Golden Spiral: Ablation Matrix.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_17:abstract} A systematic ablation study isolates the contribution of each architectural decision in the @@ -29,7 +29,7 @@ \section{Abstract}\label{abstract} the formal Coq proof obligations distributed across the dissertation. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_17:introduction} Architectural claims in neural network research are frequently confounded: multiple @@ -70,7 +70,7 @@ \section{1. Introduction}\label{introduction} \(\{F_{17}=1597, F_{18}=2584, F_{19}=4181, F_{20}=6765, F_{21}=10946, L_7=29, L_8=47\}\). \section{2. Factor Definitions and Experimental -Design}\label{factor-definitions-and-experimental-design} +Design}\label{fa_17:factor-definitions-and-experimental-design} \textbf{Definition 2.1 (Ablation factors).} The seven binary factors are: @@ -139,7 +139,7 @@ \section{2. Factor Definitions and Experimental \section{3. Analysis of Effects and Golden-Ratio -Structure}\label{analysis-of-effects-and-golden-ratio-structure} +Structure}\label{fa_17:analysis-of-effects-and-golden-ratio-structure} The full-factorial analysis identifies two dominant first-order effects and one significant @@ -215,7 +215,7 @@ \section{3. Analysis of Effects and has no first-order effect on BPB [6]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_17:results-evidence} Summary of first-order BPB effects (positive = BPB worsens when factor is removed): @@ -266,18 +266,18 @@ \section{4. Results / [7]. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_17:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_17:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_17:discussion} The ablation matrix confirms that the canonical seed selection (factor C) and the golden @@ -303,7 +303,7 @@ \section{7. Discussion}\label{discussion} (FPGA hardware detail) and Ch.34 (energy-per-token analysis). -\section{References}\label{references} +\section{References}\label{fa_17:references} [1] GOLDEN SUNFLOWERS Dissertation, Ch.28 --- \emph{FPGA hardware benchmarks}. Zenodo B002. DOI: diff --git a/docs/phd/chapters/fa_18.tex b/docs/phd/chapters/fa_18.tex index c4d37971b2..40c7879cf5 100644 --- a/docs/phd/chapters/fa_18.tex +++ b/docs/phd/chapters/fa_18.tex @@ -7,7 +7,7 @@ \chapter{Torus Geometry: Falsification \& Limitations} \caption*{Figure --- Torus Geometry: Falsification \& Limitations.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_18:abstract} No formal system is complete without an honest accounting of its boundaries. This chapter @@ -30,7 +30,7 @@ \section{Abstract}\label{abstract} path is the Coq.Interval upgrade lane described in Section 3. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_18:introduction} The GOLDEN SUNFLOWERS dissertation rests on two pillars: a formally verified arithmetic substrate @@ -65,7 +65,7 @@ \section{1. Introduction}\label{introduction} \section{2. State-of-the-Art Comparison (CLARA-SOA -Snapshot)}\label{state-of-the-art-comparison-clara-soa-snapshot} +Snapshot)}\label{fa_18:state-of-the-art-comparison-clara-soa-snapshot} The following table reflects the CLARA-SOA-COMPARISON.md snapshot taken during the @@ -170,7 +170,7 @@ \section{2. State-of-the-Art Comparison documented in Section 3. \section{3. Coq.Interval Upgrade -Lane}\label{coq.interval-upgrade-lane} +Lane}\label{fa_18:coq.interval-upgrade-lane} Of the 438 theorem statements in the Coq corpus, 297 carry \texttt{Qed} status and 41 carry @@ -232,7 +232,7 @@ \section{3. Coq.Interval Upgrade (Iris integration). \section{4. Hardware and Runtime -Limitations}\label{hardware-and-runtime-limitations} +Limitations}\label{fa_18:hardware-and-runtime-limitations} \textbf{FPGA resource ceiling.} The XC7A100T contains 101440 LUTs and 135200 FFs. The current @@ -270,18 +270,18 @@ \section{4. Hardware and Runtime until Groups A and B are closed. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_18:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_18:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_18:discussion} This chapter occupies the most uncomfortable position in a dissertation: it quantifies the @@ -310,7 +310,7 @@ \section{7. Discussion}\label{discussion} Ch.24 (scheduler liveness), and App.A (executive summary of the 297/438 proof census). -\section{References}\label{references} +\section{References}\label{fa_18:references} [1] \filepath{gHashTag/t27/proofs/canonical/} --- Coq canonical proof archive; 65 \texttt{.v} @@ -377,7 +377,7 @@ \section{References}\label{references} \url{https://arxiv.org/abs/2307.13304} \section{Falsification} -\label{sec:falsification:ch18} +\label{fa_18:sec:falsification:ch18} \paragraph{Pre-registered claim (R7).} The torus-embedded $\varphi$-distance grid produces a measurable separation diff --git a/docs/phd/chapters/fa_19.tex b/docs/phd/chapters/fa_19.tex index cba3eb4717..b182493b9e 100644 --- a/docs/phd/chapters/fa_19.tex +++ b/docs/phd/chapters/fa_19.tex @@ -7,7 +7,7 @@ \chapter{Fibonacci Tessellation: Welch-t Statistical Analysis} \caption*{Figure --- Fibonacci Tessellation: Welch-t Statistical Analysis.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_19:abstract} Empirical claims in this dissertation are substantiated through a pre-registered Welch @@ -31,7 +31,7 @@ \section{Abstract}\label{abstract} \(\varphi\)-weighted loss function whose BPB is being tested. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_19:introduction} Statistical testing in machine learning is complicated by the fact that a single training run @@ -66,7 +66,7 @@ \section{1. Introduction}\label{introduction} with the composite \(\varphi\)-weighted objective. \section{2. Test Design and -Hypotheses}\label{test-design-and-hypotheses} +Hypotheses}\label{fa_19:test-design-and-hypotheses} \textbf{Notation.} Let \(X_i\) denote the BPB achieved by the TRINITY S³AI model on the held-out @@ -102,7 +102,7 @@ \section{2. Test Design and \section{\texorpdfstring{3. Welch \(t\)-Statistic and Degrees of -Freedom}{3. Welch t-Statistic and Degrees of Freedom}}\label{welch-t-statistic-and-degrees-of-freedom} +Freedom}{3. Welch t-Statistic and Degrees of Freedom}}\label{fa_19:welch-t-statistic-and-degrees-of-freedom} The Welch \(t\)-statistic for a one-sample test against known threshold \(\mu_0\) is: @@ -165,7 +165,7 @@ \section{\texorpdfstring{3. Welch statistically significant at \(\alpha = 0.01\). \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_19:results-evidence} Three results are reported. @@ -209,12 +209,12 @@ \section{4. Results / identity \(\varphi^2 + \varphi^{-2} = 3\) [2]. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_19:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_19:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), @@ -226,7 +226,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} lattice-initialisation experiment used \(F_{19}\), \(F_{20}\), \(F_{21}\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_19:discussion} The primary limitation of the statistical analysis is \(n = 3\): with two degrees of freedom, the @@ -252,7 +252,7 @@ \section{7. Discussion}\label{discussion} (lattice initialisation), and Ch.31 (hardware evaluation). -\section{References}\label{references} +\section{References}\label{fa_19:references} [1] \texttt{igla\_assertions.json} runtime-mirror contract, key diff --git a/docs/phd/chapters/fa_20.tex b/docs/phd/chapters/fa_20.tex index 161dcd916c..acaf5914de 100644 --- a/docs/phd/chapters/fa_20.tex +++ b/docs/phd/chapters/fa_20.tex @@ -7,7 +7,7 @@ \chapter{Standard Model — Fundamental Particles} \end{figure} -\label{ch:20} +\label{fa_20:ch:20} % Lane: A % Agent: Claude % Status: COMPLETE @@ -31,7 +31,7 @@ \section{Gauge Group Structure} \subsection{Color SU(3)} \begin{definition}[QCD Gauge Group] -\label{def:su3} +\label{fa_20:def:su3} The strong interaction group: \begin{equation} SU(3) = \{U \in GL(3,\mathbb{C}) : U^\dagger U = I, \det U = 1\} @@ -44,7 +44,7 @@ \subsection{Color SU(3)} \end{definition} \begin{proposition}[SU(3) Dimension] -\label{prop:su3-dim} +\label{fa_20:prop:su3-dim} Dimension of fundamental representation: \begin{equation} \dim(\mathbf{3}) = 3 @@ -61,7 +61,7 @@ \subsection{Color SU(3)} \subsection{Weak SU(2)} \begin{definition}[Electroweak Gauge Group] -\label{def:su2} +\label{fa_20:def:su2} The weak interaction group: \begin{equation} SU(2) = \{U \in GL(2,\mathbb{C}) : U^\dagger U = I, \det U = 1\} @@ -69,7 +69,7 @@ \subsection{Weak SU(2)} \end{definition} \begin{theorem}[Pauli Matrices] -\label{thm:pauli} +\label{fa_20:thm:pauli} Generators of $SU(2)$: \begin{equation} \sigma^x = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}, \quad @@ -86,7 +86,7 @@ \subsection{Weak SU(2)} \subsection{Electromagnetic U(1)} \begin{definition}[QED Gauge Group] -\label{def:u1} +\label{fa_20:def:u1} The electromagnetic group: \begin{equation} U(1) = \{e^{i\theta} : \theta \in [0,2\pi)\} @@ -94,7 +94,7 @@ \subsection{Electromagnetic U(1)} \end{definition} \begin{proposition}[U(1) Charge Quantization] -\label{prop:u1-charge} +\label{fa_20:prop:u1-charge} Electric charge is quantized: \begin{equation} Q = \frac{1}{\sqrt{3}} \times \text{integer} @@ -110,7 +110,7 @@ \section{Fermions} \subsection{Quarks} \begin{definition}[Quark Field] -\label{def:quark} +\label{fa_20:def:quark} Quark field with color and flavor: \begin{equation} \psi_{\alpha}^f(x) = \begin{pmatrix}u_\alpha^f \\ d_\alpha^f \\ s_\alpha^f\end{pmatrix} @@ -139,7 +139,7 @@ \subsection{Quarks} \subsection{Leptons} \begin{definition}[Lepton Field] -\label{def:lepton} +\label{fa_20:def:lepton} Lepton field without color: \begin{equation} \psi^f(x) = \begin{pmatrix}e^f \\ \nu_e^f \\ \nu_\mu^f \\ \nu_\tau^f\end{pmatrix} @@ -170,7 +170,7 @@ \section{Bosons} \subsection{Gauge Bosons} \begin{definition}[Gauge Field] -\label{def:gauge-boson} +\label{fa_20:def:gauge-boson} Gauge boson field: \begin{equation} A_\mu^a(x) @@ -197,7 +197,7 @@ \subsection{Gauge Bosons} \subsection{Higgs Boson} \begin{definition}[Higgs Field] -\label{def:higgs} +\label{fa_20:def:higgs} The Higgs doublet: \begin{equation} \Phi = \begin{pmatrix}\phi^+ \\ \phi^0\end{pmatrix} @@ -210,7 +210,7 @@ \subsection{Higgs Boson} \end{definition} \begin{proposition}[Higgs Mass Generation] -\label{prop:higgs-mass} +\label{fa_20:prop:higgs-mass} Particle masses from Higgs mechanism: \begin{equation} m_f = \frac{y_f v}{\sqrt{2}} @@ -226,7 +226,7 @@ \section{Mixing Matrices} \subsection{CKM Matrix}\label{sec:ckm} \begin{definition}[Cabibbo-Kobayashi-Maskawa] -\label{def:ckm} +\label{fa_20:def:ckm} Quark mixing matrix: \begin{equation} V_{CKM} = \begin{pmatrix} @@ -238,7 +238,7 @@ \subsection{CKM Matrix}\label{sec:ckm} \end{definition} \begin{proposition}[CKM Golden Angles] -\label{prop:ckm-golden} +\label{fa_20:prop:ckm-golden} The CKM angles approximate: \begin{equation} \theta_{12} \approx 13.0^\circ = \frac{\pi}{\phi^3} @@ -251,7 +251,7 @@ \subsection{CKM Matrix}\label{sec:ckm} \subsection{PMNS Matrix} \begin{definition}[Pontecorvo-Maki-Nakagawa-Sakata] -\label{def:pmns} +\label{fa_20:def:pmns} Neutrino mixing matrix: \begin{equation} U_{PMNS} = \begin{pmatrix} @@ -263,7 +263,7 @@ \subsection{PMNS Matrix} \end{definition} \begin{proposition}[PMNS Golden Ratio] -\label{prop:pmns-golden} +\label{fa_20:prop:pmns-golden} Neutrino mixing angles: \begin{equation} \theta_{12} \approx 33.4^\circ = \frac{\pi}{\phi^2} @@ -280,7 +280,7 @@ \section{Particle Masses}\label{sec:mass} \subsection{Koide Formula} \begin{definition}[Koide Relation] -\label{def:koide} +\label{fa_20:def:koide} Charged lepton masses: \begin{equation} \frac{m_e + m_\mu + m_\tau}{\sqrt{m_e^2 + m_\mu^2 + m_\tau^2}} = \frac{2}{3} @@ -288,7 +288,7 @@ \subsection{Koide Formula} \end{definition} \begin{proposition}[Golden Koide] -\label{prop:golden-koide} +\label{fa_20:prop:golden-koide} Modified Koide with golden ratio: \begin{equation} \frac{m_e + \phi m_\mu + \phi^2 m_\tau}{\sqrt{m_e^2 + (\phi m_\mu)^2 + (\phi^2 m_\tau)^2}} = \frac{2}{3} @@ -322,14 +322,14 @@ \section{Coupling Constants} \subsection{Fine-Structure Constant} \begin{definition}[Alpha] -\label{def:alpha} +\label{fa_20:def:alpha} \begin{equation} \alpha = \frac{e^2}{4\pi\epsilon_0\hbar c} \end{equation} \end{definition} \begin{proposition}[Golden Alpha] -\label{prop:golden-alpha} +\label{fa_20:prop:golden-alpha} \begin{equation} \alpha = \frac{\phi^4}{8\pi^2} \end{equation} @@ -344,7 +344,7 @@ \subsection{Weak Coupling} \end{equation} \begin{theorem}[Weak Golden] -\label{thm:weak-golden} +\label{fa_20:thm:weak-golden} \begin{equation} \frac{g_W}{g_{\text{EM}}} = \frac{1}{\sin\theta_W} \approx \phi^{0.5} \end{equation} @@ -357,7 +357,7 @@ \subsection{Strong Coupling} \end{equation} \begin{theorem}[Strong Golden] -\label{thm:strong-golden} +\label{fa_20:thm:strong-golden} At confinement scale: \begin{equation} \frac{g_s}{g_W} \approx \phi @@ -371,7 +371,7 @@ \section{Analysis} \subsection{Gauge Unification} \begin{theorem}[Standard Model Symmetry] -\label{thm:sm-symmetry} +\label{fa_20:thm:sm-symmetry} The SM gauge group: \begin{equation} G_{SM} = SU(3)_C \times SU(2)_L \times U(1)_Y @@ -429,7 +429,7 @@ \subsection*{Open Questions} % §Falsification — added by LP-falsification lane (R7 retrofit) % ===================================================================== -\section{Falsification Criterion}\label{sec:20-falsify} +\section{Falsification Criterion}\label{fa_20:sec:20-falsify} \subsection{What Would Refute This Claim} diff --git a/docs/phd/chapters/fa_21.tex b/docs/phd/chapters/fa_21.tex index 22e0b07ec3..d15584c32e 100644 --- a/docs/phd/chapters/fa_21.tex +++ b/docs/phd/chapters/fa_21.tex @@ -10,7 +10,7 @@ \chapter{Quantum Field Theory — Fields of Nature} \end{figure} -\label{ch:21} +\label{fa_21:ch:21} % Lane: A % Agent: Claude % Status: COMPLETE @@ -34,7 +34,7 @@ \section{Classical Field Theory} \subsection{Lagrangian Density} \begin{definition}[Lagrangian] -\label{def:lagrangian} +\label{fa_21:def:lagrangian} \begin{equation} S = \int d^4x \mathcal{L}(\phi, \partial_\mu\phi) \end{equation} @@ -55,14 +55,14 @@ \subsection{Euler-Lagrange Equations} \subsection{Klein-Gordon Field} \begin{definition}[Klein-Gordon Lagrangian] -\label{def:kg} +\label{fa_21:def:kg} \begin{equation} \mathcal{L} = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}m^2\phi^2 \end{equation} \end{definition} \begin{proposition}[Klein-Gordon Equation] -\label{prop:kg-eq} +\label{fa_21:prop:kg-eq} \begin{equation} (\partial_\mu\partial^\mu + m^2)\phi = 0 \end{equation} @@ -80,7 +80,7 @@ \section{Canonical Quantization} \subsection{Canonical Commutation} \begin{definition}[Field Operators] -\label{def:field-ops} +\label{fa_21:def:field-ops} \begin{equation} [\phi(t,\mathbf{x}), \pi(t,\mathbf{y})] = i\delta^3(\mathbf{x}-\mathbf{y}) \end{equation} @@ -91,7 +91,7 @@ \subsection{Canonical Commutation} \subsection{Creation and Annihilation} \begin{theorem}[Mode Expansion] -\label{thm:mode-expansion} +\label{fa_21:thm:mode-expansion} \begin{equation} \phi(x) = \sum_{\mathbf{k}}\frac{1}{\sqrt{2\omega_k V}}(a_{\mathbf{k}} e^{-ikx} + a_{\mathbf{k}}^\dagger e^{ikx}) \end{equation} @@ -105,7 +105,7 @@ \subsection{Creation and Annihilation} \subsection{Fock Space} \begin{definition}[Fock Space] -\label{def:fock} +\label{fa_21:def:fock} \begin{equation} \mathcal{F} = \bigoplus_{n=0}^\infty \mathcal{H}^{\otimes n} \end{equation} @@ -120,7 +120,7 @@ \section{Gauge Field Theory} \subsection{U(1) Gauge Theory} \begin{definition}[QED Lagrangian] -\label{def:qed} +\label{fa_21:def:qed} \begin{equation} \mathcal{L}_{QED} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} \end{equation} @@ -131,7 +131,7 @@ \subsection{U(1) Gauge Theory} \subsection{Non-Abelian Gauge Theory} \begin{definition}[Yang-Mills Lagrangian] -\label{def:yang-mills} +\label{fa_21:def:yang-mills} \begin{equation} \mathcal{L}_{YM} = -\frac{1}{4}F_{\mu\nu}^a F^{a\mu\nu} \end{equation} @@ -143,7 +143,7 @@ \subsection{Non-Abelian Gauge Theory} \end{definition} \begin{proposition}[Non-Abelian Commutator] -\label{prop:non-abelian} +\label{fa_21:prop:non-abelian} \begin{equation} [D_\mu, D_\nu] = igF_{\mu\nu} \end{equation} @@ -158,7 +158,7 @@ \section{Renormalization} \subsection{Regularization} \begin{definition}[Dimensional Regularization] -\label{def:dim-reg} +\label{fa_21:def:dim-reg} Continue spacetime dimension to $d = 4 - \epsilon$: \begin{equation} \int \frac{d^dk}{(2\pi)^d} \frac{1}{k^2 + m^2} = \frac{m^{d-2}}{(4\pi)^{d/2}} \Gamma(1-d/2) @@ -168,7 +168,7 @@ \subsection{Regularization} \subsection{Renormalization Group} \begin{theorem}[RG Equation] -\label{thm:rg} +\label{fa_21:thm:rg} \begin{equation} \left[\mu\frac{\partial}{\partial\mu} + \beta(g)\frac{\partial}{\partial g} - n\gamma_m\right] G^{(n)}(p_i,g,\mu,m) = 0 \end{equation} @@ -179,7 +179,7 @@ \subsection{Renormalization Group} \subsection{Beta Function} \begin{proposition}[Beta Golden Ratio] -\label{prop:beta-golden} +\label{fa_21:prop:beta-golden} The QED beta function: \begin{equation} \beta(e) = \frac{e^3}{12\pi^2} @@ -198,7 +198,7 @@ \section{Path Integral Formulation} \subsection{Generating Functional} \begin{definition}[Path Integral] -\label{def:path-integral} +\label{fa_21:def:path-integral} \begin{equation} Z[J] = \int \mathcal{D}\phi \exp\left[i\int d^4x(\mathcal{L} + J\phi)\right] \end{equation} @@ -207,7 +207,7 @@ \subsection{Generating Functional} \subsection{Correlation Functions} \begin{theorem}[n-point Functions] -\label{thm:n-point} +\label{fa_21:thm:n-point} \begin{equation} \langle\phi(x_1)\cdots\phi(x_n)\rangle = \frac{1}{Z[0]}\left.\frac{\delta^n Z[J]}{i\delta J(x_1)\cdots i\delta J(x_n)}\right|_{J=0} \end{equation} @@ -220,7 +220,7 @@ \subsection{Perturbation Theory} \end{equation} \begin{proposition}[Feynman Diagrams] -\label{prop:feynman} +\label{fa_21:prop:feynman} Perturbative expansion generates Feynman diagrams: \begin{itemize} \item Lines: Propagators $1/(p^2 - m^2 + i\epsilon)$ @@ -236,14 +236,14 @@ \section{Spontaneous Symmetry Breaking} \subsection{Higgs Mechanism} \begin{definition}[Higgs Potential] -\label{def:higgs-pot} +\label{fa_21:def:higgs-pot} \begin{equation} V(\phi) = -\mu^2\phi^\dagger\phi + \lambda(\phi^\dagger\phi)^2 \end{equation} \end{definition} \begin{theorem}[Symmetry Breaking] -\label{thm:ssb} +\label{fa_21:thm:ssb} For $\mu^2 > 0$, minimum at: \begin{equation} |\phi| = \frac{v}{\sqrt{2}}, \quad v = \sqrt{\frac{\mu^2}{\lambda}} @@ -255,7 +255,7 @@ \subsection{Higgs Mechanism} \subsection{Goldstone Bosons} \begin{proposition}[Goldstone Theorem] -\label{prop:goldstone} +\label{fa_21:prop:goldstone} Spontaneously broken continuous symmetry produces massless Goldstone bosons. \end{proposition} @@ -268,7 +268,7 @@ \section{Effective Field Theory} \subsection{Operator Expansion} \begin{definition}[EFT Lagrangian] -\label{def:eft} +\label{fa_21:def:eft} \begin{equation} \mathcal{L}_{\text{EFT}} = \sum_i C_i(\mu) O_i(\mu) \end{equation} @@ -283,7 +283,7 @@ \subsection{Power Counting} \end{equation} \begin{theorem}[Weinberg Theorem] -\label{thm:weinberg} +\label{fa_21:thm:weinberg} EFTs are organized as expansion in $E/\Lambda$ where $E$ is energy scale and $\Lambda$ is cutoff. \end{theorem} @@ -353,7 +353,7 @@ \subsection*{Open Questions} % §Falsification — added by LP-falsification lane (R7 retrofit) % ===================================================================== -\section{Falsification Criterion}\label{sec:21-falsify} +\section{Falsification Criterion}\label{fa_21:sec:21-falsify} \subsection{What Would Refute This Claim} diff --git a/docs/phd/chapters/fa_22.tex b/docs/phd/chapters/fa_22.tex index 8b36db01dd..a72d0e1ace 100644 --- a/docs/phd/chapters/fa_22.tex +++ b/docs/phd/chapters/fa_22.tex @@ -7,7 +7,7 @@ \chapter{E8 Symmetry: Railway-trios Orchestration} \caption*{Figure --- E8 Symmetry: Railway-trios Orchestration.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_22:abstract} Deploying a formally verified ternary neural system at scale requires an orchestration layer @@ -34,7 +34,7 @@ \section{Abstract}\label{abstract} operational milestone requiring full Gate-3 compliance. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_22:introduction} The Trios codebase organises model training, evaluation, and deployment through a Railway-style @@ -74,7 +74,7 @@ \section{1. Introduction}\label{introduction} \section{2. Worker Pool Invariants and Falsification -Witnesses}\label{worker-pool-invariants-and-falsification-witnesses} +Witnesses}\label{fa_22:worker-pool-invariants-and-falsification-witnesses} \textbf{Definition 2.1 (Worker pool configuration).} A configuration is a triple @@ -154,7 +154,7 @@ \section{2. Worker Pool Invariants and the other components. \(\square\) \section{3. Satisfaction Witness and Victory -Predicate}\label{satisfaction-witness-and-victory-predicate} +Predicate}\label{fa_22:satisfaction-witness-and-victory-predicate} The falsification witnesses of Section 2 demonstrate that the invariant system correctly @@ -228,7 +228,7 @@ \section{3. Satisfaction Witness and Victory tiers). \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_22:results-evidence} The INV-8 composite invariant has been validated across \(F_{20} = 6765\) Railway deployment events @@ -281,7 +281,7 @@ \section{4. Results / \texttt{Qed}; no \texttt{admit} statements. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_22:qed-assertions} \begin{itemize} \tightlist @@ -324,7 +324,7 @@ \section{5. Qed gates passed, Gate-3 pending. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_22:sealed-seeds} \begin{itemize} \tightlist @@ -338,7 +338,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci/Lucas reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_22:discussion} The primary limitation of the INV-8 composite invariant is that it checks configuration values @@ -364,7 +364,7 @@ \section{7. Discussion}\label{discussion} (Ch.28, Ch.31, Ch.34) to create a closed-loop formally-verified deployment pipeline. -\section{References}\label{references} +\section{References}\label{fa_22:references} [1] GOLDEN SUNFLOWERS dissertation, Ch.3 --- Ternary Arithmetic Foundations. This volume. diff --git a/docs/phd/chapters/fa_23.tex b/docs/phd/chapters/fa_23.tex index 5cdc93cc74..4532391140 100644 --- a/docs/phd/chapters/fa_23.tex +++ b/docs/phd/chapters/fa_23.tex @@ -9,7 +9,7 @@ \chapter{GF(16) Algebra: MCP Integration} \caption*{Figure --- GF(16) Algebra: MCP Integration.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_23:abstract} The Model Context Protocol (MCP) provides a standardised interface for connecting language @@ -32,7 +32,7 @@ \section{Abstract}\label{abstract} relative to the baseline 63 tokens/sec rate when MCP overhead is included. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_23:introduction} Large-scale deployment of neural inference engines increasingly relies on agentic architectures in @@ -71,7 +71,7 @@ \section{1. Introduction}\label{introduction} exceeding \(N + L\). \section{2. MCP Adapter Layer -Architecture}\label{mcp-adapter-layer-architecture} +Architecture}\label{fa_23:mcp-adapter-layer-architecture} \textbf{Definition 2.1 (MCP context boundary).} A \emph{canonical boundary} is a token position @@ -134,7 +134,7 @@ \section{2. MCP Adapter Layer regardless of padding content [4]. \section{3. Protocol Implementation and Latency -Analysis}\label{protocol-implementation-and-latency-analysis} +Analysis}\label{fa_23:protocol-implementation-and-latency-analysis} The MCP adapter is implemented as a thin Rust layer sitting between the FPGA token stream and @@ -199,7 +199,7 @@ \section{3. Protocol Implementation and Latency [7]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_23:results-evidence} Performance measurements on QMTech XC7A100T FPGA (0 DSP slices, 92 MHz clock, 1 W): @@ -247,18 +247,18 @@ \section{4. Results / that include MCP integration (cf.~Ch.17). \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_23:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_23:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_23:discussion} The MCP integration chapter demonstrates that the \(\varphi\)-structured inference architecture can @@ -280,7 +280,7 @@ \section{7. Discussion}\label{discussion} (GLN normalisation), Ch.27 (TRI-27 verifiable VM) and App.F (FPGA bitstream distribution). -\section{References}\label{references} +\section{References}\label{fa_23:references} [1] Anthropic. (2024). Model Context Protocol Specification v1.0. diff --git a/docs/phd/chapters/fa_24.tex b/docs/phd/chapters/fa_24.tex index 798d793e96..181c511b44 100644 --- a/docs/phd/chapters/fa_24.tex +++ b/docs/phd/chapters/fa_24.tex @@ -11,7 +11,7 @@ \chapter{IGLA Architecture: Period-locked Runtime Monitor} \caption*{Figure --- IGLA Architecture: Period-locked Runtime Monitor.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_24:abstract} The Period-Locked Runtime Monitor (PLRM) is a scheduling and watchdog component of the IGLA RACE @@ -32,7 +32,7 @@ \section{Abstract}\label{abstract} pending Iris integration (Ch.18); all safety properties are Qed-proved. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_24:introduction} A multi-agent inference runtime operating on shared hardware must guarantee two properties @@ -84,9 +84,9 @@ \section{1. Introduction}\label{introduction} structure. \section{2. Formal Model of the Period-Locked -Monitor}\label{formal-model-of-the-period-locked-monitor} +Monitor}\label{fa_24:formal-model-of-the-period-locked-monitor} -\subsection{2.1 Agent Model}\label{agent-model} +\subsection{2.1 Agent Model}\label{fa_24:agent-model} Let \(\mathcal{A} = \{a_1, \ldots, a_k\}\) be the set of IGLA RACE agents. Each agent \(a_i\) is @@ -112,7 +112,7 @@ \subsection{2.1 Agent Model}\label{agent-model} ACTIVE. \subsection{2.2 Coq -Encoding}\label{coq-encoding} +Encoding}\label{fa_24:coq-encoding} The PLRM is formalised in \filepath{t27/proofs/canonical/} as a @@ -139,7 +139,7 @@ \subsection{2.2 Coq has been identified as the mechanisation target. \subsection{2.3 Period Ratio and -Non-Resonance}\label{period-ratio-and-non-resonance} +Non-Resonance}\label{fa_24:period-ratio-and-non-resonance} \textbf{Proposition 2.3} (Non-resonance). \emph{The period clocks \(L_7 = 29\) and @@ -166,7 +166,7 @@ \subsection{2.3 Period Ratio and rather than constituting a blackout. \subsection{2.4 Priority Queue and Phi-Weighted -Scheduling}\label{priority-queue-and-phi-weighted-scheduling} +Scheduling}\label{fa_24:priority-queue-and-phi-weighted-scheduling} When the PLRM preempts an agent, the scheduler selects the next ACTIVE candidate from a binary @@ -186,10 +186,10 @@ \subsection{2.4 Priority Queue and Phi-Weighted remaining bounded without saturation. \section{3. Implementation and Hardware -Interface}\label{implementation-and-hardware-interface} +Interface}\label{fa_24:implementation-and-hardware-interface} \subsection{3.1 RTL -Implementation}\label{rtl-implementation} +Implementation}\label{fa_24:rtl-implementation} The PLRM is implemented as a two-counter module in FPGA RTL: - \textbf{Counter A} @@ -209,7 +209,7 @@ \subsection{3.1 RTL \subsection{3.2 Interrupt Interface with the Hardware -Bridge}\label{interrupt-interface-with-the-hardware-bridge} +Bridge}\label{fa_24:interrupt-interface-with-the-hardware-bridge} The PLRM exposes a 3-bit interrupt line to the Hardware Bridge (Ch.12): @@ -236,7 +236,7 @@ \subsection{3.2 Interrupt Interface with the overhead bytes). Qed. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_24:results-evidence} The PLRM was evaluated on the IGLA RACE simulation bench running the 1003-token HSLM sequence: @@ -284,7 +284,7 @@ \section{4. Results / period bounds verified above. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_24:qed-assertions} No Coq theorems are anchored specifically to this chapter in the input JSON; obligations are tracked @@ -297,13 +297,13 @@ \section{5. Qed SCH-5 carry Admitted status pending Iris integration as detailed in Ch.18.) -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_24:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_24:discussion} The Period-Locked Runtime Monitor is a compact but structurally essential component: without it, the @@ -336,7 +336,7 @@ \section{7. Discussion}\label{discussion} adds vector-symbolic agents to the IGLA RACE pool). -\section{References}\label{references} +\section{References}\label{fa_24:references} [1] \filepath{gHashTag/trios\#418} --- Ch.24 Period-Locked Runtime Monitor scope issue. @@ -388,7 +388,7 @@ \section{References}\label{references} --- 92 MHz clock domain, 0 DSP constraint. \section{Falsification} -\label{sec:falsification:ch24} +\label{fa_24:sec:falsification:ch24} \paragraph{Pre-registered claim (R7).} The Period-Locked Runtime Monitor (PLRM) must guarantee bounded preemption diff --git a/docs/phd/chapters/fa_25.tex b/docs/phd/chapters/fa_25.tex index e7e978dd87..2f157fca49 100644 --- a/docs/phd/chapters/fa_25.tex +++ b/docs/phd/chapters/fa_25.tex @@ -9,7 +9,7 @@ \chapter{Benchmarks: Period Cycles} \caption*{Figure --- Benchmarks: Period Cycles.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_25:abstract} This chapter develops the theory of \(\varphi\)-period cycles --- periodic orbits in @@ -30,7 +30,7 @@ \section{Abstract}\label{abstract} the statistical periodicity of the training loss (Ch.19). -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_25:introduction} Periodic behaviour in gradient-descent optimisation is usually treated as a pathology: @@ -73,7 +73,7 @@ \section{1. Introduction}\label{introduction} \section{\texorpdfstring{2. \(\varphi\)-Lattice Structure and the Cycle -Map}{2. \textbackslash varphi-Lattice Structure and the Cycle Map}}\label{varphi-lattice-structure-and-the-cycle-map} +Map}{2. \textbackslash varphi-Lattice Structure and the Cycle Map}}\label{fa_25:varphi-lattice-structure-and-the-cycle-map} \textbf{Definition 2.1 (\(\varphi\)-quantised lattice).} The one-dimensional @@ -135,7 +135,7 @@ \section{\texorpdfstring{2. \(\varphi\)-Lattice relevant orbit lengths. \section{3. Cycle Classification and Attention -Periodicity}\label{cycle-classification-and-attention-periodicity} +Periodicity}\label{fa_25:cycle-classification-and-attention-periodicity} The cycle structure of \(\Phi\) on \(\Lambda_\varphi^{(1)}\) for small lattice sizes @@ -214,7 +214,7 @@ \section{3. Cycle Classification and Attention by the periodicity of the encoding. \(\square\) \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_25:results-evidence} \textbf{Evidence 1 --- Loss periodicity.} Training loss curves for all three primary replicates @@ -256,12 +256,12 @@ \section{4. Results / \(t\), \(p = 0.008\)). \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_25:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_25:sealed-seeds} Inherits the canonical seed pool \(F_{17}=1597\), \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), @@ -272,7 +272,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} counts at \(|\Lambda| = F_{17}\) are \(L_7\) and \(L_8\) for orders 29 and 47 respectively. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_25:discussion} The \(\varphi\)-cycle theory developed here is a novel contribution: to the authors' knowledge, no @@ -302,7 +302,7 @@ \section{7. Discussion}\label{discussion} (seed admissibility), and Ch.19 (loss periodicity) are tight. -\section{References}\label{references} +\section{References}\label{fa_25:references} [1] This dissertation, Ch.7 --- Vogel Phyllotaxis \(137.5^\circ = 360^\circ/\varphi^2\). @@ -356,7 +356,7 @@ \section{References}\label{references} phyllotaxis). \section{Falsification} -\label{sec:falsification:ch25} +\label{fa_25:sec:falsification:ch25} \paragraph{Pre-registered claim (R7).} The $\varphi$-period benchmark suite exhibits a Gate-2 BPB threshold at diff --git a/docs/phd/chapters/fa_26.tex b/docs/phd/chapters/fa_26.tex index 23cf47d7b1..561ab5759d 100644 --- a/docs/phd/chapters/fa_26.tex +++ b/docs/phd/chapters/fa_26.tex @@ -8,7 +8,7 @@ \chapter{Data Analysis: Koschei Coprocessor ISA} \caption*{Figure --- Data Analysis: Koschei Coprocessor ISA.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_26:abstract} The KOSCHEI coprocessor extends the QMTech XC7A100T FPGA with a φ-numeric instruction set @@ -28,7 +28,7 @@ \section{Abstract}\label{abstract} \texttt{Trinity.Canonical.Kernel.FlowerE8Embedding}. The ISA achieves 63 tokens/sec at 92 MHz and 1 W. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_26:introduction} A coprocessor ISA for φ-numeric computation must satisfy three simultaneous constraints that are @@ -72,10 +72,10 @@ \section{1. Introduction}\label{introduction} ISA. \section{2. ISA Register File and -Encoding}\label{isa-register-file-and-encoding} +Encoding}\label{fa_26:isa-register-file-and-encoding} \subsection{2.1 Register -File}\label{register-file} +File}\label{fa_26:register-file} KOSCHEI has 16 general-purpose registers \(r_0\)--\(r_{15}\), each 64 bits wide. The @@ -122,7 +122,7 @@ \subsection{2.1 Register integer powers of \(\varphi\). \subsection{2.2 Instruction -Encoding}\label{instruction-encoding} +Encoding}\label{fa_26:instruction-encoding} Instructions are 32 bits: 7-bit opcode, 4-bit destination, 4-bit source A, 4-bit source B, @@ -138,10 +138,10 @@ \subsection{2.2 Instruction 0x01--0x07. \section{3. Opcode -Specifications}\label{opcode-specifications} +Specifications}\label{fa_26:opcode-specifications} \subsection{3.1 TF3\_ADD --- Ternary -Addition}\label{tf3_add-ternary-addition} +Addition}\label{fa_26:tf3_add-ternary-addition} \begin{verbatim} TF3_ADD RD, RA, RB @@ -163,7 +163,7 @@ \subsection{3.1 TF3\_ADD --- Ternary [1]. \subsection{3.2 TF3\_MUL --- Ternary -Multiplication}\label{tf3_mul-ternary-multiplication} +Multiplication}\label{fa_26:tf3_mul-ternary-multiplication} \begin{verbatim} TF3_MUL RD, RA, RB @@ -185,7 +185,7 @@ \subsection{3.2 TF3\_MUL --- Ternary two LUT-4 primitives per bit [2]. \subsection{3.3 VSA\_BIND --- Hyperdimensional -Binding}\label{vsa_bind-hyperdimensional-binding} +Binding}\label{fa_26:vsa_bind-hyperdimensional-binding} \begin{verbatim} VSA_BIND RD, RA, RB @@ -204,7 +204,7 @@ \subsection{3.3 VSA\_BIND --- Hyperdimensional \subsection{3.4 VSA\_UNBIND --- Hyperdimensional -Unbinding}\label{vsa_unbind-hyperdimensional-unbinding} +Unbinding}\label{fa_26:vsa_unbind-hyperdimensional-unbinding} \begin{verbatim} VSA_UNBIND RD, RA, RB @@ -221,7 +221,7 @@ \subsection{3.4 VSA\_UNBIND --- \subsection{3.5 VSA\_BUNDLE --- Hyperdimensional -Bundling}\label{vsa_bundle-hyperdimensional-bundling} +Bundling}\label{fa_26:vsa_bundle-hyperdimensional-bundling} \begin{verbatim} VSA_BUNDLE RD, RA, RB @@ -240,7 +240,7 @@ \subsection{3.5 VSA\_BUNDLE --- (status: Qed) [4]. \subsection{3.6 GF16\_QUANT --- Galois Field 16 -Quantisation}\label{gf16_quant-galois-field-16-quantisation} +Quantisation}\label{fa_26:gf16_quant-galois-field-16-quantisation} \begin{verbatim} GF16_QUANT RD, RA, IMM[3:0] @@ -262,7 +262,7 @@ \subsection{3.6 GF16\_QUANT --- Galois Field 16 LUT-6 primitives. \subsection{3.7 PHI\_ROPE --- φ-Rotary Position -Encoding}\label{phi_rope-ux3c6-rotary-position-encoding} +Encoding}\label{fa_26:phi_rope-ux3c6-rotary-position-encoding} \begin{verbatim} PHI_ROPE RD, RA, IMM[12:0] @@ -290,7 +290,7 @@ \subsection{3.7 PHI\_ROPE --- φ-Rotary Position [7]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_26:results-evidence} Synthesis on the QMTech XC7A100T (Vivado 2023.2, seed \(F_{17}=1597\)) yields: @@ -314,7 +314,7 @@ \section{4. Results / the HSLM 1003-token sequence. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_26:qed-assertions} No Coq theorems are anchored directly to this chapter; the ISA semantics are certified by the @@ -349,13 +349,13 @@ \section{5. Qed \filepath{gHashTag/t27/proofs/canonical/} and contribute to the 297 Qed census [8]. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_26:sealed-seeds} Inherits the canonical seed pool F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_26:discussion} The KOSCHEI ISA demonstrates that a φ-lattice arithmetic unit can be implemented entirely in LUT @@ -381,7 +381,7 @@ \section{7. Discussion}\label{discussion} obligations in the current census and is prioritised for the Gate-3 submission. -\section{References}\label{references} +\section{References}\label{fa_26:references} [1] Trinity Canonical Coq Home. \texttt{Trinity.Canonical.Kernel.Phi} --- 16 Qed. diff --git a/docs/phd/chapters/fa_27.tex b/docs/phd/chapters/fa_27.tex index 2a3d564e79..07a93b4732 100644 --- a/docs/phd/chapters/fa_27.tex +++ b/docs/phd/chapters/fa_27.tex @@ -9,7 +9,7 @@ \chapter{Trinity Identity: tri27 DSL} \caption*{Figure --- Trinity Identity: tri27 DSL.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_27:abstract} TRI27 is the domain-specific language (DSL) of the Trinity S³AI kernel, typed over a balanced-ternary @@ -28,7 +28,7 @@ \section{Abstract}\label{abstract} by construction. The Zenodo artifact B003 archives the verifiable VM implementation. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_27:introduction} The arithmetic core of Trinity S³AI processes weights and activations represented as @@ -67,10 +67,10 @@ \section{1. Introduction}\label{introduction} evaluation results and artifact metadata. \section{2. TRI27 Syntax and Denotational -Semantics}\label{tri27-syntax-and-denotational-semantics} +Semantics}\label{fa_27:tri27-syntax-and-denotational-semantics} \subsection{2.1 Abstract -Syntax}\label{abstract-syntax} +Syntax}\label{fa_27:abstract-syntax} The TRI27 expression language is defined by the following inductive type in Coq: @@ -112,7 +112,7 @@ \subsection{2.1 Abstract \texttt{trit\_exhaustive}. \subsection{2.2 Environments and -Evaluation}\label{environments-and-evaluation} +Evaluation}\label{fa_27:environments-and-evaluation} An environment \texttt{rho\ :\ env} is a total function \texttt{nat\ -\textgreater{}\ trit} @@ -133,7 +133,7 @@ \subsection{2.2 Environments and \texttt{Some\ v}. \subsection{2.3 Ternary -Arithmetic}\label{ternary-arithmetic} +Arithmetic}\label{fa_27:ternary-arithmetic} The fundamental ternary operations are defined by the \(3 \times 3\) tables: @@ -178,7 +178,7 @@ \subsection{2.3 Ternary \subsection{\texorpdfstring{2.4 Relation to GF16 and -\(\varphi\)-Arithmetic}{2.4 Relation to GF16 and \textbackslash varphi-Arithmetic}}\label{relation-to-gf16-and-varphi-arithmetic} +\(\varphi\)-Arithmetic}{2.4 Relation to GF16 and \textbackslash varphi-Arithmetic}}\label{fa_27:relation-to-gf16-and-varphi-arithmetic} The GF16 field elements (Ch.9 [2]) are pairs of trit-register values under the embedding @@ -198,11 +198,11 @@ \subsection{\texorpdfstring{2.4 Relation to \texttt{trit} value is bounded. \section{3. Mechanised Proofs: Determinism and -Exhaustiveness}\label{mechanised-proofs-determinism-and-exhaustiveness} +Exhaustiveness}\label{fa_27:mechanised-proofs-determinism-and-exhaustiveness} \subsection{\texorpdfstring{3.1 Theorem \texttt{eval\_det}: -Determinism}{3.1 Theorem eval\_det: Determinism}}\label{theorem-eval_det-determinism} +Determinism}{3.1 Theorem eval\_det: Determinism}}\label{fa_27:theorem-eval_det-determinism} \textbf{Statement} (KER-4, \filepath{gHashTag/t27/proofs/canonical/kernel/Semantics.v} @@ -245,7 +245,7 @@ \subsection{\texorpdfstring{3.1 Theorem \subsection{\texorpdfstring{3.2 Theorem \texttt{trit\_exhaustive}: -Exhaustiveness}{3.2 Theorem trit\_exhaustive: Exhaustiveness}}\label{theorem-trit_exhaustive-exhaustiveness} +Exhaustiveness}{3.2 Theorem trit\_exhaustive: Exhaustiveness}}\label{fa_27:theorem-trit_exhaustive-exhaustiveness} \textbf{Statement} (KER-5, \filepath{gHashTag/t27/proofs/canonical/kernel/Trit.v} @@ -278,7 +278,7 @@ \subsection{\texorpdfstring{3.2 Theorem \(\varphi^2 + \varphi^{-2} = 3\) [1]. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_27:results-evidence} \begin{itemize} \tightlist @@ -318,7 +318,7 @@ \section{4. Results / \end{itemize} \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_27:qed-assertions} \begin{itemize} \tightlist @@ -336,7 +336,7 @@ \section{5. Qed \texttt{Neg}, \texttt{Zero}, or \texttt{Pos}. \end{itemize} -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_27:sealed-seeds} \begin{itemize} \tightlist @@ -348,7 +348,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} notes: TRI-27 Verifiable VM artifact. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_27:discussion} The TRI27 DSL formalised here is intentionally minimal. The present two theorems establish only @@ -373,7 +373,7 @@ \section{7. Discussion}\label{discussion} (FPGA implementation) and App.H (VM specification) build directly on the TRI27 kernel defined here. -\section{References}\label{references} +\section{References}\label{fa_27:references} [1] \emph{Golden Sunflowers} dissertation, Ch.3 --- Trinity Identity diff --git a/docs/phd/chapters/fa_28.tex b/docs/phd/chapters/fa_28.tex index ba62fc9f46..19defe947d 100644 --- a/docs/phd/chapters/fa_28.tex +++ b/docs/phd/chapters/fa_28.tex @@ -9,7 +9,7 @@ \chapter{Momentum Algebra: QMTech XC7A100T FPGA} \caption*{Figure --- Momentum Algebra: QMTech XC7A100T FPGA.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_28:abstract} The QMTech XC7A100T development board hosts the primary hardware realisation of the Trinity S³AI @@ -32,7 +32,7 @@ \section{Abstract}\label{abstract} bitstreams B001 and B002 as the primary evidence artefacts. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_28:introduction} Field-Programmable Gate Arrays offer a path to energy-efficient neural inference that complements @@ -80,7 +80,7 @@ \section{1. Introduction}\label{introduction} slack. \section{2. Architecture: Zero-DSP Ternary -Datapath}\label{architecture-zero-dsp-ternary-datapath} +Datapath}\label{fa_28:architecture-zero-dsp-ternary-datapath} \textbf{Definition 2.1 (Ternary accumulator).} A ternary accumulator for a vector of \(N\) inputs @@ -132,7 +132,7 @@ \section{2. Architecture: Zero-DSP Ternary constraint. \section{3. Resource Utilisation and Timing -Closure}\label{resource-utilisation-and-timing-closure} +Closure}\label{fa_28:resource-utilisation-and-timing-closure} \textbf{Resource utilisation (post-implementation).} @@ -206,7 +206,7 @@ \section{3. Resource Utilisation and Timing inspection in the B002 artefact.} \(\square\) \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_28:results-evidence} The primary evidence artefacts are: @@ -282,7 +282,7 @@ \section{4. Results / design methodology described in this chapter. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_28:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. The @@ -291,7 +291,7 @@ \section{5. Qed \texttt{TernarySufficiency.v}) from Ch.4 as architectural pre-conditions. -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_28:sealed-seeds} \begin{itemize} \tightlist @@ -326,7 +326,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} Fibonacci/Lucas reference: F₁₇=1597, F₁₈=2584, F₁₉=4181, F₂₀=6765, F₂₁=10946, L₇=29, L₈=47. -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_28:discussion} Three limitations bound the current implementation. First, BRAM utilisation at 91.5\% @@ -350,7 +350,7 @@ \section{7. Discussion}\label{discussion} 1 W power figure within the 3000$\times$ DARPA energy efficiency target. -\section{References}\label{references} +\section{References}\label{fa_28:references} [1] QMTech XC7A100T product specification. Xilinx Artix-7 FPGA datasheet, DS181 Rev.~1.31 diff --git a/docs/phd/chapters/fa_29.tex b/docs/phd/chapters/fa_29.tex index 859c37b0de..adae918e9f 100644 --- a/docs/phd/chapters/fa_29.tex +++ b/docs/phd/chapters/fa_29.tex @@ -10,7 +10,7 @@ \chapter{Lucas Closure — Number Theory} \end{figure} -\label{ch:29} +\label{fa_29:ch:29} % Lane: A % Agent: Claude % Status: COMPLETE @@ -34,7 +34,7 @@ \section{Lucas Sequence Definition} \subsection{Recursive Definition} \begin{definition}[Lucas Sequence] -\label{def:lucas} +\label{fa_29:def:lucas} \begin{equation} L_0 = 2, \quad L_1 = 1 \end{equation} @@ -75,7 +75,7 @@ \section{Golden Ratio Connections} \subsection{Lucas-Fibonacci Relation} \begin{theorem}[Ratio Relation] -\label{thm:lucas-fibo} +\label{fa_29:thm:lucas-fibo} \begin{equation} \lim_{n\to\infty} \frac{L_n}{F_n} = \phi \end{equation} @@ -91,7 +91,7 @@ \subsection{Lucas-Fibonacci Relation} \subsection{Lucas Golden Approximation} \begin{proposition}[Lucas Convergence] -\label{prop:lucas-golden} +\label{fa_29:prop:lucas-golden} \begin{equation} \frac{L_{n+1}}{L_n} \to \phi \end{equation} @@ -102,7 +102,7 @@ \subsection{Lucas Golden Approximation} \subsection{Trinity Identity Extension} \begin{definition}[Lucas Trinity] -\label{def:lucas-trinity} +\label{fa_29:def:lucas-trinity} \begin{equation} L_n^2 + L_n^{-2} = \phi^{2n} \end{equation} @@ -122,7 +122,7 @@ \section{Number Theoretic Properties} \subsection{Divisibility} \begin{theorem}[Divisibility Pattern] -\label{thm:lucas-div} +\label{fa_29:thm:lucas-div} \begin{equation} n | L_n \implies n | F_n \text{ and } n | L_n \text{ if and only if } n \equiv 2 \pmod{4} \end{equation} @@ -131,14 +131,14 @@ \subsection{Divisibility} \subsection{Primes} \begin{definition}[Lucas Primes] -\label{def:lucas-primes} +\label{fa_29:def:lucas-primes} \begin{equation} p \text{ is Lucas prime if } p = L_n \text{ for some } n \text{ and } \gcd(L_n, n) = 1 \end{equation} \end{definition} \begin{theorem}[Lucas Prime Density] -\label{thm:lucas-prime-density} +\label{fa_29:thm:lucas-prime-density} \begin{equation} \lim_{N\to\infty} \frac{\pi_L(N)}{N} = \frac{1}{\ln\phi} \end{equation} @@ -147,7 +147,7 @@ \subsection{Primes} \subsection{Modular Properties} \begin{proposition}[Lucas Modulo] -\label{prop:lucas-mod} +\label{fa_29:prop:lucas-mod} \begin{equation} L_n \equiv L_m \pmod{N} \end{equation} @@ -184,7 +184,7 @@ \subsection{Quark Mixing} \end{equation} \begin{proposition}[Golden-Lucas Mixing] -\label{prop:golden-lucas-mixing} +\label{fa_29:prop:golden-lucas-mixing} \begin{equation} \theta_{12} \approx \frac{\pi}{L_3/L_2} \end{equation} @@ -199,7 +199,7 @@ \subsection{Neutrino Mass Hierarchy} \end{equation} \begin{theorem}[Neutrino Lucas] -\label{thm:neutrino-lucas} +\label{fa_29:thm:neutrino-lucas} \begin{equation} m_{\nu_1}:m_{\nu_2}:m_{\nu_3} \approx 1:\frac{L_{n+1}}{L_n}:\left(\frac{L_{n+1}}{L_n}\right)^2 \end{equation} @@ -212,7 +212,7 @@ \section{Geometric Applications} \subsection{Lucas Tilings} \begin{definition}[Lucas Tiling] -\label{def:lucas-tiling} +\label{fa_29:def:lucas-tiling} Tiling using Lucas numbers as basis: \begin{equation} N_n = L_n @@ -220,7 +220,7 @@ \subsection{Lucas Tilings} \end{definition} \begin{proposition}[Lucas Tiling Properties] -\label{prop:lucas-tiling} +\label{fa_29:prop:lucas-tiling} \begin{enumerate} \item Self-similar by Lucas ratio \item Quasiperiodic order @@ -232,14 +232,14 @@ \subsection{Lucas Tilings} \subsection{Lucas Spirals} \begin{definition}[Lucas Spiral] -\label{def:lucas-spiral} +\label{fa_29:def:lucas-spiral} \begin{equation} r(\theta) = a L_n^{\theta/\pi} \end{equation} \end{definition} \begin{theorem}[Lucas Spiral Growth] -\label{thm:lucas-spiral} +\label{fa_29:thm:lucas-spiral} \begin{equation} \frac{r(\theta + 2\pi)}{r(\theta)} = L_{n+1} \end{equation} @@ -254,7 +254,7 @@ \section{Algebraic Closure} \subsection{Fibonacci-Lucas Identities} \begin{theorem}[Product Formula] -\label{thm:product} +\label{fa_29:thm:product} \begin{equation} F_n L_{n+1} - F_{n+1} L_n = 5(-1)^n \end{equation} @@ -274,7 +274,7 @@ \subsection{Fibonacci-Lucas Identities} \subsection{Cassini-Lucas Identities} \begin{theorem}[Cassini-like] -\label{thm:cassini} +\label{fa_29:thm:cassini} \begin{equation} L_n L_{n+1} - L_{n-1} L_{n+2} = 5(-1)^n \end{equation} @@ -346,7 +346,7 @@ \subsection*{Open Questions} % §Falsification — added by LP-falsification lane (R7 retrofit) % ===================================================================== -\section{Falsification Criterion}\label{sec:29-falsify} +\section{Falsification Criterion}\label{fa_29:sec:29-falsify} \subsection{What Would Refute This Claim} diff --git a/docs/phd/chapters/fa_30.tex b/docs/phd/chapters/fa_30.tex index 5e5fb09291..53100755a5 100644 --- a/docs/phd/chapters/fa_30.tex +++ b/docs/phd/chapters/fa_30.tex @@ -6,7 +6,7 @@ \chapter{Golden Imagery: Trinity S3AI --- VSA-AR} \caption*{Figure --- Golden Imagery: Trinity S3AI --- VSA-AR.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_30:abstract} Trinity SAI (Structured Artificial Intelligence) integrates a Vector Symbolic Architecture (VSA) @@ -26,7 +26,7 @@ \section{Abstract}\label{abstract} toks/sec at 92 MHz with 0 DSP slices, consistent with the system-wide power budget of 1 W. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_30:introduction} The third pillar of the Trinity S³AI architecture is the symbolic layer. The first pillar is the @@ -71,10 +71,10 @@ \section{1. Introduction}\label{introduction} pathway. \section{2. Ternary VSA over the GoldenFloat -Substrate}\label{ternary-vsa-over-the-goldenfloat-substrate} +Substrate}\label{fa_30:ternary-vsa-over-the-goldenfloat-substrate} \subsection{2.1 Hypervector -Definition}\label{hypervector-definition} +Definition}\label{fa_30:hypervector-definition} \textbf{Definition 2.1 (Ternary hypervector).} A ternary hypervector of dimension \(D\) is a vector @@ -118,7 +118,7 @@ \subsection{2.1 Hypervector for each \(i\). Qed. \subsection{2.2 Associative Recall -Memory}\label{associative-recall-memory} +Memory}\label{fa_30:associative-recall-memory} The AR memory is a content-addressable store of \(M\) hypervectors @@ -142,7 +142,7 @@ \subsection{2.2 Associative Recall overwhelming probability [3]. \subsection{2.3 GoldenFloat Encoding of -Hypervectors}\label{goldenfloat-encoding-of-hypervectors} +Hypervectors}\label{fa_30:goldenfloat-encoding-of-hypervectors} Each component \(v_i \in \{-1, 0, +1\}\) is stored in GF16 as the canonical constants @@ -162,7 +162,7 @@ \subsection{2.3 GoldenFloat Encoding of \section{3. Phi-Rotary Position Encoding (phi-RoPE) in VSA -Context}\label{phi-rotary-position-encoding-phi-rope-in-vsa-context} +Context}\label{fa_30:phi-rotary-position-encoding-phi-rope-in-vsa-context} The phi-RoPE encoding (Zenodo Z05 [6]) assigns to token position \(p\) the angle @@ -205,7 +205,7 @@ \section{3. Phi-Rotary Position Encoding Qed. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_30:results-evidence} The Trinity SAI VSA+AR module was evaluated on the HSLM 1003-token benchmark using the IGLA RACE @@ -258,7 +258,7 @@ \section{4. Results / primary symbolic reasoning pathway. \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_30:qed-assertions} No Coq theorems are anchored to this chapter; obligations are tracked in the Golden Ledger. @@ -272,7 +272,7 @@ \section{5. Qed planned as part of the Iris/Coq.Interval upgrade lane described in Ch.18.) -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_30:sealed-seeds} \begin{itemize} \tightlist @@ -288,7 +288,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} \(F_{18}=2584\), \(F_{19}=4181\), \(F_{20}=6765\), \(F_{21}=10946\), \(L_7=29\), \(L_8=47\). -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_30:discussion} The Trinity SAI VSA+AR component extends the GOLDEN SUNFLOWERS framework from pure @@ -321,7 +321,7 @@ \section{7. Discussion}\label{discussion} Ch.28 (hardware throughput), and App.H (Zenodo DOI registry for the B007 anchor). -\section{References}\label{references} +\section{References}\label{fa_30:references} [1] Kanerva, P. (2009). Hyperdimensional Computing: An Introduction to Computing in diff --git a/docs/phd/chapters/fa_31.tex b/docs/phd/chapters/fa_31.tex index 874bd3a426..ab1ea1873f 100644 --- a/docs/phd/chapters/fa_31.tex +++ b/docs/phd/chapters/fa_31.tex @@ -6,7 +6,7 @@ \chapter{Philosophy — Mathematical Foundations} \end{figure} -\label{ch:31} +\label{fa_31:ch:31} % Lane: A % Agent: Claude % Status: COMPLETE @@ -30,12 +30,12 @@ \section{Mathematical Platonism} \subsection{Platonic Realism} \begin{definition}[Platonism] -\label{def:platonism} +\label{fa_31:def:platonism} Mathematical objects exist independently of human thought in a realm of abstract forms. \end{definition} \begin{theorem}[Platonic Golden Ratio] -\label{thm:platonic-golden} +\label{fa_31:thm:platonic-golden} The golden ratio exists as a Platonic form: \begin{equation} \phi \in \mathcal{P} @@ -54,7 +54,7 @@ \subsection{Arguments for Platonism} \end{enumerate} \begin{proposition}[Golden Platonism] -\label{prop:golden-platonism} +\label{fa_31:prop:golden-platonism} \begin{equation} \phi = \frac{1+\sqrt{5}}{2} \in \mathbb{R} \cap \mathcal{P} \end{equation} @@ -69,12 +69,12 @@ \section{Mathematical Structuralism} \subsection{Structuralist Definition} \begin{definition}[Structuralism] -\label{def:structuralism} +\label{fa_31:def:structuralism} Mathematical objects are defined by their relations within structures, not intrinsically. \end{definition} \begin{theorem}[Golden Structuralism] -\label{thm:golden-struct} +\label{fa_31:thm:golden-struct} \begin{equation} \phi = \{x : x^2 = x + 1\} \end{equation} @@ -89,7 +89,7 @@ \subsection{Category Theory} \end{equation} \begin{proposition}[Golden Category] -\label{prop:golden-cat} +\label{fa_31:prop:golden-cat} The golden ratio emerges from category structure: \begin{equation} \phi = \lim_{n\to\infty} \frac{\text{Hom}(A^n, B)}{\text{Hom}(A^{n-1}, B)} @@ -104,12 +104,12 @@ \section{Scientific Realism} \subsection{Realist Position} \begin{definition}[Scientific Realism] -\label{def:realism} +\label{fa_31:def:realism} Scientific theories are approximately true descriptions of both observable and unobservable reality. \end{definition} \begin{theorem}[Realist Golden Ratio] -\label{thm:realist-golden} +\label{fa_31:thm:realist-golden} \begin{equation} \phi \text{ exists in physical reality} \end{equation} @@ -120,12 +120,12 @@ \subsection{Realist Position} \subsection{Anti-Realism} \begin{definition}[Anti-Realism] -\label{def:antirealism} +\label{fa_31:def:antirealism} Scientific theories are instruments for prediction, not descriptions of reality. \end{definition} \begin{proposition}[Instrumentalist Golden Ratio] -\label{prop:instrumental-golden} +\label{fa_31:prop:instrumental-golden} \begin{equation} \phi \text{ is useful for calculation, not real} \end{equation} @@ -140,14 +140,14 @@ \section{Philosophy of Physics} \subsection{Constancy Question} \begin{definition}[Physical Constants] -\label{def:constants} +\label{fa_31:def:constants} \begin{equation} \alpha, G, c, h \text{ are fundamental parameters} \end{equation} \end{definition} \begin{theorem}[Golden Constants] -\label{thm:golden-constants} +\label{fa_31:thm:golden-constants} \begin{equation} \alpha = \frac{\phi^4}{8\pi^2} \end{equation} @@ -162,7 +162,7 @@ \subsection{Anthropic Reasoning} \end{equation} \begin{proposition}[Golden Anthropic] -\label{prop:golden-anthropic} +\label{fa_31:prop:golden-anthropic} \begin{equation} \frac{\Omega_\Lambda}{\Omega_m} \approx \phi^3 \end{equation} @@ -177,12 +177,12 @@ \section{Epistemology} \subsection{A Priori Knowledge} \begin{definition}[A Priori] -\label{def:apriori} +\label{fa_31:def:apriori} Knowledge independent of experience. \end{definition} \begin{theorem}[A Priori Golden] -\label{thm:apriori-golden} +\label{fa_31:thm:apriori-golden} \begin{equation} \phi = \frac{1+\sqrt{5}}{2} \text{ known a priori} \end{equation} @@ -193,12 +193,12 @@ \subsection{A Priori Knowledge} \subsection{Empiricism} \begin{definition}[Empiricism] -\label{def:empiricism} +\label{fa_31:def:empiricism} All knowledge comes from sensory experience. \end{definition} \begin{proposition}[Empirical Golden] -\label{prop:empirical-golden} +\label{fa_31:prop:empirical-golden} \begin{equation} \phi \text{ discovered in nature through measurement} \end{equation} @@ -213,12 +213,12 @@ \section{Aesthetics and Truth} \subsection{Mathematical Beauty} \begin{definition}[Mathematical Beauty] -\label{def:beauty} +\label{fa_31:def:beauty} Elegance, simplicity, and depth in mathematical structure. \end{definition} \begin{theorem}[Golden Beauty] -\label{thm:golden-beauty} +\label{fa_31:thm:golden-beauty} \begin{equation} \text{Beauty}(\phi) = \max_{x \in \mathbb{R}} \text{Beauty}(x) \end{equation} @@ -234,7 +234,7 @@ \subsection{Wigner's Unreasonable Effectiveness} — Eugene Wigner \begin{proposition}[Golden Effectiveness] -\label{prop:golden-effective} +\label{fa_31:prop:golden-effective} \begin{equation} \text{Effectiveness}(\phi) = \phi \times \text{Effectiveness}(\text{other}) \end{equation} @@ -249,14 +249,14 @@ \section{Metaphysics} \subsection{Pythagoreanism} \begin{definition}[Pythagorean Principle] -\label{def:pythagorean} +\label{fa_31:def:pythagorean} \begin{equation} \text{All is number} \end{equation} \end{definition} \begin{theorem}[Pythagorean Golden] -\label{thm:pythagorean-golden} +\label{fa_31:thm:pythagorean-golden} \begin{equation} \text{Reality} \cong \phi \end{equation} @@ -267,14 +267,14 @@ \subsection{Pythagoreanism} \subsection{Mathematical Universe Hypothesis} \begin{definition}[MUH] -\label{def:muh} +\label{fa_31:def:muh} \begin{equation} \text{Physical reality is mathematical structure} \end{equation} \end{definition} \begin{proposition}[Golden MUH] -\label{prop:golden-muh} +\label{fa_31:prop:golden-muh} \begin{equation} \mathcal{R} = \{\phi\text{-based mathematical structure}\} \end{equation} diff --git a/docs/phd/chapters/fa_32.tex b/docs/phd/chapters/fa_32.tex index 78f95b3383..96e731e224 100644 --- a/docs/phd/chapters/fa_32.tex +++ b/docs/phd/chapters/fa_32.tex @@ -30,7 +30,7 @@ \section{Summary of Contributions} \subsection{Mathematical Contributions} \begin{theorem}[Trinity Identity] -\label{thm:trinity-summary} +\label{fa_32:thm:trinity-summary} \begin{equation} \phi^2 + \phi^{-2} = 3 \end{equation} @@ -39,7 +39,7 @@ \subsection{Mathematical Contributions} \end{theorem} \begin{theorem}[Alpha Derivation] -\label{thm:alpha-summary} +\label{fa_32:thm:alpha-summary} \begin{equation} \alpha = \frac{\phi^4}{8\pi^2} = 0.007297... \end{equation} @@ -48,7 +48,7 @@ \subsection{Mathematical Contributions} \end{theorem} \begin{theorem}[E₈ Mass Predictions] -\label{thm:e8-summary} +\label{fa_32:thm:e8-summary} \begin{equation} \frac{m_2}{m_1} \approx \phi^{3.5}, \quad \frac{m_3}{m_2} \approx \phi^{1.5} \end{equation} @@ -113,7 +113,7 @@ \section{Theoretical Implications} \subsection{Unification Principle} \begin{theorem}[Golden Unification] -\label{thm:golden-unif} +\label{fa_32:thm:golden-unif} \begin{equation} G_{GUT} = SU(3) \times SU(2) \times U(1) \xrightarrow{\phi} E_8 \end{equation} @@ -124,7 +124,7 @@ \subsection{Unification Principle} \subsection{Optimization Principle} \begin{proposition}[Golden Optimization] -\label{prop:golden-opt} +\label{fa_32:prop:golden-opt} \begin{equation} \min_{x} f(x) \implies x \propto \phi \end{equation} @@ -139,7 +139,7 @@ \subsection{Information Theory} \end{equation} \begin{theorem}[Golden Entropy] -\label{thm:golden-entropy} +\label{fa_32:thm:golden-entropy} \begin{equation} H_{\max} \propto \phi \end{equation} diff --git a/docs/phd/chapters/fa_33.tex b/docs/phd/chapters/fa_33.tex index bc03c1a450..3631246273 100644 --- a/docs/phd/chapters/fa_33.tex +++ b/docs/phd/chapters/fa_33.tex @@ -8,7 +8,7 @@ \chapter{Epilogue: JTAG macOS BLK-001 Resolved} \caption*{Figure --- Epilogue: JTAG macOS BLK-001 Resolved.} \end{figure} -\section{Abstract}\label{abstract} +\section{Abstract}\label{fa_33:abstract} Blocker BLK-001 was a hardware bring-up failure in which the Xilinx Platform Cable USB II JTAG @@ -31,7 +31,7 @@ \section{Abstract}\label{abstract} Update-DR) mirrors the ternary structure of the Trinity kernel. -\section{1. Introduction}\label{introduction} +\section{1. Introduction}\label{fa_33:introduction} The QMTech XC7A100T FPGA board (Xilinx Artix-7, 100K LUT, 0 DSP in the Trinity configuration) is @@ -73,10 +73,10 @@ \section{1. Introduction}\label{introduction} hardware interface layer. \section{2. Diagnosis and Root -Cause}\label{diagnosis-and-root-cause} +Cause}\label{fa_33:diagnosis-and-root-cause} \subsection{2.1 USB Enumeration on -macOS-ARM}\label{usb-enumeration-on-macos-arm} +macOS-ARM}\label{fa_33:usb-enumeration-on-macos-arm} The Xilinx Platform Cable USB II uses a Cypress EZ-USB FX2LP microcontroller (CY7C68013A) that @@ -106,7 +106,7 @@ \subsection{2.1 USB Enumeration on re-enumerated with \texttt{idProduct\ =\ 0x0008}. \subsection{2.2 fxload -Cross-Compilation}\label{fxload-cross-compilation} +Cross-Compilation}\label{fa_33:fxload-cross-compilation} \texttt{fxload} 0.0.1 was cross-compiled for macOS-ARM (\texttt{aarch64-apple-darwin}) using: @@ -126,7 +126,7 @@ \subsection{2.2 fxload issues. \subsection{2.3 -flash\_no\_sudo.sh}\label{flash_no_sudo.sh} +flash\_no\_sudo.sh}\label{fa_33:flash_no_sudo.sh} The resolution script performs the following steps: @@ -155,7 +155,7 @@ \subsection{2.3 the M2 host. \section{3. Verified Hardware Configuration -Post-BLK-001}\label{verified-hardware-configuration-post-blk-001} +Post-BLK-001}\label{fa_33:verified-hardware-configuration-post-blk-001} After BLK-001 resolution, the following configuration was verified and is now the @@ -199,7 +199,7 @@ \section{3. Verified Hardware Configuration resolution did not affect the performance profile. \section{4. Results / -Evidence}\label{results-evidence} +Evidence}\label{fa_33:results-evidence} \begin{itemize} \tightlist @@ -235,7 +235,7 @@ \section{4. Results / \end{itemize} \section{5. Qed -Assertions}\label{qed-assertions} +Assertions}\label{fa_33:qed-assertions} No Coq theorems are anchored to this chapter; the BLK-001 resolution is a hardware procedure with no @@ -243,7 +243,7 @@ \section{5. Qed in the Golden Ledger under hardware blocker BLK-001 (status: RESOLVED). -\section{6. Sealed Seeds}\label{sealed-seeds} +\section{6. Sealed Seeds}\label{fa_33:sealed-seeds} \begin{itemize} \tightlist @@ -263,7 +263,7 @@ \section{6. Sealed Seeds}\label{sealed-seeds} RESOLVED 2026-03-14. \end{itemize} -\section{7. Discussion}\label{discussion} +\section{7. Discussion}\label{fa_33:discussion} BLK-001 was a low-level hardware integration issue with no bearing on the formal proof tree or the @@ -289,7 +289,7 @@ \section{7. Discussion}\label{discussion} removing the firmware dependency would complete the fully open-source bring-up path. -\section{References}\label{references} +\section{References}\label{fa_33:references} [1] Xilinx, ``Platform Cable USB II Data Sheet,'' DS593, Xilinx Inc., 2013. diff --git a/docs/phd/cross-ref-audit.md b/docs/phd/cross-ref-audit.md new file mode 100644 index 0000000000..fe74e6fe37 --- /dev/null +++ b/docs/phd/cross-ref-audit.md @@ -0,0 +1,1323 @@ +# Cross-Reference Audit — Phase 1 UNIFY task 1.5 + +**Branch:** `feat/phd-phase1-unify-1-5` (stacked on `feat/phd-phase1-unify-1-2`, PR #595) +**Issue:** [trios#380](https://github.com/gHashTag/trios/issues/380) task 1.5 +**Anchor:** φ² + φ⁻² = 3 · DOI [10.5281/zenodo.19227877](https://zenodo.org/records/19227877) + +## Summary + +- **Total `\label{}` sites:** 1145 +- **Unique label keys:** 1145 +- **Duplicate label keys:** 0 (was 126 pre-patch — all eliminated) +- **Dangling refs:** 0 (was 0 — preserved) +- **Original referenced keys still resolved:** 119/119 (no breakage) + +## Acceptance criteria (#380 task 1.5) + +| Criterion | Status | +|---|---| +| Label→file map produced | ✅ this document | +| All `\ref` resolve to a label | ✅ 0 dangling | +| No duplicate labels remain | ✅ 0 duplicates | +| No referenced label was broken | ✅ 119/119 resolved | + +## Patch logic + +Originally, 7 structural keys (`abstract`, `introduction`, `results-evidence`, `qed-assertions`, `sealed-seeds`, `discussion`, `references`) plus ~119 unreferenced content-section keys were defined identically across 70 chapter files in `docs/phd/chapters/`. Because none were consumed by `\ref`/`\autoref`/`\eqref`/`\Cref`/`\pageref`, they produced LaTeX duplicate-label warnings without breaking any cross-reference. + +**Rule:** for every `\label{KEY}` in a chapter file `.tex`: +- if `KEY` appears in any `\ref{KEY}` in the corpus → **leave bare** (protected) +- otherwise → rewrite to `\label{:KEY}` (idempotent: skip if already prefixed) + +This eliminates collisions with zero risk of breaking cross-refs because protected keys (the 119 referenced ones) are already namespaced and unique. + +## Label → File map + +Total entries: 1145 + +
Click to expand full map (1145 keys) + +| Label key | File(s) | +|---|---| +| `app:A` | `appendix/A-catalogue.tex` | +| `app:F` | `appendix/F-coq-citation-map.tex` | +| `app:acm-ae` | `appendix/H-acm-ae-checklist.tex` | +| `app:data-availability` | `appendix/G-data-availability.tex` | +| `app:falsification` | `appendix/B-falsification.tex` | +| `app:fpga-bitstream` | `appendix/F-fpga-bitstream.tex` | +| `app:golden-benchmark` | `appendix/C-golden-benchmark.tex` | +| `app:golden-mirror` | `appendix/D-golden-mirror.tex` | +| `app:troubleshooting` | `appendix/J-troubleshooting.tex` | +| `app:xdc-pin-map` | `appendix/I-xdc-pin-map.tex` | +| `app:zenodo-doi` | `appendix/H-zenodo-doi.tex` | +| `ch:1` | `chapters/fa_01.tex` | +| `ch:11` | `chapters/fa_11.tex` | +| `ch:13` | `chapters/fa_13.tex` | +| `ch:15` | `chapters/fa_15.tex` | +| `ch:17-spiral` | `chapters/fa_17.tex` | +| `ch:18` | `chapters/fa_18.tex` | +| `ch:19` | `chapters/fa_19.tex` | +| `ch:21-experiments-jepa` | `chapters/fa_21.tex` | +| `ch:23-gf16-algebra` | `chapters/fa_23.tex` | +| `ch:24` | `chapters/fa_24.tex` | +| `ch:24-igla-arch` | `chapters/fa_24.tex` | +| `ch:25` | `chapters/fa_25.tex` | +| `ch:25-benchmarks` | `chapters/fa_25.tex` | +| `ch:26-data-analysis` | `chapters/fa_26.tex` | +| `ch:28` | `chapters/fa_28.tex` | +| `ch:28-momentum-algebra` | `chapters/fa_28.tex` | +| `ch:32` | `chapters/fa_32.tex` | +| `ch:33` | `chapters/fa_33.tex` | +| `ch:34` | `chapters/fa_33.tex` | +| `ch:6` | `chapters/fa_06.tex` | +| `ch:9` | `chapters/fa_09.tex` | +| `ch:benchmarks` | `chapters/fa_25.tex` | +| `ch:data-analysis` | `chapters/fa_26.tex` | +| `ch:e8-symmetry` | `chapters/fa_22.tex` | +| `ch:energy` | `chapters/fa_28.tex` | +| `ch:experiments-asha` | `chapters/fa_21.tex` | +| `ch:experiments-bpb` | `chapters/fa_21.tex` | +| `ch:experiments-gf16` | `chapters/fa_23.tex` | +| `ch:fibonacci` | `chapters/fa_07.tex` | +| `ch:fibonacci-tesselation` | `chapters/fa_07.tex` | +| `ch:gf16-algebra` | `chapters/fa_23.tex` | +| `ch:golden-egg` | `chapters/fa_01.tex` | +| `ch:golden-seed` | `chapters/fa_01.tex` | +| `ch:igla-architecture` | `chapters/fa_24.tex` | +| `ch:igla-race` | `chapters/fa_24.tex` | +| `ch:jepa` | `chapters/fa_21.tex` | +| `ch:lucas-closure` | `chapters/fa_29.tex` | +| `ch:lucas-ladder` | `chapters/fa_29.tex` | +| `ch:lucas-ring` | `chapters/fa_27.tex` | +| `ch:monad` | `chapters/fa_00.tex` | +| `ch:nca` | `chapters/fa_29.tex` | +| `ch:plm` | `chapters/fa_24.tex` | +| `ch:standard-model` | `chapters/fa_20.tex` | +| `ch:three-strands` | `chapters/fa_27.tex` | +| `ch:trinity-identity` | `chapters/fa_27.tex` | +| `ch:vesica-piscis` | `chapters/fa_11.tex` | +| `ch:vsa` | `chapters/fa_29.tex` | +| `ch_00:ch:0` | `chapters/ch_00.tex` | +| `ch_00:thm:0:1` | `chapters/ch_00.tex` | +| `ch_00:thm:0:2` | `chapters/ch_00.tex` | +| `ch_01:abstract` | `chapters/ch_01.tex` | +| `ch_01:ch1-s1-vision-extended` | `chapters/ch_01.tex` | +| `ch_01:ch1-s2-contributions` | `chapters/ch_01.tex` | +| `ch_01:ch1-s3-lineage` | `chapters/ch_01.tex` | +| `ch_01:ch1-s4-theorem-xref` | `chapters/ch_01.tex` | +| `ch_01:ch1-s5-roadmap` | `chapters/ch_01.tex` | +| `ch_01:ch1-s6-notation` | `chapters/ch_01.tex` | +| `ch_01:discussion` | `chapters/ch_01.tex` | +| `ch_01:introduction` | `chapters/ch_01.tex` | +| `ch_01:qed-assertions` | `chapters/ch_01.tex` | +| `ch_01:references` | `chapters/ch_01.tex` | +| `ch_01:research-questions-and-scope` | `chapters/ch_01.tex` | +| `ch_01:results-evidence` | `chapters/ch_01.tex` | +| `ch_01:sealed-seeds` | `chapters/ch_01.tex` | +| `ch_01:tab:ch1-falsification-matrix` | `chapters/ch_01.tex` | +| `ch_01:the-trinity-architecture-and-its-algebraic-substrate` | `chapters/ch_01.tex` | +| `ch_01:thm:ch1-alpha-phi-closed` | `chapters/ch_01.tex` | +| `ch_01:thm:ch1-lucas-closure` | `chapters/ch_01.tex` | +| `ch_02:abstract` | `chapters/ch_02.tex` | +| `ch_02:ch2-s1-kart-kan` | `chapters/ch_02.tex` | +| `ch_02:ch2-s2-finite-field` | `chapters/ch_02.tex` | +| `ch_02:ch2-s3-ternary` | `chapters/ch_02.tex` | +| `ch_02:ch2-s4-vsa` | `chapters/ch_02.tex` | +| `ch_02:ch2-s5-ltn` | `chapters/ch_02.tex` | +| `ch_02:ch2-s6-cliffs` | `chapters/ch_02.tex` | +| `ch_02:ch2-s7-gap` | `chapters/ch_02.tex` | +| `ch_02:ch2-s8-theorems` | `chapters/ch_02.tex` | +| `ch_02:discussion` | `chapters/ch_02.tex` | +| `ch_02:early-symbolicconnectionist-hybrids` | `chapters/ch_02.tex` | +| `ch_02:fibonacci-and-lucas-lattices-as-basis-sets` | `chapters/ch_02.tex` | +| `ch_02:gap-in-prior-art` | `chapters/ch_02.tex` | +| `ch_02:introduction` | `chapters/ch_02.tex` | +| `ch_02:logic-tensor-networks-and-differentiable-reasoning` | `chapters/ch_02.tex` | +| `ch_02:qed-assertions` | `chapters/ch_02.tex` | +| `ch_02:references` | `chapters/ch_02.tex` | +| `ch_02:representational-bottleneck-and-the-ux3c6-structural-prior` | `chapters/ch_02.tex` | +| `ch_02:results-evidence` | `chapters/ch_02.tex` | +| `ch_02:sealed-seeds` | `chapters/ch_02.tex` | +| `ch_02:sparse-and-ternary-neural-computation` | `chapters/ch_02.tex` | +| `ch_02:taxonomy-of-neuro-symbolic-paradigms` | `chapters/ch_02.tex` | +| `ch_02:the-normalisation-problem` | `chapters/ch_02.tex` | +| `ch_02:thm:ch2-phi-square` | `chapters/ch_02.tex` | +| `ch_02:thm:ch2-trinity` | `chapters/ch_02.tex` | +| `ch_02:vector-symbolic-architectures` | `chapters/ch_02.tex` | +| `ch_03:abstract` | `chapters/ch_03.tex` | +| `ch_03:ch3-s1-trinity-detail` | `chapters/ch_03.tex` | +| `ch_03:ch3-s2-phi-family` | `chapters/ch_03.tex` | +| `ch_03:ch3-s3-coq-listing` | `chapters/ch_03.tex` | +| `ch_03:ch3-s4-numeric` | `chapters/ch_03.tex` | +| `ch_03:ch3-s5-arch` | `chapters/ch_03.tex` | +| `ch_03:coq-mechanisation-and-sac-0-invariant` | `chapters/ch_03.tex` | +| `ch_03:derivation-of-the-anchor-identity` | `chapters/ch_03.tex` | +| `ch_03:discussion` | `chapters/ch_03.tex` | +| `ch_03:introduction` | `chapters/ch_03.tex` | +| `ch_03:invariant-sac-0` | `chapters/ch_03.tex` | +| `ch_03:minimal-polynomial-and-basic-consequences` | `chapters/ch_03.tex` | +| `ch_03:power-survey` | `chapters/ch_03.tex` | +| `ch_03:proof-architecture` | `chapters/ch_03.tex` | +| `ch_03:qed-assertions` | `chapters/ch_03.tex` | +| `ch_03:references` | `chapters/ch_03.tex` | +| `ch_03:relation-to-fibonacci-arithmetic` | `chapters/ch_03.tex` | +| `ch_03:results-evidence` | `chapters/ch_03.tex` | +| `ch_03:sealed-seeds` | `chapters/ch_03.tex` | +| `ch_03:the-integer-3-coincidence` | `chapters/ch_03.tex` | +| `ch_04:abstract` | `chapters/ch_04.tex` | +| `ch_04:ch4-s1-alpha-phi` | `chapters/ch_04.tex` | +| `ch_04:ch4-s2-dimensional` | `chapters/ch_04.tex` | +| `ch_04:ch4-s3-alpha-qed` | `chapters/ch_04.tex` | +| `ch_04:ch4-s4-derivation-levels` | `chapters/ch_04.tex` | +| `ch_04:ch4-s5-runtime` | `chapters/ch_04.tex` | +| `ch_04:ch4-s6-gate` | `chapters/ch_04.tex` | +| `ch_04:derivation-of-the-closed-form` | `chapters/ch_04.tex` | +| `ch_04:discussion` | `chapters/ch_04.tex` | +| `ch_04:introduction` | `chapters/ch_04.tex` | +| `ch_04:multiplicative-identity-and-kernel-integration` | `chapters/ch_04.tex` | +| `ch_04:qed-assertions` | `chapters/ch_04.tex` | +| `ch_04:references` | `chapters/ch_04.tex` | +| `ch_04:results-evidence` | `chapters/ch_04.tex` | +| `ch_04:sealed-seeds` | `chapters/ch_04.tex` | +| `ch_04:tab:ch4-dimensional` | `chapters/ch_04.tex` | +| `ch_05:abstract` | `chapters/ch_05.tex` | +| `ch_05:ch5-s1-lucas-closure` | `chapters/ch_05.tex` | +| `ch_05:ch5-s2-basin` | `chapters/ch_05.tex` | +| `ch_05:ch5-s3-seeds` | `chapters/ch_05.tex` | +| `ch_05:ch5-s4-coq-listing` | `chapters/ch_05.tex` | +| `ch_05:ch5-s5-admissibility` | `chapters/ch_05.tex` | +| `ch_05:ch5-s6-arch` | `chapters/ch_05.tex` | +| `ch_05:discussion` | `chapters/ch_05.tex` | +| `ch_05:fibonacci-lucas-seeds-and-their-contractive-basin` | `chapters/ch_05.tex` | +| `ch_05:introduction` | `chapters/ch_05.tex` | +| `ch_05:qed-assertions` | `chapters/ch_05.tex` | +| `ch_05:references` | `chapters/ch_05.tex` | +| `ch_05:results-evidence` | `chapters/ch_05.tex` | +| `ch_05:sealed-seeds` | `chapters/ch_05.tex` | +| `ch_05:the-ux3c6-distance-metric-and-the-balancing-fixed-point` | `chapters/ch_05.tex` | +| `ch_06:abstract` | `chapters/ch_06.tex` | +| `ch_06:coq-encoding` | `chapters/ch_06.tex` | +| `ch_06:discussion` | `chapters/ch_06.tex` | +| `ch_06:goldenfloat-format-definitions` | `chapters/ch_06.tex` | +| `ch_06:introduction` | `chapters/ch_06.tex` | +| `ch_06:key-theorems-and-proof-sketches` | `chapters/ch_06.tex` | +| `ch_06:lucas-closure-on-gf16` | `chapters/ch_06.tex` | +| `ch_06:preliminaries` | `chapters/ch_06.tex` | +| `ch_06:qed-assertions` | `chapters/ch_06.tex` | +| `ch_06:references` | `chapters/ch_06.tex` | +| `ch_06:results-evidence` | `chapters/ch_06.tex` | +| `ch_06:sealed-seeds` | `chapters/ch_06.tex` | +| `ch_07:abstract` | `chapters/ch_07.tex` | +| `ch_07:discussion` | `chapters/ch_07.tex` | +| `ch_07:from-the-trinity-identity-to-the-golden-angle` | `chapters/ch_07.tex` | +| `ch_07:h4-root-system-e8-lattice-and-the-varphi-scaled-block-decomposition` | `chapters/ch_07.tex` | +| `ch_07:introduction` | `chapters/ch_07.tex` | +| `ch_07:qed-assertions` | `chapters/ch_07.tex` | +| `ch_07:references` | `chapters/ch_07.tex` | +| `ch_07:results-evidence` | `chapters/ch_07.tex` | +| `ch_07:sealed-seeds` | `chapters/ch_07.tex` | +| `ch_08:abstract` | `chapters/ch_08.tex` | +| `ch_08:discussion` | `chapters/ch_08.tex` | +| `ch_08:gain-admissibility` | `chapters/ch_08.tex` | +| `ch_08:hybrid-qk-gain-invariant-inv-6` | `chapters/ch_08.tex` | +| `ch_08:introduction` | `chapters/ch_08.tex` | +| `ch_08:proof-sketch-for-admit_phi_sq` | `chapters/ch_08.tex` | +| `ch_08:qed-assertions` | `chapters/ch_08.tex` | +| `ch_08:references` | `chapters/ch_08.tex` | +| `ch_08:results-evidence` | `chapters/ch_08.tex` | +| `ch_08:sealed-seeds` | `chapters/ch_08.tex` | +| `ch_08:tf3-and-tf9-algebraic-structure` | `chapters/ch_08.tex` | +| `ch_08:tf9-product-encoding` | `chapters/ch_08.tex` | +| `ch_08:trit-encoding` | `chapters/ch_08.tex` | +| `ch_08:ux3c6-normalisation` | `chapters/ch_08.tex` | +| `ch_09:ablation-matrix-tier-abc-m1m6` | `chapters/ch_09.tex` | +| `ch_09:abstract` | `chapters/ch_09.tex` | +| `ch_09:competitor-format-summaries` | `chapters/ch_09.tex` | +| `ch_09:discussion` | `chapters/ch_09.tex` | +| `ch_09:gf16-format-specification` | `chapters/ch_09.tex` | +| `ch_09:gf16-phi_bias60-and-the-inv-3-safe-domain` | `chapters/ch_09.tex` | +| `ch_09:introduction` | `chapters/ch_09.tex` | +| `ch_09:inv-3-nine-coq-precision-bounds` | `chapters/ch_09.tex` | +| `ch_09:qed-assertions` | `chapters/ch_09.tex` | +| `ch_09:references` | `chapters/ch_09.tex` | +| `ch_09:results-evidence` | `chapters/ch_09.tex` | +| `ch_09:sealed-seeds` | `chapters/ch_09.tex` | +| `ch_10:abstract` | `chapters/ch_10.tex` | +| `ch_10:discussion` | `chapters/ch_10.tex` | +| `ch_10:gf16-range-and-precision-formalisation` | `chapters/ch_10.tex` | +| `ch_10:introduction` | `chapters/ch_10.tex` | +| `ch_10:qed-assertions` | `chapters/ch_10.tex` | +| `ch_10:references` | `chapters/ch_10.tex` | +| `ch_10:results-evidence` | `chapters/ch_10.tex` | +| `ch_10:sealed-seeds` | `chapters/ch_10.tex` | +| `ch_10:the-pareto-frontier-and-conjecture-c1` | `chapters/ch_10.tex` | +| `ch_11:abstract` | `chapters/ch_11.tex` | +| `ch_11:discussion` | `chapters/ch_11.tex` | +| `ch_11:hypothesis-formalisation-and-registration-protocol` | `chapters/ch_11.tex` | +| `ch_11:introduction` | `chapters/ch_11.tex` | +| `ch_11:inv-7-invariant-and-coq-formalisation` | `chapters/ch_11.tex` | +| `ch_11:qed-assertions` | `chapters/ch_11.tex` | +| `ch_11:references` | `chapters/ch_11.tex` | +| `ch_11:results-evidence` | `chapters/ch_11.tex` | +| `ch_11:sealed-seeds` | `chapters/ch_11.tex` | +| `ch_12:abstract` | `chapters/ch_12.tex` | +| `ch_12:bridge-architecture-and-interface-contracts` | `chapters/ch_12.tex` | +| `ch_12:clock-domain-analysis-and-timing` | `chapters/ch_12.tex` | +| `ch_12:discussion` | `chapters/ch_12.tex` | +| `ch_12:error-handling-protocol` | `chapters/ch_12.tex` | +| `ch_12:frequency-ratios-and-the-golden-ratio` | `chapters/ch_12.tex` | +| `ch_12:introduction` | `chapters/ch_12.tex` | +| `ch_12:logical-structure` | `chapters/ch_12.tex` | +| `ch_12:power-accounting` | `chapters/ch_12.tex` | +| `ch_12:qed-assertions` | `chapters/ch_12.tex` | +| `ch_12:references` | `chapters/ch_12.tex` | +| `ch_12:results-evidence` | `chapters/ch_12.tex` | +| `ch_12:sealed-seeds` | `chapters/ch_12.tex` | +| `ch_12:signal-naming-convention` | `chapters/ch_12.tex` | +| `ch_12:throughput-budget` | `chapters/ch_12.tex` | +| `ch_13:abstract` | `chapters/ch_13.tex` | +| `ch_13:discussion` | `chapters/ch_13.tex` | +| `ch_13:introduction` | `chapters/ch_13.tex` | +| `ch_13:qed-assertions` | `chapters/ch_13.tex` | +| `ch_13:references` | `chapters/ch_13.tex` | +| `ch_13:results-evidence` | `chapters/ch_13.tex` | +| `ch_13:sealed-seeds` | `chapters/ch_13.tex` | +| `ch_13:the-runtime-mirror-contract-and-igla_assertions.json` | `chapters/ch_13.tex` | +| `ch_13:the-strobe-seed-admissibility-criterion` | `chapters/ch_13.tex` | +| `ch_14:abstract` | `chapters/ch_14.tex` | +| `ch_14:bpb-definition-and-algebraic-properties` | `chapters/ch_14.tex` | +| `ch_14:byte-level-normalisation` | `chapters/ch_14.tex` | +| `ch_14:cross-entropy-and-perplexity` | `chapters/ch_14.tex` | +| `ch_14:discussion` | `chapters/ch_14.tex` | +| `ch_14:gate-2-bpb-1.85` | `chapters/ch_14.tex` | +| `ch_14:gate-3-bpb-1.50` | `chapters/ch_14.tex` | +| `ch_14:gate-thresholds-and-their-derivation` | `chapters/ch_14.tex` | +| `ch_14:introduction` | `chapters/ch_14.tex` | +| `ch_14:qed-assertions` | `chapters/ch_14.tex` | +| `ch_14:references` | `chapters/ch_14.tex` | +| `ch_14:relationship-to-the-darpa-energy-goal` | `chapters/ch_14.tex` | +| `ch_14:results-evidence` | `chapters/ch_14.tex` | +| `ch_14:sealed-seeds` | `chapters/ch_14.tex` | +| `ch_14:ux3c6-weighted-bpb` | `chapters/ch_14.tex` | +| `ch_15:abstract` | `chapters/ch_15.tex` | +| `ch_15:bpb-protocol-and-monotone-backward-invariant-inv-1` | `chapters/ch_15.tex` | +| `ch_15:database-schema` | `chapters/ch_15.tex` | +| `ch_15:discussion` | `chapters/ch_15.tex` | +| `ch_15:evaluation-protocol` | `chapters/ch_15.tex` | +| `ch_15:gate-evaluation` | `chapters/ch_15.tex` | +| `ch_15:introduction` | `chapters/ch_15.tex` | +| `ch_15:inv-1-bpb-monotone-backward` | `chapters/ch_15.tex` | +| `ch_15:qed-assertions` | `chapters/ch_15.tex` | +| `ch_15:railway-write-back-architecture` | `chapters/ch_15.tex` | +| `ch_15:references` | `chapters/ch_15.tex` | +| `ch_15:results-evidence` | `chapters/ch_15.tex` | +| `ch_15:sealed-seeds` | `chapters/ch_15.tex` | +| `ch_15:warmup-gate` | `chapters/ch_15.tex` | +| `ch_15:write-back-protocol` | `chapters/ch_15.tex` | +| `ch_16:abstract` | `chapters/ch_16.tex` | +| `ch_16:discussion` | `chapters/ch_16.tex` | +| `ch_16:grid-construction-and-sparsity-analysis` | `chapters/ch_16.tex` | +| `ch_16:introduction` | `chapters/ch_16.tex` | +| `ch_16:qed-assertions` | `chapters/ch_16.tex` | +| `ch_16:references` | `chapters/ch_16.tex` | +| `ch_16:results-evidence` | `chapters/ch_16.tex` | +| `ch_16:sealed-seeds` | `chapters/ch_16.tex` | +| `ch_16:the-phi-distance-function` | `chapters/ch_16.tex` | +| `ch_17:abstract` | `chapters/ch_17.tex` | +| `ch_17:analysis-of-effects-and-golden-ratio-structure` | `chapters/ch_17.tex` | +| `ch_17:discussion` | `chapters/ch_17.tex` | +| `ch_17:factor-definitions-and-experimental-design` | `chapters/ch_17.tex` | +| `ch_17:introduction` | `chapters/ch_17.tex` | +| `ch_17:qed-assertions` | `chapters/ch_17.tex` | +| `ch_17:references` | `chapters/ch_17.tex` | +| `ch_17:results-evidence` | `chapters/ch_17.tex` | +| `ch_17:sealed-seeds` | `chapters/ch_17.tex` | +| `ch_18:abstract` | `chapters/ch_18.tex` | +| `ch_18:coq.interval-upgrade-lane` | `chapters/ch_18.tex` | +| `ch_18:discussion` | `chapters/ch_18.tex` | +| `ch_18:hardware-and-runtime-limitations` | `chapters/ch_18.tex` | +| `ch_18:introduction` | `chapters/ch_18.tex` | +| `ch_18:qed-assertions` | `chapters/ch_18.tex` | +| `ch_18:references` | `chapters/ch_18.tex` | +| `ch_18:sealed-seeds` | `chapters/ch_18.tex` | +| `ch_18:state-of-the-art-comparison-clara-soa-snapshot` | `chapters/ch_18.tex` | +| `ch_19:abstract` | `chapters/ch_19.tex` | +| `ch_19:discussion` | `chapters/ch_19.tex` | +| `ch_19:introduction` | `chapters/ch_19.tex` | +| `ch_19:qed-assertions` | `chapters/ch_19.tex` | +| `ch_19:references` | `chapters/ch_19.tex` | +| `ch_19:results-evidence` | `chapters/ch_19.tex` | +| `ch_19:sealed-seeds` | `chapters/ch_19.tex` | +| `ch_19:test-design-and-hypotheses` | `chapters/ch_19.tex` | +| `ch_19:welch-t-statistic-and-degrees-of-freedom` | `chapters/ch_19.tex` | +| `ch_20:abstract` | `chapters/ch_20.tex` | +| `ch_20:algebraic-basis` | `chapters/ch_20.tex` | +| `ch_20:discussion` | `chapters/ch_20.tex` | +| `ch_20:hardware-and-software-specification` | `chapters/ch_20.tex` | +| `ch_20:hardware-pinning` | `chapters/ch_20.tex` | +| `ch_20:introduction` | `chapters/ch_20.tex` | +| `ch_20:non-determinism-budget` | `chapters/ch_20.tex` | +| `ch_20:qed-assertions` | `chapters/ch_20.tex` | +| `ch_20:references` | `chapters/ch_20.tex` | +| `ch_20:results-evidence` | `chapters/ch_20.tex` | +| `ch_20:sanctioned-seed-protocol` | `chapters/ch_20.tex` | +| `ch_20:sealed-seeds` | `chapters/ch_20.tex` | +| `ch_20:seed-assignment-to-experiments` | `chapters/ch_20.tex` | +| `ch_20:seed-verification` | `chapters/ch_20.tex` | +| `ch_20:software-environment` | `chapters/ch_20.tex` | +| `ch_21:abstract` | `chapters/ch_21.tex` | +| `ch_21:agent-topology` | `chapters/ch_21.tex` | +| `ch_21:definitions` | `chapters/ch_21.tex` | +| `ch_21:discussion` | `chapters/ch_21.tex` | +| `ch_21:formal-victory-criterion-inv-7` | `chapters/ch_21.tex` | +| `ch_21:introduction` | `chapters/ch_21.tex` | +| `ch_21:multi-agent-fleet-architecture` | `chapters/ch_21.tex` | +| `ch_21:qed-assertions` | `chapters/ch_21.tex` | +| `ch_21:rainbow-bridge-consistency-inv-7b` | `chapters/ch_21.tex` | +| `ch_21:references` | `chapters/ch_21.tex` | +| `ch_21:relation-to-varphi2-varphi-2-3` | `chapters/ch_21.tex` | +| `ch_21:results-evidence` | `chapters/ch_21.tex` | +| `ch_21:sealed-seeds` | `chapters/ch_21.tex` | +| `ch_21:six-refutation-theorems` | `chapters/ch_21.tex` | +| `ch_21:victory-declaration-protocol` | `chapters/ch_21.tex` | +| `ch_22:abstract` | `chapters/ch_22.tex` | +| `ch_22:discussion` | `chapters/ch_22.tex` | +| `ch_22:introduction` | `chapters/ch_22.tex` | +| `ch_22:qed-assertions` | `chapters/ch_22.tex` | +| `ch_22:references` | `chapters/ch_22.tex` | +| `ch_22:results-evidence` | `chapters/ch_22.tex` | +| `ch_22:satisfaction-witness-and-victory-predicate` | `chapters/ch_22.tex` | +| `ch_22:sealed-seeds` | `chapters/ch_22.tex` | +| `ch_22:worker-pool-invariants-and-falsification-witnesses` | `chapters/ch_22.tex` | +| `ch_23:abstract` | `chapters/ch_23.tex` | +| `ch_23:discussion` | `chapters/ch_23.tex` | +| `ch_23:introduction` | `chapters/ch_23.tex` | +| `ch_23:mcp-adapter-layer-architecture` | `chapters/ch_23.tex` | +| `ch_23:protocol-implementation-and-latency-analysis` | `chapters/ch_23.tex` | +| `ch_23:qed-assertions` | `chapters/ch_23.tex` | +| `ch_23:references` | `chapters/ch_23.tex` | +| `ch_23:results-evidence` | `chapters/ch_23.tex` | +| `ch_23:sealed-seeds` | `chapters/ch_23.tex` | +| `ch_24:abstract` | `chapters/ch_24.tex` | +| `ch_24:agent-model` | `chapters/ch_24.tex` | +| `ch_24:coq-encoding` | `chapters/ch_24.tex` | +| `ch_24:discussion` | `chapters/ch_24.tex` | +| `ch_24:formal-model-of-the-period-locked-monitor` | `chapters/ch_24.tex` | +| `ch_24:implementation-and-hardware-interface` | `chapters/ch_24.tex` | +| `ch_24:interrupt-interface-with-the-hardware-bridge` | `chapters/ch_24.tex` | +| `ch_24:introduction` | `chapters/ch_24.tex` | +| `ch_24:period-ratio-and-non-resonance` | `chapters/ch_24.tex` | +| `ch_24:priority-queue-and-phi-weighted-scheduling` | `chapters/ch_24.tex` | +| `ch_24:qed-assertions` | `chapters/ch_24.tex` | +| `ch_24:references` | `chapters/ch_24.tex` | +| `ch_24:results-evidence` | `chapters/ch_24.tex` | +| `ch_24:rtl-implementation` | `chapters/ch_24.tex` | +| `ch_24:sealed-seeds` | `chapters/ch_24.tex` | +| `ch_25:abstract` | `chapters/ch_25.tex` | +| `ch_25:cycle-classification-and-attention-periodicity` | `chapters/ch_25.tex` | +| `ch_25:discussion` | `chapters/ch_25.tex` | +| `ch_25:introduction` | `chapters/ch_25.tex` | +| `ch_25:qed-assertions` | `chapters/ch_25.tex` | +| `ch_25:references` | `chapters/ch_25.tex` | +| `ch_25:results-evidence` | `chapters/ch_25.tex` | +| `ch_25:sealed-seeds` | `chapters/ch_25.tex` | +| `ch_25:varphi-lattice-structure-and-the-cycle-map` | `chapters/ch_25.tex` | +| `ch_26:abstract` | `chapters/ch_26.tex` | +| `ch_26:discussion` | `chapters/ch_26.tex` | +| `ch_26:gf16_quant-galois-field-16-quantisation` | `chapters/ch_26.tex` | +| `ch_26:instruction-encoding` | `chapters/ch_26.tex` | +| `ch_26:introduction` | `chapters/ch_26.tex` | +| `ch_26:isa-register-file-and-encoding` | `chapters/ch_26.tex` | +| `ch_26:opcode-specifications` | `chapters/ch_26.tex` | +| `ch_26:phi_rope-ux3c6-rotary-position-encoding` | `chapters/ch_26.tex` | +| `ch_26:qed-assertions` | `chapters/ch_26.tex` | +| `ch_26:references` | `chapters/ch_26.tex` | +| `ch_26:register-file` | `chapters/ch_26.tex` | +| `ch_26:results-evidence` | `chapters/ch_26.tex` | +| `ch_26:sealed-seeds` | `chapters/ch_26.tex` | +| `ch_26:tf3_add-ternary-addition` | `chapters/ch_26.tex` | +| `ch_26:tf3_mul-ternary-multiplication` | `chapters/ch_26.tex` | +| `ch_26:vsa_bind-hyperdimensional-binding` | `chapters/ch_26.tex` | +| `ch_26:vsa_bundle-hyperdimensional-bundling` | `chapters/ch_26.tex` | +| `ch_26:vsa_unbind-hyperdimensional-unbinding` | `chapters/ch_26.tex` | +| `ch_27:abstract` | `chapters/ch_27.tex` | +| `ch_27:abstract-syntax` | `chapters/ch_27.tex` | +| `ch_27:discussion` | `chapters/ch_27.tex` | +| `ch_27:environments-and-evaluation` | `chapters/ch_27.tex` | +| `ch_27:introduction` | `chapters/ch_27.tex` | +| `ch_27:mechanised-proofs-determinism-and-exhaustiveness` | `chapters/ch_27.tex` | +| `ch_27:qed-assertions` | `chapters/ch_27.tex` | +| `ch_27:references` | `chapters/ch_27.tex` | +| `ch_27:relation-to-gf16-and-varphi-arithmetic` | `chapters/ch_27.tex` | +| `ch_27:results-evidence` | `chapters/ch_27.tex` | +| `ch_27:sealed-seeds` | `chapters/ch_27.tex` | +| `ch_27:ternary-arithmetic` | `chapters/ch_27.tex` | +| `ch_27:theorem-eval_det-determinism` | `chapters/ch_27.tex` | +| `ch_27:theorem-trit_exhaustive-exhaustiveness` | `chapters/ch_27.tex` | +| `ch_27:tri27-syntax-and-denotational-semantics` | `chapters/ch_27.tex` | +| `ch_28:abstract` | `chapters/ch_28.tex` | +| `ch_28:architecture-zero-dsp-ternary-datapath` | `chapters/ch_28.tex` | +| `ch_28:discussion` | `chapters/ch_28.tex` | +| `ch_28:introduction` | `chapters/ch_28.tex` | +| `ch_28:qed-assertions` | `chapters/ch_28.tex` | +| `ch_28:references` | `chapters/ch_28.tex` | +| `ch_28:resource-utilisation-and-timing-closure` | `chapters/ch_28.tex` | +| `ch_28:results-evidence` | `chapters/ch_28.tex` | +| `ch_28:sealed-seeds` | `chapters/ch_28.tex` | +| `ch_29:abstract` | `chapters/ch_29.tex` | +| `ch_29:coq-formalisation-and-ckm-unitarity-seed` | `chapters/ch_29.tex` | +| `ch_29:discussion` | `chapters/ch_29.tex` | +| `ch_29:introduction` | `chapters/ch_29.tex` | +| `ch_29:qed-assertions` | `chapters/ch_29.tex` | +| `ch_29:references` | `chapters/ch_29.tex` | +| `ch_29:results-evidence` | `chapters/ch_29.tex` | +| `ch_29:sealed-seeds` | `chapters/ch_29.tex` | +| `ch_29:the-sacred-formula-v-conjecture-and-ux3c6-monomial-parameterisation` | `chapters/ch_29.tex` | +| `ch_30:abstract` | `chapters/ch_30.tex` | +| `ch_30:associative-recall-memory` | `chapters/ch_30.tex` | +| `ch_30:discussion` | `chapters/ch_30.tex` | +| `ch_30:goldenfloat-encoding-of-hypervectors` | `chapters/ch_30.tex` | +| `ch_30:hypervector-definition` | `chapters/ch_30.tex` | +| `ch_30:introduction` | `chapters/ch_30.tex` | +| `ch_30:phi-rotary-position-encoding-phi-rope-in-vsa-context` | `chapters/ch_30.tex` | +| `ch_30:qed-assertions` | `chapters/ch_30.tex` | +| `ch_30:references` | `chapters/ch_30.tex` | +| `ch_30:results-evidence` | `chapters/ch_30.tex` | +| `ch_30:sealed-seeds` | `chapters/ch_30.tex` | +| `ch_30:ternary-vsa-over-the-goldenfloat-substrate` | `chapters/ch_30.tex` | +| `ch_31:abstract` | `chapters/ch_31.tex` | +| `ch_31:discussion` | `chapters/ch_31.tex` | +| `ch_31:formal-seal-297-coq-theorems` | `chapters/ch_31.tex` | +| `ch_31:hardware-architecture` | `chapters/ch_31.tex` | +| `ch_31:introduction` | `chapters/ch_31.tex` | +| `ch_31:qed-assertions` | `chapters/ch_31.tex` | +| `ch_31:references` | `chapters/ch_31.tex` | +| `ch_31:results-evidence` | `chapters/ch_31.tex` | +| `ch_31:sealed-seeds` | `chapters/ch_31.tex` | +| `ch_32:abstract` | `chapters/ch_32.tex` | +| `ch_32:crc-16ccitt-polynomial` | `chapters/ch_32.tex` | +| `ch_32:discussion` | `chapters/ch_32.tex` | +| `ch_32:error-recovery-automaton` | `chapters/ch_32.tex` | +| `ch_32:frame-grammar` | `chapters/ch_32.tex` | +| `ch_32:frame-structure-and-grammar` | `chapters/ch_32.tex` | +| `ch_32:introduction` | `chapters/ch_32.tex` | +| `ch_32:physical-layer` | `chapters/ch_32.tex` | +| `ch_32:qed-assertions` | `chapters/ch_32.tex` | +| `ch_32:references` | `chapters/ch_32.tex` | +| `ch_32:results-evidence` | `chapters/ch_32.tex` | +| `ch_32:sealed-seeds` | `chapters/ch_32.tex` | +| `ch_32:sync-frame-payload` | `chapters/ch_32.tex` | +| `ch_32:sync-frame-trigger` | `chapters/ch_32.tex` | +| `ch_32:ux3c6-synchronisation-frames` | `chapters/ch_32.tex` | +| `ch_33:abstract` | `chapters/ch_33.tex` | +| `ch_33:diagnosis-and-root-cause` | `chapters/ch_33.tex` | +| `ch_33:discussion` | `chapters/ch_33.tex` | +| `ch_33:flash_no_sudo.sh` | `chapters/ch_33.tex` | +| `ch_33:fxload-cross-compilation` | `chapters/ch_33.tex` | +| `ch_33:introduction` | `chapters/ch_33.tex` | +| `ch_33:qed-assertions` | `chapters/ch_33.tex` | +| `ch_33:references` | `chapters/ch_33.tex` | +| `ch_33:results-evidence` | `chapters/ch_33.tex` | +| `ch_33:sealed-seeds` | `chapters/ch_33.tex` | +| `ch_33:usb-enumeration-on-macos-arm` | `chapters/ch_33.tex` | +| `ch_33:verified-hardware-configuration-post-blk-001` | `chapters/ch_33.tex` | +| `ch_34:abstract` | `chapters/ch_34.tex` | +| `ch_34:discussion` | `chapters/ch_34.tex` | +| `ch_34:energy-accounting-framework` | `chapters/ch_34.tex` | +| `ch_34:introduction` | `chapters/ch_34.tex` | +| `ch_34:qed-assertions` | `chapters/ch_34.tex` | +| `ch_34:references` | `chapters/ch_34.tex` | +| `ch_34:results-evidence` | `chapters/ch_34.tex` | +| `ch_34:sealed-seeds` | `chapters/ch_34.tex` | +| `ch_34:ternary-mechanism-analysis` | `chapters/ch_34.tex` | +| `ch_35_mesh_node:ch:mesh-node` | `chapters/ch_35_mesh_node.tex` | +| `ch_35_mesh_node:fig:asic-block` | `chapters/ch_35_mesh_node.tex` | +| `ch_35_mesh_node:tab:comparison` | `chapters/ch_35_mesh_node.tex` | +| `ch_35_mesh_node:tab:rns-packets` | `chapters/ch_35_mesh_node.tex` | +| `ch_35_mesh_node:thm:mru-liveness` | `chapters/ch_35_mesh_node.tex` | +| `ch_35_mesh_node:thm:phi-id` | `chapters/ch_35_mesh_node.tex` | +| `ch_35_mesh_node:thm:power-budget` | `chapters/ch_35_mesh_node.tex` | +| `cor:01-l2-three` | `chapters/fa_01.tex` | +| `cor:01-lucas-as-trace` | `chapters/fa_01.tex` | +| `cor:01-reciprocal` | `chapters/fa_01.tex` | +| `cor:05-asymptotic-rate` | `chapters/fa_05.tex` | +| `cor:05-binet` | `chapters/fa_05.tex` | +| `def:13-lucas-12` | `chapters/fa_13.tex` | +| `eq:ch0-fit` | `chapters/ch_00.tex` | +| `eq:gf-def` | `appendix/C-golden-benchmark.tex` | +| `fa_00:thm:trinity-identity-prologue` | `chapters/fa_00.tex` | +| `fa_01:cor:01-approx-quality` | `chapters/fa_01.tex` | +| `fa_01:cor:01-cascade` | `chapters/fa_01.tex` | +| `fa_01:cor:01-fp-rate` | `chapters/fa_01.tex` | +| `fa_01:cor:01-pentagon-vesica` | `chapters/fa_01.tex` | +| `fa_01:fig:vesica` | `chapters/fa_01.tex` | +| `fa_01:lem:01-gm-limit` | `chapters/fa_01.tex` | +| `fa_01:lem:01-golden-angle` | `chapters/fa_01.tex` | +| `fa_01:lem:01-hex-vesica` | `chapters/fa_01.tex` | +| `fa_01:lem:01-newton-phi` | `chapters/fa_01.tex` | +| `fa_01:lem:01-pentagram-self` | `chapters/fa_01.tex` | +| `fa_01:lem:01-small-n` | `chapters/fa_01.tex` | +| `fa_01:lem:01-vesica-area` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-A` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-AA` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-AB` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-AC` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-AD` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-AE` | 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`chapters/fa_01.tex` | +| `fa_01:sec:01-app-G` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-H` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-I` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-J` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-K` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-L` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-M` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-N` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-O` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-P` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-Q` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-R` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-S` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-T` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-U` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-V` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-W` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-X` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-Y` | `chapters/fa_01.tex` | +| `fa_01:sec:01-app-Z` | `chapters/fa_01.tex` | +| `fa_01:sec:01-strand-I` | `chapters/fa_01.tex` | +| `fa_01:sec:01-strand-II` | `chapters/fa_01.tex` | +| `fa_01:sec:01-strand-III` | `chapters/fa_01.tex` | +| `fa_01:thm:01-dodec-circ` | `chapters/fa_01.tex` | +| `fa_01:thm:01-lucas-anchor` | `chapters/fa_01.tex` | +| `fa_01:thm:01-lucas-as-trace` | `chapters/fa_01.tex` | +| `fa_01:thm:01-ring-int` | `chapters/fa_01.tex` | +| `fa_01:thm:01-three-witnesses` | `chapters/fa_01.tex` | +| `fa_01:thm:01-trace-as-Z` | `chapters/fa_01.tex` | +| `fa_01:thm:01-universal-anchor` | `chapters/fa_01.tex` | +| `fa_02:abstract` | `chapters/fa_02.tex` | +| `fa_02:discussion` | `chapters/fa_02.tex` | +| `fa_02:early-symbolicconnectionist-hybrids` | `chapters/fa_02.tex` | +| `fa_02:fibonacci-and-lucas-lattices-as-basis-sets` | `chapters/fa_02.tex` | +| `fa_02:gap-in-prior-art` | `chapters/fa_02.tex` | +| `fa_02:introduction` | `chapters/fa_02.tex` | +| `fa_02:logic-tensor-networks-and-differentiable-reasoning` | `chapters/fa_02.tex` | +| `fa_02:qed-assertions` | `chapters/fa_02.tex` | +| `fa_02:references` | `chapters/fa_02.tex` | +| `fa_02:representational-bottleneck-and-the-ux3c6-structural-prior` | `chapters/fa_02.tex` | +| `fa_02:results-evidence` | `chapters/fa_02.tex` | +| `fa_02:sealed-seeds` | `chapters/fa_02.tex` | +| `fa_02:sparse-and-ternary-neural-computation` | `chapters/fa_02.tex` | +| `fa_02:taxonomy-of-neuro-symbolic-paradigms` | `chapters/fa_02.tex` | +| `fa_02:the-normalisation-problem` | `chapters/fa_02.tex` | +| `fa_02:vector-symbolic-architectures` | `chapters/fa_02.tex` | +| `fa_03:abstract` | `chapters/fa_03.tex` | +| `fa_03:coq-mechanisation-and-sac-0-invariant` | `chapters/fa_03.tex` | +| `fa_03:derivation-of-the-anchor-identity` | `chapters/fa_03.tex` | +| `fa_03:discussion` | `chapters/fa_03.tex` | +| `fa_03:introduction` | `chapters/fa_03.tex` | +| `fa_03:invariant-sac-0` | `chapters/fa_03.tex` | +| `fa_03:minimal-polynomial-and-basic-consequences` | `chapters/fa_03.tex` | +| `fa_03:power-survey` | `chapters/fa_03.tex` | +| `fa_03:proof-architecture` | `chapters/fa_03.tex` | +| `fa_03:qed-assertions` | `chapters/fa_03.tex` | +| `fa_03:references` | `chapters/fa_03.tex` | +| `fa_03:relation-to-fibonacci-arithmetic` | `chapters/fa_03.tex` | +| `fa_03:results-evidence` | `chapters/fa_03.tex` | +| `fa_03:sealed-seeds` | `chapters/fa_03.tex` | +| `fa_03:the-integer-3-coincidence` | `chapters/fa_03.tex` | +| `fa_04:abstract` | `chapters/fa_04.tex` | +| `fa_04:derivation-of-the-closed-form` | `chapters/fa_04.tex` | +| `fa_04:discussion` | `chapters/fa_04.tex` | +| `fa_04:introduction` | `chapters/fa_04.tex` | +| `fa_04:multiplicative-identity-and-kernel-integration` | `chapters/fa_04.tex` | +| `fa_04:qed-assertions` | `chapters/fa_04.tex` | +| `fa_04:references` | `chapters/fa_04.tex` | +| `fa_04:results-evidence` | `chapters/fa_04.tex` | +| `fa_04:sealed-seeds` | `chapters/fa_04.tex` | +| `fa_05:ch:golden-bridge` | `chapters/fa_05.tex` | +| 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+| `fa_05:lem:05-rational` | `chapters/fa_05.tex` | +| `fa_05:lem:05-riordan-small` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-A` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-AA` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-AB` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-AC` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-AD` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-AE` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-AF` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-AG` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-AH` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-B` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-C` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-D` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-E` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-G` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-H` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-I` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-J` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-L` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-M` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-N` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-O` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-P` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-Q` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-R` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-S` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-T` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-U` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-V` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-W` | `chapters/fa_05.tex` | +| `fa_05:sec:05-app-Z` | `chapters/fa_05.tex` | +| `fa_05:sec:05-closing` | `chapters/fa_05.tex` | +| `fa_05:sec:05-intro` | `chapters/fa_05.tex` | +| `fa_05:sec:05-library` | `chapters/fa_05.tex` | +| `fa_05:thm:05-bridge-to-l4` | `chapters/fa_05.tex` | +| `fa_05:thm:05-bridge-to-l6` | `chapters/fa_05.tex` | +| `fa_05:thm:05-cassini-triangle` | `chapters/fa_05.tex` | +| `fa_05:thm:05-gf-product` | `chapters/fa_05.tex` | +| `fa_06:abstract` | `chapters/fa_06.tex` | +| `fa_06:coq-encoding` | `chapters/fa_06.tex` | +| `fa_06:discussion` | `chapters/fa_06.tex` | +| `fa_06:goldenfloat-format-definitions` | `chapters/fa_06.tex` | +| `fa_06:introduction` | `chapters/fa_06.tex` | +| `fa_06:key-theorems-and-proof-sketches` | `chapters/fa_06.tex` | +| `fa_06:lucas-closure-on-gf16` | `chapters/fa_06.tex` | +| `fa_06:preliminaries` | `chapters/fa_06.tex` | +| `fa_06:qed-assertions` | `chapters/fa_06.tex` | +| `fa_06:references` | `chapters/fa_06.tex` | +| `fa_06:results-evidence` | `chapters/fa_06.tex` | +| `fa_06:sealed-seeds` | `chapters/fa_06.tex` | +| `fa_07:abstract` | `chapters/fa_07.tex` | +| `fa_07:discussion` | `chapters/fa_07.tex` | +| `fa_07:from-the-trinity-identity-to-the-golden-angle` | `chapters/fa_07.tex` | +| `fa_07:h4-root-system-e8-lattice-and-the-varphi-scaled-block-decomposition` | `chapters/fa_07.tex` | +| `fa_07:introduction` | `chapters/fa_07.tex` | +| `fa_07:qed-assertions` | `chapters/fa_07.tex` | +| `fa_07:references` | `chapters/fa_07.tex` | +| `fa_07:results-evidence` | `chapters/fa_07.tex` | +| `fa_07:sealed-seeds` | `chapters/fa_07.tex` | +| `fa_08:abstract` | `chapters/fa_08.tex` | +| `fa_08:discussion` | `chapters/fa_08.tex` | +| `fa_08:gain-admissibility` | `chapters/fa_08.tex` | +| `fa_08:hybrid-qk-gain-invariant-inv-6` | `chapters/fa_08.tex` | +| `fa_08:introduction` | `chapters/fa_08.tex` | +| `fa_08:proof-sketch-for-admit_phi_sq` | `chapters/fa_08.tex` | +| `fa_08:qed-assertions` | `chapters/fa_08.tex` | +| `fa_08:references` | `chapters/fa_08.tex` | +| `fa_08:results-evidence` | `chapters/fa_08.tex` | +| `fa_08:sealed-seeds` | `chapters/fa_08.tex` | +| `fa_08:sec:falsification:ch08` | `chapters/fa_08.tex` | +| `fa_08:tf3-and-tf9-algebraic-structure` | `chapters/fa_08.tex` | +| `fa_08:tf9-product-encoding` | `chapters/fa_08.tex` | +| `fa_08:trit-encoding` | `chapters/fa_08.tex` | +| `fa_08:ux3c6-normalisation` | `chapters/fa_08.tex` | +| `fa_09:ablation-matrix-tier-abc-m1m6` | `chapters/fa_09.tex` | +| `fa_09:abstract` | `chapters/fa_09.tex` | +| `fa_09:competitor-format-summaries` | `chapters/fa_09.tex` | +| `fa_09:discussion` | `chapters/fa_09.tex` | +| `fa_09:gf16-format-specification` | `chapters/fa_09.tex` | +| `fa_09:gf16-phi_bias60-and-the-inv-3-safe-domain` | `chapters/fa_09.tex` | +| `fa_09:introduction` | `chapters/fa_09.tex` | +| `fa_09:inv-3-nine-coq-precision-bounds` | `chapters/fa_09.tex` | +| `fa_09:qed-assertions` | `chapters/fa_09.tex` | +| `fa_09:references` | `chapters/fa_09.tex` | +| `fa_09:results-evidence` | `chapters/fa_09.tex` | +| `fa_09:sealed-seeds` | `chapters/fa_09.tex` | +| `fa_10:abstract` | `chapters/fa_10.tex` | +| `fa_10:discussion` | `chapters/fa_10.tex` | +| `fa_10:gf16-range-and-precision-formalisation` | `chapters/fa_10.tex` | +| `fa_10:introduction` | `chapters/fa_10.tex` | +| `fa_10:qed-assertions` | `chapters/fa_10.tex` | +| `fa_10:references` | `chapters/fa_10.tex` | +| `fa_10:results-evidence` | `chapters/fa_10.tex` | +| `fa_10:sealed-seeds` | `chapters/fa_10.tex` | +| `fa_10:the-pareto-frontier-and-conjecture-c1` | `chapters/fa_10.tex` | +| `fa_11:abstract` | `chapters/fa_11.tex` | +| `fa_11:discussion` | `chapters/fa_11.tex` | +| `fa_11:hypothesis-formalisation-and-registration-protocol` | `chapters/fa_11.tex` | +| `fa_11:introduction` | `chapters/fa_11.tex` | +| `fa_11:inv-7-invariant-and-coq-formalisation` | `chapters/fa_11.tex` | +| `fa_11:qed-assertions` | `chapters/fa_11.tex` | +| `fa_11:references` | `chapters/fa_11.tex` | +| `fa_11:results-evidence` | `chapters/fa_11.tex` | +| `fa_11:sealed-seeds` | `chapters/fa_11.tex` | +| `fa_12:abstract` | `chapters/fa_12.tex` | +| `fa_12:bridge-architecture-and-interface-contracts` | `chapters/fa_12.tex` | +| `fa_12:clock-domain-analysis-and-timing` | `chapters/fa_12.tex` | +| `fa_12:discussion` | `chapters/fa_12.tex` | +| `fa_12:error-handling-protocol` | `chapters/fa_12.tex` | +| `fa_12:frequency-ratios-and-the-golden-ratio` | `chapters/fa_12.tex` | +| `fa_12:introduction` | `chapters/fa_12.tex` | +| `fa_12:logical-structure` | `chapters/fa_12.tex` | +| `fa_12:power-accounting` | `chapters/fa_12.tex` | +| `fa_12:qed-assertions` | `chapters/fa_12.tex` | +| `fa_12:references` | `chapters/fa_12.tex` | +| `fa_12:results-evidence` | `chapters/fa_12.tex` | +| `fa_12:sealed-seeds` | `chapters/fa_12.tex` | +| `fa_12:signal-naming-convention` | `chapters/fa_12.tex` | +| `fa_12:throughput-budget` | `chapters/fa_12.tex` | +| `fa_13:ch:13-metatron` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appD` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appE` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appF` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appG` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appI` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appJ` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appK` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appL` | `chapters/fa_13.tex` | +| `fa_13:sec:13-appM` | `chapters/fa_13.tex` | +| `fa_13:sec:13-arch` | `chapters/fa_13.tex` | +| `fa_13:sec:13-arch-summary` | `chapters/fa_13.tex` | +| `fa_13:sec:13-cartesian-rim` | `chapters/fa_13.tex` | +| `fa_13:sec:13-conn-23` | `chapters/fa_13.tex` | +| `fa_13:sec:13-connection-to-17` | `chapters/fa_13.tex` | +| `fa_13:sec:13-coords-bookkeeping` | `chapters/fa_13.tex` | +| `fa_13:sec:13-coq-map` | `chapters/fa_13.tex` | +| `fa_13:sec:13-counting-arch` | `chapters/fa_13.tex` | +| `fa_13:sec:13-cube-vs-spiral` | `chapters/fa_13.tex` | +| `fa_13:sec:13-diagram` | `chapters/fa_13.tex` | +| `fa_13:sec:13-disc-est` | `chapters/fa_13.tex` | +| `fa_13:sec:13-disc-not` | `chapters/fa_13.tex` | +| `fa_13:sec:13-disc-open` | `chapters/fa_13.tex` | +| `fa_13:sec:13-disc-summary` | `chapters/fa_13.tex` | +| `fa_13:sec:13-discussion` | `chapters/fa_13.tex` | +| `fa_13:sec:13-edge-counts` | `chapters/fa_13.tex` | +| `fa_13:sec:13-emp-26` | `chapters/fa_13.tex` | +| `fa_13:sec:13-filt-def` | `chapters/fa_13.tex` | +| `fa_13:sec:13-filt-quotients` | `chapters/fa_13.tex` | +| `fa_13:sec:13-filt-why` | `chapters/fa_13.tex` | +| `fa_13:sec:13-five-platonic` | `chapters/fa_13.tex` | +| `fa_13:sec:13-gf16` | `chapters/fa_13.tex` | +| `fa_13:sec:13-identities` | `chapters/fa_13.tex` | +| `fa_13:sec:13-layer-distances` | `chapters/fa_13.tex` | +| `fa_13:sec:13-lucas-12-orbit` | `chapters/fa_13.tex` | +| `fa_13:sec:13-lucas-ring-coords` | `chapters/fa_13.tex` | +| `fa_13:sec:13-notation` | `chapters/fa_13.tex` | +| `fa_13:sec:13-origin` | `chapters/fa_13.tex` | +| `fa_13:sec:13-polar` | `chapters/fa_13.tex` | +| `fa_13:sec:13-projection` | `chapters/fa_13.tex` | +| `fa_13:sec:13-strand-I` | `chapters/fa_13.tex` | +| `fa_13:sec:13-strand-I-takeaway` | `chapters/fa_13.tex` | +| `fa_13:sec:13-strand-II` | `chapters/fa_13.tex` | +| `fa_13:sec:13-strand-II-wrap` | `chapters/fa_13.tex` | +| `fa_13:sec:13-strand-III` | `chapters/fa_13.tex` | +| `fa_13:sec:13-strand-III-wrap` | `chapters/fa_13.tex` | +| `fa_13:sec:13-thirteen` | `chapters/fa_13.tex` | +| `fa_13:sec:13-three-cubes` | `chapters/fa_13.tex` | +| `fa_13:sec:13-three-strands` | `chapters/fa_13.tex` | +| `fa_13:sec:13-trinity-plane` | `chapters/fa_13.tex` | +| `fa_14:abstract` | `chapters/fa_14.tex` | +| `fa_14:bpb-definition-and-algebraic-properties` | `chapters/fa_14.tex` | +| `fa_14:byte-level-normalisation` | `chapters/fa_14.tex` | +| `fa_14:cross-entropy-and-perplexity` | `chapters/fa_14.tex` | +| `fa_14:discussion` | `chapters/fa_14.tex` | +| `fa_14:gate-2-bpb-1.85` | `chapters/fa_14.tex` | +| `fa_14:gate-3-bpb-1.50` | `chapters/fa_14.tex` | +| `fa_14:gate-thresholds-and-their-derivation` | `chapters/fa_14.tex` | +| `fa_14:introduction` | `chapters/fa_14.tex` | +| `fa_14:qed-assertions` | `chapters/fa_14.tex` | +| `fa_14:references` | `chapters/fa_14.tex` | +| `fa_14:relationship-to-the-darpa-energy-goal` | `chapters/fa_14.tex` | +| `fa_14:results-evidence` | `chapters/fa_14.tex` | +| `fa_14:sealed-seeds` | `chapters/fa_14.tex` | +| `fa_14:ux3c6-weighted-bpb` | `chapters/fa_14.tex` | +| `fa_15:abstract` | `chapters/fa_15.tex` | +| `fa_15:bpb-protocol-and-monotone-backward-invariant-inv-1` | `chapters/fa_15.tex` | +| `fa_15:database-schema` | `chapters/fa_15.tex` | +| `fa_15:discussion` | `chapters/fa_15.tex` | +| `fa_15:evaluation-protocol` | `chapters/fa_15.tex` | +| `fa_15:gate-evaluation` | `chapters/fa_15.tex` | +| `fa_15:introduction` | `chapters/fa_15.tex` | +| `fa_15:inv-1-bpb-monotone-backward` | `chapters/fa_15.tex` | +| `fa_15:qed-assertions` | `chapters/fa_15.tex` | +| `fa_15:railway-write-back-architecture` | `chapters/fa_15.tex` | +| `fa_15:references` | `chapters/fa_15.tex` | +| `fa_15:results-evidence` | `chapters/fa_15.tex` | +| `fa_15:sealed-seeds` | `chapters/fa_15.tex` | +| `fa_15:warmup-gate` | `chapters/fa_15.tex` | +| `fa_15:write-back-protocol` | `chapters/fa_15.tex` | +| `fa_16:abstract` | `chapters/fa_16.tex` | +| `fa_16:discussion` | `chapters/fa_16.tex` | +| `fa_16:grid-construction-and-sparsity-analysis` | `chapters/fa_16.tex` | +| `fa_16:introduction` | `chapters/fa_16.tex` | +| `fa_16:qed-assertions` | `chapters/fa_16.tex` | +| `fa_16:references` | `chapters/fa_16.tex` | +| `fa_16:results-evidence` | `chapters/fa_16.tex` | +| `fa_16:sealed-seeds` | `chapters/fa_16.tex` | +| `fa_16:the-phi-distance-function` | `chapters/fa_16.tex` | +| `fa_17:abstract` | `chapters/fa_17.tex` | +| `fa_17:analysis-of-effects-and-golden-ratio-structure` | `chapters/fa_17.tex` | +| `fa_17:discussion` | `chapters/fa_17.tex` | +| `fa_17:factor-definitions-and-experimental-design` | `chapters/fa_17.tex` | +| `fa_17:introduction` | `chapters/fa_17.tex` | +| `fa_17:qed-assertions` | `chapters/fa_17.tex` | +| `fa_17:references` | `chapters/fa_17.tex` | +| `fa_17:results-evidence` | `chapters/fa_17.tex` | +| `fa_17:sealed-seeds` | `chapters/fa_17.tex` | +| `fa_18:abstract` | `chapters/fa_18.tex` | +| `fa_18:coq.interval-upgrade-lane` | `chapters/fa_18.tex` | +| `fa_18:discussion` | `chapters/fa_18.tex` | +| `fa_18:hardware-and-runtime-limitations` | `chapters/fa_18.tex` | +| `fa_18:introduction` | `chapters/fa_18.tex` | +| `fa_18:qed-assertions` | `chapters/fa_18.tex` | +| `fa_18:references` | `chapters/fa_18.tex` | +| `fa_18:sealed-seeds` | `chapters/fa_18.tex` | +| `fa_18:sec:falsification:ch18` | `chapters/fa_18.tex` | +| `fa_18:state-of-the-art-comparison-clara-soa-snapshot` | `chapters/fa_18.tex` | +| `fa_19:abstract` | `chapters/fa_19.tex` | +| `fa_19:discussion` | `chapters/fa_19.tex` | +| `fa_19:introduction` | `chapters/fa_19.tex` | +| `fa_19:qed-assertions` | `chapters/fa_19.tex` | +| `fa_19:references` | `chapters/fa_19.tex` | +| `fa_19:results-evidence` | `chapters/fa_19.tex` | +| `fa_19:sealed-seeds` | `chapters/fa_19.tex` | +| `fa_19:test-design-and-hypotheses` | `chapters/fa_19.tex` | +| `fa_19:welch-t-statistic-and-degrees-of-freedom` | `chapters/fa_19.tex` | +| `fa_20:ch:20` | `chapters/fa_20.tex` | +| `fa_20:def:alpha` | `chapters/fa_20.tex` | +| `fa_20:def:ckm` | `chapters/fa_20.tex` | +| `fa_20:def:gauge-boson` | `chapters/fa_20.tex` | +| `fa_20:def:higgs` | `chapters/fa_20.tex` | +| `fa_20:def:koide` | `chapters/fa_20.tex` | +| `fa_20:def:lepton` | `chapters/fa_20.tex` | +| `fa_20:def:pmns` | `chapters/fa_20.tex` | +| `fa_20:def:quark` | `chapters/fa_20.tex` | +| `fa_20:def:su2` | `chapters/fa_20.tex` | +| `fa_20:def:su3` | `chapters/fa_20.tex` | +| `fa_20:def:u1` | `chapters/fa_20.tex` | +| `fa_20:prop:ckm-golden` | `chapters/fa_20.tex` | +| `fa_20:prop:golden-alpha` | `chapters/fa_20.tex` | +| `fa_20:prop:golden-koide` | `chapters/fa_20.tex` | +| `fa_20:prop:higgs-mass` | `chapters/fa_20.tex` | +| `fa_20:prop:pmns-golden` | `chapters/fa_20.tex` | +| `fa_20:prop:su3-dim` | `chapters/fa_20.tex` | +| `fa_20:prop:u1-charge` | `chapters/fa_20.tex` | +| `fa_20:sec:20-falsify` | `chapters/fa_20.tex` | +| `fa_20:thm:pauli` | `chapters/fa_20.tex` | +| `fa_20:thm:sm-symmetry` | `chapters/fa_20.tex` | +| `fa_20:thm:strong-golden` | `chapters/fa_20.tex` | +| `fa_20:thm:weak-golden` | `chapters/fa_20.tex` | +| `fa_21:ch:21` | `chapters/fa_21.tex` | +| `fa_21:def:dim-reg` | `chapters/fa_21.tex` | +| `fa_21:def:eft` | `chapters/fa_21.tex` | +| `fa_21:def:field-ops` | `chapters/fa_21.tex` | +| `fa_21:def:fock` | `chapters/fa_21.tex` | +| `fa_21:def:higgs-pot` | `chapters/fa_21.tex` | +| `fa_21:def:kg` | `chapters/fa_21.tex` | +| `fa_21:def:lagrangian` | `chapters/fa_21.tex` | +| `fa_21:def:path-integral` | `chapters/fa_21.tex` | +| `fa_21:def:qed` | `chapters/fa_21.tex` | +| `fa_21:def:yang-mills` | `chapters/fa_21.tex` | +| `fa_21:prop:beta-golden` | `chapters/fa_21.tex` | +| `fa_21:prop:feynman` | `chapters/fa_21.tex` | +| `fa_21:prop:goldstone` | `chapters/fa_21.tex` | +| `fa_21:prop:kg-eq` | `chapters/fa_21.tex` | +| `fa_21:prop:non-abelian` | `chapters/fa_21.tex` | +| `fa_21:sec:21-falsify` | `chapters/fa_21.tex` | +| `fa_21:thm:mode-expansion` | `chapters/fa_21.tex` | +| `fa_21:thm:n-point` | `chapters/fa_21.tex` | +| `fa_21:thm:rg` | `chapters/fa_21.tex` | +| `fa_21:thm:ssb` | `chapters/fa_21.tex` | +| `fa_21:thm:weinberg` | `chapters/fa_21.tex` | +| `fa_22:abstract` | `chapters/fa_22.tex` | +| `fa_22:discussion` | `chapters/fa_22.tex` | +| `fa_22:introduction` | `chapters/fa_22.tex` | +| `fa_22:qed-assertions` | `chapters/fa_22.tex` | +| `fa_22:references` | `chapters/fa_22.tex` | +| `fa_22:results-evidence` | `chapters/fa_22.tex` | +| `fa_22:satisfaction-witness-and-victory-predicate` | `chapters/fa_22.tex` | +| `fa_22:sealed-seeds` | `chapters/fa_22.tex` | +| `fa_22:worker-pool-invariants-and-falsification-witnesses` | `chapters/fa_22.tex` | +| `fa_23:abstract` | `chapters/fa_23.tex` | +| `fa_23:discussion` | `chapters/fa_23.tex` | +| `fa_23:introduction` | `chapters/fa_23.tex` | +| `fa_23:mcp-adapter-layer-architecture` | `chapters/fa_23.tex` | +| `fa_23:protocol-implementation-and-latency-analysis` | `chapters/fa_23.tex` | +| `fa_23:qed-assertions` | `chapters/fa_23.tex` | +| `fa_23:references` | `chapters/fa_23.tex` | +| `fa_23:results-evidence` | `chapters/fa_23.tex` | +| `fa_23:sealed-seeds` | `chapters/fa_23.tex` | +| `fa_24:abstract` | `chapters/fa_24.tex` | +| `fa_24:agent-model` | `chapters/fa_24.tex` | +| `fa_24:coq-encoding` | `chapters/fa_24.tex` | +| `fa_24:discussion` | `chapters/fa_24.tex` | +| `fa_24:formal-model-of-the-period-locked-monitor` | `chapters/fa_24.tex` | +| `fa_24:implementation-and-hardware-interface` | `chapters/fa_24.tex` | +| `fa_24:interrupt-interface-with-the-hardware-bridge` | `chapters/fa_24.tex` | +| `fa_24:introduction` | `chapters/fa_24.tex` | +| `fa_24:period-ratio-and-non-resonance` | `chapters/fa_24.tex` | +| `fa_24:priority-queue-and-phi-weighted-scheduling` | `chapters/fa_24.tex` | +| `fa_24:qed-assertions` | `chapters/fa_24.tex` | +| `fa_24:references` | `chapters/fa_24.tex` | +| `fa_24:results-evidence` | `chapters/fa_24.tex` | +| `fa_24:rtl-implementation` | `chapters/fa_24.tex` | +| `fa_24:sealed-seeds` | `chapters/fa_24.tex` | +| `fa_24:sec:falsification:ch24` | `chapters/fa_24.tex` | +| `fa_25:abstract` | `chapters/fa_25.tex` | +| `fa_25:cycle-classification-and-attention-periodicity` | `chapters/fa_25.tex` | +| `fa_25:discussion` | `chapters/fa_25.tex` | +| `fa_25:introduction` | `chapters/fa_25.tex` | +| `fa_25:qed-assertions` | `chapters/fa_25.tex` | +| `fa_25:references` | `chapters/fa_25.tex` | +| `fa_25:results-evidence` | `chapters/fa_25.tex` | +| `fa_25:sealed-seeds` | `chapters/fa_25.tex` | +| `fa_25:sec:falsification:ch25` | `chapters/fa_25.tex` | +| `fa_25:varphi-lattice-structure-and-the-cycle-map` | `chapters/fa_25.tex` | +| `fa_26:abstract` | `chapters/fa_26.tex` | +| `fa_26:discussion` | `chapters/fa_26.tex` | +| `fa_26:gf16_quant-galois-field-16-quantisation` | `chapters/fa_26.tex` | +| `fa_26:instruction-encoding` | `chapters/fa_26.tex` | +| `fa_26:introduction` | `chapters/fa_26.tex` | +| `fa_26:isa-register-file-and-encoding` | `chapters/fa_26.tex` | +| `fa_26:opcode-specifications` | `chapters/fa_26.tex` | +| `fa_26:phi_rope-ux3c6-rotary-position-encoding` | `chapters/fa_26.tex` | +| `fa_26:qed-assertions` | `chapters/fa_26.tex` | +| `fa_26:references` | `chapters/fa_26.tex` | +| `fa_26:register-file` | `chapters/fa_26.tex` | +| `fa_26:results-evidence` | `chapters/fa_26.tex` | +| `fa_26:sealed-seeds` | `chapters/fa_26.tex` | +| `fa_26:tf3_add-ternary-addition` | `chapters/fa_26.tex` | +| `fa_26:tf3_mul-ternary-multiplication` | `chapters/fa_26.tex` | +| `fa_26:vsa_bind-hyperdimensional-binding` | `chapters/fa_26.tex` | +| `fa_26:vsa_bundle-hyperdimensional-bundling` | `chapters/fa_26.tex` | +| `fa_26:vsa_unbind-hyperdimensional-unbinding` | `chapters/fa_26.tex` | +| `fa_27:abstract` | `chapters/fa_27.tex` | +| `fa_27:abstract-syntax` | `chapters/fa_27.tex` | +| `fa_27:discussion` | `chapters/fa_27.tex` | +| `fa_27:environments-and-evaluation` | `chapters/fa_27.tex` | +| `fa_27:introduction` | `chapters/fa_27.tex` | +| `fa_27:mechanised-proofs-determinism-and-exhaustiveness` | `chapters/fa_27.tex` | +| `fa_27:qed-assertions` | `chapters/fa_27.tex` | +| `fa_27:references` | `chapters/fa_27.tex` | +| `fa_27:relation-to-gf16-and-varphi-arithmetic` | `chapters/fa_27.tex` | +| `fa_27:results-evidence` | `chapters/fa_27.tex` | +| `fa_27:sealed-seeds` | `chapters/fa_27.tex` | +| `fa_27:ternary-arithmetic` | `chapters/fa_27.tex` | +| `fa_27:theorem-eval_det-determinism` | `chapters/fa_27.tex` | +| `fa_27:theorem-trit_exhaustive-exhaustiveness` | `chapters/fa_27.tex` | +| `fa_27:tri27-syntax-and-denotational-semantics` | `chapters/fa_27.tex` | +| `fa_28:abstract` | `chapters/fa_28.tex` | +| `fa_28:architecture-zero-dsp-ternary-datapath` | `chapters/fa_28.tex` | +| `fa_28:discussion` | `chapters/fa_28.tex` | +| `fa_28:introduction` | `chapters/fa_28.tex` | +| `fa_28:qed-assertions` | `chapters/fa_28.tex` | +| `fa_28:references` | `chapters/fa_28.tex` | +| `fa_28:resource-utilisation-and-timing-closure` | `chapters/fa_28.tex` | +| `fa_28:results-evidence` | `chapters/fa_28.tex` | +| `fa_28:sealed-seeds` | `chapters/fa_28.tex` | +| `fa_29:ch:29` | `chapters/fa_29.tex` | +| `fa_29:def:lucas` | `chapters/fa_29.tex` | +| `fa_29:def:lucas-primes` | `chapters/fa_29.tex` | +| `fa_29:def:lucas-spiral` | `chapters/fa_29.tex` | +| `fa_29:def:lucas-tiling` | `chapters/fa_29.tex` | +| `fa_29:def:lucas-trinity` | `chapters/fa_29.tex` | +| `fa_29:prop:golden-lucas-mixing` | `chapters/fa_29.tex` | +| `fa_29:prop:lucas-golden` | `chapters/fa_29.tex` | +| `fa_29:prop:lucas-mod` | `chapters/fa_29.tex` | +| `fa_29:prop:lucas-tiling` | `chapters/fa_29.tex` | +| `fa_29:sec:29-falsify` | `chapters/fa_29.tex` | +| `fa_29:thm:cassini` | `chapters/fa_29.tex` | +| `fa_29:thm:lucas-div` | `chapters/fa_29.tex` | +| `fa_29:thm:lucas-fibo` | `chapters/fa_29.tex` | +| `fa_29:thm:lucas-prime-density` | `chapters/fa_29.tex` | +| `fa_29:thm:lucas-spiral` | `chapters/fa_29.tex` | +| `fa_29:thm:neutrino-lucas` | `chapters/fa_29.tex` | +| `fa_29:thm:product` | `chapters/fa_29.tex` | +| `fa_30:abstract` | `chapters/fa_30.tex` | +| `fa_30:associative-recall-memory` | `chapters/fa_30.tex` | +| `fa_30:discussion` | `chapters/fa_30.tex` | +| `fa_30:goldenfloat-encoding-of-hypervectors` | `chapters/fa_30.tex` | +| `fa_30:hypervector-definition` | `chapters/fa_30.tex` | +| `fa_30:introduction` | `chapters/fa_30.tex` | +| `fa_30:phi-rotary-position-encoding-phi-rope-in-vsa-context` | `chapters/fa_30.tex` | +| `fa_30:qed-assertions` | `chapters/fa_30.tex` | +| `fa_30:references` | `chapters/fa_30.tex` | +| `fa_30:results-evidence` | `chapters/fa_30.tex` | +| `fa_30:sealed-seeds` | `chapters/fa_30.tex` | +| `fa_30:ternary-vsa-over-the-goldenfloat-substrate` | `chapters/fa_30.tex` | +| `fa_31:ch:31` | `chapters/fa_31.tex` | +| `fa_31:def:antirealism` | `chapters/fa_31.tex` | +| `fa_31:def:apriori` | `chapters/fa_31.tex` | +| `fa_31:def:beauty` | `chapters/fa_31.tex` | +| `fa_31:def:constants` | `chapters/fa_31.tex` | +| `fa_31:def:empiricism` | `chapters/fa_31.tex` | +| `fa_31:def:muh` | `chapters/fa_31.tex` | +| `fa_31:def:platonism` | `chapters/fa_31.tex` | +| `fa_31:def:pythagorean` | `chapters/fa_31.tex` | +| `fa_31:def:realism` | `chapters/fa_31.tex` | +| `fa_31:def:structuralism` | `chapters/fa_31.tex` | +| `fa_31:prop:empirical-golden` | `chapters/fa_31.tex` | +| `fa_31:prop:golden-anthropic` | `chapters/fa_31.tex` | +| `fa_31:prop:golden-cat` | `chapters/fa_31.tex` | +| `fa_31:prop:golden-effective` | `chapters/fa_31.tex` | +| `fa_31:prop:golden-muh` | `chapters/fa_31.tex` | +| `fa_31:prop:golden-platonism` | `chapters/fa_31.tex` | +| `fa_31:prop:instrumental-golden` | `chapters/fa_31.tex` | +| `fa_31:thm:apriori-golden` | `chapters/fa_31.tex` | +| `fa_31:thm:golden-beauty` | `chapters/fa_31.tex` | +| `fa_31:thm:golden-constants` | `chapters/fa_31.tex` | +| `fa_31:thm:golden-struct` | `chapters/fa_31.tex` | +| `fa_31:thm:platonic-golden` | `chapters/fa_31.tex` | +| `fa_31:thm:pythagorean-golden` | `chapters/fa_31.tex` | +| `fa_31:thm:realist-golden` | `chapters/fa_31.tex` | +| `fa_32:prop:golden-opt` | `chapters/fa_32.tex` | +| `fa_32:thm:alpha-summary` | `chapters/fa_32.tex` | +| `fa_32:thm:e8-summary` | `chapters/fa_32.tex` | +| `fa_32:thm:golden-entropy` | `chapters/fa_32.tex` | +| `fa_32:thm:golden-unif` | `chapters/fa_32.tex` | +| `fa_32:thm:trinity-summary` | `chapters/fa_32.tex` | +| `fa_33:abstract` | `chapters/fa_33.tex` | +| `fa_33:diagnosis-and-root-cause` | `chapters/fa_33.tex` | +| `fa_33:discussion` | `chapters/fa_33.tex` | +| `fa_33:flash_no_sudo.sh` | `chapters/fa_33.tex` | +| `fa_33:fxload-cross-compilation` | `chapters/fa_33.tex` | +| `fa_33:introduction` | `chapters/fa_33.tex` | +| `fa_33:qed-assertions` | `chapters/fa_33.tex` | +| `fa_33:references` | `chapters/fa_33.tex` | +| `fa_33:results-evidence` | `chapters/fa_33.tex` | +| `fa_33:sealed-seeds` | `chapters/fa_33.tex` | +| `fa_33:usb-enumeration-on-macos-arm` | `chapters/fa_33.tex` | +| `fa_33:verified-hardware-configuration-post-blk-001` | `chapters/fa_33.tex` | +| `fig:-` | `frontmatter/list-of-figures.tex` | +| `lem:01-best-rational` | `chapters/fa_01.tex` | +| `lem:01-cf-rec` | `chapters/fa_01.tex` | +| `lem:05-coef-limit` | `chapters/fa_05.tex` | +| `lem:05-luc-hankel` | `chapters/fa_05.tex` | +| `lem:05-matrix-power` | `chapters/fa_05.tex` | +| `lem:13-galois` | `chapters/fa_13.tex` | +| `lem:13-gf16-floor` | `chapters/fa_13.tex` | +| `lem:13-primary` | `chapters/fa_13.tex` | +| `lem:13-secondary` | `chapters/fa_13.tex` | +| `lem:13-tertiary` | `chapters/fa_13.tex` | +| `lem:13-trinity` | `chapters/fa_13.tex` | +| `sec:05-anchor-coeff` | `chapters/fa_05.tex` | +| `sec:05-app-F` | `chapters/fa_05.tex` | +| `sec:05-app-K` | `chapters/fa_05.tex` | +| `sec:05-app-X` | `chapters/fa_05.tex` | +| `sec:05-app-Y` | `chapters/fa_05.tex` | +| `sec:05-closed-form` | `chapters/fa_05.tex` | +| `sec:05-coupling` | `chapters/fa_05.tex` | +| `sec:05-falsification` | `chapters/fa_05.tex` | +| `sec:05-partial-frac` | `chapters/fa_05.tex` | +| `sec:05-prelim` | `chapters/fa_05.tex` | +| `sec:05-radius` | `chapters/fa_05.tex` | +| `sec:05-strand-i` | `chapters/fa_05.tex` | +| `sec:05-strand-ii` | `chapters/fa_05.tex` | +| `sec:05-strand-iii` | `chapters/fa_05.tex` | +| `sec:13-appA` | `chapters/fa_13.tex` | +| `sec:13-appB` | `chapters/fa_13.tex` | +| `sec:13-appC` | `chapters/fa_13.tex` | +| `sec:13-appH` | `chapters/fa_13.tex` | +| `sec:13-arch-scaffold` | `chapters/fa_13.tex` | +| `sec:13-filt-coq` | `chapters/fa_13.tex` | +| `sec:13-filtration` | `chapters/fa_13.tex` | +| `sec:13-seventy-eight` | `chapters/fa_13.tex` | +| `sec:13-symmetry-group` | `chapters/fa_13.tex` | +| `sec:13-trinity-bookkeeping` | `chapters/fa_13.tex` | +| `sec:ckm` | `chapters/fa_20.tex` | +| `sec:mass` | `chapters/fa_20.tex` | +| `sec:mesh-roadmap` | `chapters/ch_35_mesh_node.tex` | +| `sec:xvc-bridge` | `appendix/F-fpga-bitstream.tex` | +| `tab:-` | `frontmatter/list-of-tables.tex` | +| `tab:ch0-fits` | `chapters/ch_00.tex` | +| `tab:power` | `chapters/ch_35_mesh_node.tex` | +| `thm:01-anchor` | `chapters/fa_01.tex` | +| `thm:01-convergent-fib` | `chapters/fa_01.tex` | +| `thm:01-fixed` | `chapters/fa_01.tex` | +| `thm:01-pentagon` | `chapters/fa_01.tex` | +| `thm:01-pentagon-alg` | `chapters/fa_01.tex` | +| `thm:01-quadratic` | `chapters/fa_01.tex` | +| `thm:01-vesica-lens` | `chapters/fa_01.tex` | +| `thm:05-anchor-as-coeff` | `chapters/fa_05.tex` | +| `thm:05-asymptotic` | `chapters/fa_05.tex` | +| `thm:05-bridge` | `chapters/fa_05.tex` | +| `thm:05-cassini-fib` | `chapters/fa_05.tex` | +| `thm:05-cassini-luc` | `chapters/fa_05.tex` | +| `thm:05-coupling` | `chapters/fa_05.tex` | +| `thm:05-fl-conv` | `chapters/fa_05.tex` | +| `thm:05-genfn-closed` | `chapters/fa_05.tex` | +| `thm:05-partial-frac` | `chapters/fa_05.tex` | +| `thm:05-radius` | `chapters/fa_05.tex` | +| `thm:13-projection` | `chapters/fa_13.tex` | +| `thm:13-total-edges` | `chapters/fa_13.tex` | +| `thm:D:1` | `appendix/D-golden-mirror.tex` | +| `thm:ch1-trinity-identity` | `chapters/ch_01.tex` | +| `thm:ch3-trinity-canonical` | `chapters/ch_03.tex` | +| `thm:euler-lagrange` | `chapters/fa_21.tex` | +| `thm:lucas-binet` | `chapters/fa_29.tex` | +| `thm:lucas-trinity` | `chapters/fa_29.tex` | + +
+ +## Referenced keys (119) + +These keys are consumed by at least one `\ref`/`\autoref`/`\eqref`/`\Cref`/`\pageref` and were preserved in their bare form: + +
Click to expand + +| Key | Defined in | Referenced from | +|---|---|---| +| `ch:1` | `chapters/fa_01.tex` | `chapters/ch_00.tex` | +| `ch:11` | `chapters/fa_11.tex` | `chapters/ch_00.tex` | +| `ch:13` | `chapters/fa_13.tex` | `appendix/B-falsification.tex`, `appendix/J-troubleshooting.tex` | +| `ch:15` | `chapters/fa_15.tex` | `appendix/B-falsification.tex`, `appendix/G-data-availability.tex` | +| `ch:17-spiral` | `chapters/fa_17.tex` | `chapters/fa_13.tex` | +| `ch:18` | `chapters/fa_18.tex` | `appendix/B-falsification.tex`, `appendix/G-data-availability.tex` | +| `ch:19` | `chapters/fa_19.tex` | `chapters/ch_00.tex` | +| `ch:21-experiments-jepa` | `chapters/fa_21.tex` | `chapters/fa_13.tex` | +| `ch:23-gf16-algebra` | `chapters/fa_23.tex` | `chapters/fa_13.tex` | +| `ch:24` | `chapters/fa_24.tex` | `appendix/B-falsification.tex` | +| `ch:24-igla-arch` | `chapters/fa_24.tex` | `chapters/fa_13.tex` | +| `ch:25` | `chapters/fa_25.tex` | `appendix/B-falsification.tex` | +| `ch:25-benchmarks` | `chapters/fa_25.tex` | `chapters/fa_13.tex` | +| `ch:26-data-analysis` | `chapters/fa_26.tex` | `chapters/fa_13.tex` | +| `ch:28` | `chapters/fa_28.tex` | `appendix/F-fpga-bitstream.tex` | +| `ch:28-momentum-algebra` | `chapters/fa_28.tex` | `chapters/fa_13.tex` | +| `ch:32` | `chapters/fa_32.tex` | `appendix/I-xdc-pin-map.tex` | +| `ch:33` | `chapters/fa_33.tex` | `appendix/F-fpga-bitstream.tex` | +| `ch:34` | `chapters/fa_33.tex` | `appendix/B-falsification.tex`, `appendix/F-fpga-bitstream.tex` | +| `ch:6` | `chapters/fa_06.tex` | `appendix/C-golden-benchmark.tex` | +| `ch:9` | `chapters/fa_09.tex` | `appendix/B-falsification.tex`, `appendix/C-golden-benchmark.tex` | +| `ch:benchmarks` | `chapters/fa_25.tex` | `chapters/fa_00.tex` | +| `ch:data-analysis` | `chapters/fa_26.tex` | `chapters/fa_00.tex` | +| `ch:e8-symmetry` | `chapters/fa_22.tex` | `chapters/fa_00.tex` | +| `ch:energy` | `chapters/fa_28.tex` | `frontmatter/notation.tex` | +| `ch:experiments-asha` | `chapters/fa_21.tex` | `frontmatter/notation.tex` | +| `ch:experiments-bpb` | `chapters/fa_21.tex` | `frontmatter/notation.tex` | +| `ch:experiments-gf16` | `chapters/fa_23.tex` | `frontmatter/notation.tex` | +| `ch:fibonacci` | `chapters/fa_07.tex` | `frontmatter/notation.tex` | +| `ch:fibonacci-tesselation` | `chapters/fa_07.tex` | `chapters/fa_00.tex` | +| `ch:gf16-algebra` | `chapters/fa_23.tex` | `chapters/fa_00.tex` | +| `ch:golden-egg` | `chapters/fa_01.tex` | `frontmatter/notation.tex` | +| `ch:golden-seed` | `chapters/fa_01.tex` | `frontmatter/notation.tex` | +| `ch:igla-architecture` | `chapters/fa_24.tex` | `chapters/fa_00.tex` | +| `ch:igla-race` | `chapters/fa_24.tex` | `frontmatter/notation.tex` | +| `ch:jepa` | `chapters/fa_21.tex` | `frontmatter/notation.tex` | +| `ch:lucas-closure` | `chapters/fa_29.tex` | `chapters/fa_00.tex` | +| `ch:lucas-ladder` | `chapters/fa_29.tex` | `chapters/fa_05.tex` | +| `ch:lucas-ring` | `chapters/fa_27.tex` | `chapters/fa_00.tex`, `chapters/fa_05.tex`, `frontmatter/notation.tex` | +| `ch:monad` | `chapters/fa_00.tex` | `chapters/fa_00.tex` | +| `ch:nca` | `chapters/fa_29.tex` | `frontmatter/notation.tex` | +| `ch:plm` | `chapters/fa_24.tex` | `appendix/F-fpga-bitstream.tex` | +| `ch:standard-model` | `chapters/fa_20.tex` | `chapters/fa_00.tex` | +| `ch:three-strands` | `chapters/fa_27.tex` | `chapters/fa_00.tex`, `frontmatter/notation.tex` | +| `ch:trinity-identity` | `chapters/fa_27.tex` | `chapters/fa_00.tex` | +| `ch:vesica-piscis` | `chapters/fa_11.tex` | `chapters/fa_00.tex` | +| `ch:vsa` | `chapters/fa_29.tex` | `frontmatter/notation.tex` | +| `cor:01-l2-three` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `cor:01-lucas-as-trace` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `cor:01-reciprocal` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `cor:05-asymptotic-rate` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `cor:05-binet` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `def:13-lucas-12` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `eq:ch0-fit` | `chapters/ch_00.tex` | `chapters/ch_00.tex` | +| `lem:01-best-rational` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `lem:01-cf-rec` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `lem:05-coef-limit` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `lem:05-luc-hankel` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `lem:05-matrix-power` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `lem:13-galois` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `lem:13-gf16-floor` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `lem:13-primary` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `lem:13-secondary` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `lem:13-tertiary` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `lem:13-trinity` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:05-anchor-coeff` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-app-F` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-app-K` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-app-X` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-app-Y` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-closed-form` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-coupling` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-falsification` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-partial-frac` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-prelim` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-radius` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-strand-i` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-strand-ii` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:05-strand-iii` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `sec:13-appA` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-appB` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-appC` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-appH` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-arch-scaffold` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-filt-coq` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-filtration` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-seventy-eight` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-symmetry-group` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:13-trinity-bookkeeping` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `sec:ckm` | `chapters/fa_20.tex` | `chapters/fa_20.tex` | +| `sec:mass` | `chapters/fa_20.tex` | `chapters/fa_20.tex` | +| `sec:mesh-roadmap` | `chapters/ch_35_mesh_node.tex` | `chapters/ch_35_mesh_node.tex` | +| `sec:xvc-bridge` | `appendix/F-fpga-bitstream.tex` | `appendix/F-fpga-bitstream.tex` | +| `tab:ch0-fits` | `chapters/ch_00.tex` | `chapters/ch_00.tex` | +| `tab:power` | `chapters/ch_35_mesh_node.tex` | `chapters/ch_35_mesh_node.tex` | +| `thm:01-anchor` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `thm:01-convergent-fib` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `thm:01-fixed` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `thm:01-pentagon` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `thm:01-pentagon-alg` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `thm:01-quadratic` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `thm:01-vesica-lens` | `chapters/fa_01.tex` | `chapters/fa_01.tex` | +| `thm:05-anchor-as-coeff` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-asymptotic` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-bridge` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-cassini-fib` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-cassini-luc` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-coupling` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-fl-conv` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-genfn-closed` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-partial-frac` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:05-radius` | `chapters/fa_05.tex` | `chapters/fa_05.tex` | +| `thm:13-projection` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `thm:13-total-edges` | `chapters/fa_13.tex` | `chapters/fa_13.tex` | +| `thm:ch1-trinity-identity` | `chapters/ch_01.tex` | `chapters/ch_01.tex` | +| `thm:ch3-trinity-canonical` | `chapters/ch_03.tex` | `chapters/ch_03.tex` | +| `thm:euler-lagrange` | `chapters/fa_21.tex` | `chapters/fa_21.tex` | +| `thm:lucas-binet` | `chapters/fa_29.tex` | `chapters/fa_05.tex` | +| `thm:lucas-trinity` | `chapters/fa_29.tex` | `chapters/fa_29.tex` | + +
+ +## Skill provenance + +Authored under skills `phd-chapter-author v1.1` + `phd-monograph-auditor v1.2`. +Per R5 (honesty): all renames are mechanical, none flip Admitted↔Proven; no `.py`/`.sh` were committed (R1).