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6 changes: 3 additions & 3 deletions docs/phd/chapters/ch_00.tex
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Expand Up @@ -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
Expand Down Expand Up @@ -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
Expand All @@ -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
Expand Down
36 changes: 18 additions & 18 deletions docs/phd/chapters/ch_01.tex
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Expand Up @@ -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

Expand All @@ -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:

Expand All @@ -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.

Expand All @@ -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/}.

Expand Down Expand Up @@ -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
Expand Down Expand Up @@ -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
Expand All @@ -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
Expand Down Expand Up @@ -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
Expand Down Expand Up @@ -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
Expand All @@ -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.
Expand All @@ -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
Expand All @@ -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
Expand Down Expand Up @@ -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
Expand Down
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