Sharp row-energy packed radix for the retained HPE equality compiler

[Arham-Tahir64]

DONE — PR1906, head 788848032c16bbe0a543c1186c6691c66131c090. Claim released.

Established: for arbitrary finite Gaussian-integer coefficient rows of squared energy at most R², integer B>=2R is injective; every smaller positive integer radix fails for the whole ball when R is a positive integer and dimension is at least two. A short least-nonzero-digit proof shows nonzero kernel energy >=B²+1; the parallelogram identity bounds the difference energy by4R². This sharp ball threshold is not an optimality claim for the specific unitary rows.

For the guarded normalized Clifford grammar, scaled rows have R=2^K, yielding K+1 radix bits, one less than credited PR1859's sufficient entry-bound result. The three pinned HPE literal pairs pass at10 bits instead of11/20; largest packed input5,111 bits instead of5,622/10,221. No timing claim.

Evidence: registered replay passes on Windows/CPython3.12.10. All3,072 scaled rows/energies,30 differential interpreter executions,5,265 complete Gaussian-ball vectors,11,493 short-kernel vectors,1,024 unsafe radix cases,15 source controls, and four CLI/source/receipt boundary test groups pass. Deterministic receipt generation uses explicit LF; source exports replay under optimized Python and reject source/receipt drift and duplicate keys. Metadata/source pins and git diff checks pass. Reproduce: python scripts/research_package.py --package research/hpe-energy-packed-radix. Windows sandbox TEMP/TMP were set to the checkout root; standard tempfile behavior is retained.

Final overlap:554 open PRs including drafts at2026-10-14T13:37:04Z;535 Discussion/comment/reply records at13:38:12Z, complete pagination plus local notes/ledger. PR1859 head stayed f5e18c0. Search coverage excludes every diff/nested source and is not a novelty certificate.

Handoff: formalize the packing-kernel energy argument, parallelogram implication and guarded-gate row norms in Lean before changing the retained checker; kernel timing remains unmeasured. No new Lean proof, selected checker/frontier change, cold replay, supplier, precision change or kappa claim.

Credit Jeremy Kothe/jeremykothe2 PR593, commdata2338's reused pinned literal parser, woklerx PR1859, Sagar Thakrar's distinct PR1781 census, and the retained Clifford/Lemma A/G lineage. Arham Tahir with substantial OpenAI Codex assistance; Apache-2.0. No subagents or independent human review.

Original claim and method plan (the final proof simplifies the proposed Cauchy-Schwarz route):

CLAIM — uniform-transfer-audit, refined; independently derived successor to PR1859.

Base current main: 0b1cccf; selected-record SHA256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32. Draw d17af0bfdda8dde69736e5942d5a7563, priority revision 3, adversarial audit. The positive research_baseline source-binding control passed.

Question: for arbitrary finite Gaussian-integer coefficient rows u,v with squared Euclidean energy at most R^2 (integer R>=1), is evaluation at B=2R injective, and is that integer threshold sharp? Apply it to validated unitary Clifford circuits scaled to R=2^K. This would reduce PR1859's K+2-bit sufficient radix to K+1 bits without changing the retained Lean checker.

Method/domain: a dimension-independent highest-digit/Cauchy-Schwarz argument; exhaustive small Gaussian-integer balls; adjacent unsafe B=2R-1 collisions; exact gate-valid source replay of only the three frozen HPE literal pairs, with phase, character, normalization, norm/row and radix failure controls. Expected artifact: research/hpe-energy-packed-radix, proof + standard-library verifier + source pins + deterministic receipt. No new kernel proof, supplier admission, speed benchmark or kappa claim.

Predecessors: woklerx PR1859 at f5e18c0 gives the K+2-bit unitary entry-bound argument; Jeremy Kothe PR593 supplies retained semantics/theorem/literals; commdata2338's parser is reused and pinned. Sagar Thakrar PR1781 is a distinct selected-recipe census with the original radix. No pending implementation is imported. The new ingredient is whole-row energy, rather than the per-component bound.

Entry overlap: 543 open PRs including drafts, 529 Discussion/comment/reply records, complete pagination plus local notes/ledger. Titles/bodies and package metadata only, not every diff or nested file; no novelty guarantee. A final refresh is required. Claim expires 2026-10-14T15:29:31.117047+00:00.

Prepared by Arham Tahir with substantial OpenAI Codex assistance; Apache-2.0. No subagents or independent human review.

[RossRichardson]

DONE / HANDOFF — independent audit complete; CLAIM RELEASED.

PR1934, exact head a055aaf34b012e04714066aafbfc7057ee1f3e9f, independently corroborates PR1906 at 788848032c16bbe0a543c1186c6691c66131c090. All six retained HPE literal circuits were reconstructed using a separate closed parser and direct integer realification, without predecessor code, adjoint-derived rows or packed equality. All 3,072 complex rows / 6,291,456 real matrix entries satisfy the normalized equalities; the three paired whole-matrix row digests and common energies match PR1906. No discrepancy or new radix bound was found.

A separate exact-circuit specialist independently checked the all-dimension Gaussian kernel-energy/sharpness argument, gate-validity-to-row-energy implication, normalization and new code. This is AI review, not human or Lean verification. Scope: a written argument review plus complete finite literal-matrix evidence; not a general Lean parser, compiler/resource proof or supplier adoption.

Reproduce: python3.12 -I -B scripts/research_package.py --package research/hpe-energy-independent-audit. Final committed local replay passed on CPython 3.12.12/macOS arm64: seven test groups; all 585 words of length 0..3 over eight two-qubit gates compared to dense Gaussian-rational matrices; phase/character/scale faults; equivalent presentations; seven scientific receipt corruptions with checksum guards bypassed; detector-removal sensitivity; missing/duplicate evidence, source drift, explicit regeneration and optimized read-only relocation. Final-head optimized Python 3.14.6 direct replay and selected-record integrity pass. The exact PR1906 direct verifier also passes separately. PR1934 was read back OPEN with the exact head/body; hosted checks are 61 QUEUED, not passed.

Remaining adoption step, already identified by PR1906: formalize the Gaussian energy-kernel lemma and valid-gate row norms in Lean before changing the retained checker; kernel timing and compiler resource bounds remain separate. No new kappa, Lean build, full supplier cold replay, selected-frontier change, merge or closure of other work.

Final overlap at 14:20–14:22 UTC covered 582 open PRs and 563 Discussion/comment/reply records, complete pagination and no warnings, plus local notes/ledger. Nearest matches were the intentionally audited PR1906, predecessor PR1859 and this thread; this is not a novelty proof. Main and PR1906 remain at their pinned heads. Credit Arham Tahir/PR1906, woklerx/PR1859, Jeremy Kothe/PR593, commdata2338, Sagar Thakrar/PR1781 and the retained Clifford/Lemma A/G lineage. Prepared for Ross with substantial OpenAI Codex assistance, Apache-2.0.

Original claim and precise scope:

CLAIM — independent verification of PR1906, formal-interface-lemma (scope refined).

Exact main base 0b1cccf; PR1906 head 7888480; selector nonce cfbe139628a1920458d65ac59391c82d, priority revision 3; selected-record SHA256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question: independently audit the written Gaussian energy-ball packing implication and its binding to all six retained HPE literal circuits. This deliberately corroborates PR1906, not a new radix proposal. Method: an independently written closed literal parser plus direct full Gaussian-integer matrix propagation, with no predecessor interpreter, packed comparison, or adjoint-derived rows; separately review the all-dimension kernel-energy/sharpness argument and valid-gate normalization. Domain: the three frozen pairs (512 coordinates), all matrix entries/row energies, meaningful phase/scale/direction/row-bound controls. No new kappa, supplier or Lean proof. Lean tooling is unavailable on this host.

Expected artifact: compact source-pinned independent audit and reproduction script, or a precise counterexample/handoff. Current targeted PR/discussion read found PR1859 and PR1781 as related predecessors; this thread's PR1906 claim is released. Full overlap helper refresh is running with an explicitly writable external cache after its default cache was denied. Any coverage limits will be recorded. One exact-circuit specialist independently checks the proof; parent implements the matrix oracle. Prepared for Ross with substantial OpenAI Codex assistance, Apache-2.0.

Claim expires 2026-10-14T16:20:00Z.