Independently audit PR #186 and refine its conditional exponent

[M6LI]

Change and mathematical scope

Independently audit Dugongue's PR #186 at 166a34d and refine its fixed conditional arithmetic. Under the retained interfaces,

T(n) = O(n (log n)^(1-kappa)), with kappa = 330942774629799608347/500000000000000000000000 = 0.000661885549259599216694.

The log exponent is 0.999338114450740400783306. The gain over that pinned #186 headline is 608347/500000000000000000000000 = 1.216694e-18. The complex supplier binds. This is a small precision refinement and independent finite audit. Later submitted constructions, including #191 and #194, report stronger conditional headlines; this PR makes no current-frontier or global-optimality claim.

The new arithmetic reconstructs both paid supplier moments with exact rational log/exp enclosures and the complete rare-class fallback, the strict stopped-atom threshold, the conservative scalar/group/router/row bridge, all 47 inequalities and seven margins. All scalar reserves remain paid. Supplier, atom and headline grids are 10^24; eta, beta and phase_gap are 10^-24. Mathematical controls reject both supplier successors, the preceding atom point, the next headline and omitted charges. Adjacent-grid exclusions concern these fixed witnesses and parameters only.

The bank audit independently checks complete 144-column address identities and restoration over Z and F2, disjoint block coverage, shared-chart examples, omitted/repeated-block controls and allocation accounting. Nine physical replicas give integral stock 177432. Separately, multiplying normalized stock 59144/3, every child count, rank mass, deficit and full fallback edge count by 3 preserves the moment equation. Only 2200 rank-60 entrance exteriors are removed; rank-21 recipients remain internal splices and all other children and full scalar/row bills stay paid.

The unchanged independent signed scalar core checks the current 46-sink word, all 13326 source/initial-target/dirty formal columns, arbitrary restoration of 10686 dirty registers, all 2310 aliases and 330 reflection gates. Five mathematical corruptions are rejected. Fresh geometry/frame chronology and independent arithmetic were reviewed separately from native replay. The native target zero-rank diagnostic is 6270; the literal-event recount is 6325, a difference of 55. Every positive paid histogram bin agrees. The diagnostic difference is retained explicitly; zero-rank events add no child/rank/moment toll, and full scalar work remains paid.

The branch preserves all 1700 native PR #181 files and the prior 26 independent audit files byte-for-byte. It imports all 51 original PR #186 bundle files, including immutable archive parts and notices, without changing their bytes. A narrowly scoped research/.gitattributes preserves two literal-CRLF witness files on checkout. Sixteen new package files and the extended owned workflow add portable verification; native Makefile targets and native CI are unchanged.

Validation

The final local package is commit 7fef2373b06a5622b249155194f397f52ac85d4f, tree 31d26ccd4bea8be61600bf8096e8b2636eb6e78b. Published commit 33f3f1e preserves the existing PR history as its first parent and the pinned #186 head as an additional parent. Its tree is byte-identical to the tested local tree; validation binds these exact file bytes.

• Canonical pinned Shared-edge supplier, coordinated frames and entrance banks: conditional kappa = 6.61885549259598e-4 #186 replay passed in 158.703 seconds, including all complete native word/bank/moment/assembly checks. All 51 Git package blobs and all 1700 native archive files independently matched exact source Git bytes before and after.
• Fresh independent exact arithmetic, geometry and bank checks passed in normal and optimized Python. A different AI agent's full normal scalar result matched the independently run optimized result, including the event digest and corruption controls. No human peer review or new formal theorem proof is implied.
• All 14 native Makefile groups passed in private isolated copies of the earlier f5ab0230 tree (75b7b0e4), with all 1727 tracked bytes unchanged and clean diffs. Dependency review confirms that the native inputs and group recipes are unchanged in this additive package; this is transferred component evidence, not a claim that all 14 were freshly rerun on the new tree.
• Final source-exact Linux workflow replay passed in 671.718 seconds on the new tree: the required selected Makefile gate, all eight literal workflow blocks, normal/optimized full scalar checks, source bytes before/after and clean tracked diff all passed. Separate final package review approved the exact source tree, with all 38 arithmetic/source fixtures passing independently in normal and optimized Python. Keep this draft pending hosted CI approval. GitHub CI requires maintainer approval for this first-time contributor; no green PR workflow or fresh local Lean build is claimed.

Earlier runtime-capped attempts are retained in the research receipts and are not counted as passes. The original byte-preserved CRLF files and one inherited EOF blank line trigger default whitespace diagnostics; authored files pass the whitespace check. No original source is silently rewritten to clear those diagnostics.

Reproduce with Python 3.12 or later from the repository root:

python -B research/coordinated-frames-and-entrance-banks/verify.py
python -B research/frontier-refinement186/verify.py
python -B research/frontier-refinement186/check_banks.py
python -B research/frontier-refinement186/test_refinement.py
python -B research/frontier-refinement186/test_source.py
python -B research/frontier-refinement186/scalar.py

Repeat the five independent commands with python -O -B. The original native aggregate requires assertions enabled. READMEs describe the preserved #181 checks and exact source closures.

General tensor/Clifford realization, uniform weighted compilation, existence of common charts, paid multibank routing/restoration and stage/orbit coverage, prime supply, precision/recovery, analytic reduction, setup and fixed-tape all-size interfaces remain explicit hypotheses. Complete finite address/scalar checks do not prove those interfaces or an unconditional integer-multiplication theorem.

Attribution

Prepared by Muhammed Ali Mehmood with substantial OpenAI Codex assistance. PR #186's coordinated frames, compensated rematching and completed entrance banks are Dugongue's contribution with its OpenAI assistance disclosure preserved. The retained PR #181 shared-edge selection is Chafik Boukhalfa's work with substantial Anthropic Claude and OpenAI Codex assistance. PR #168 v4, its modules, physical bit word and gauge/sink framework retain eumemic's Claude/OpenAI disclosures. Credits and licenses for icekylinx (#144/#130), DaysSky (#162), James Chang (#166/#124), Rohan Arun (#178), the PR #175 overlay, Zhihao Chen, Swapnil Jain, RaD/hipotures, OpenAI #109 and other predecessors remain in the unchanged notices and source archive. No exclusive-priority claim is made.