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Certify explicit-word polynomials through exact H₀ bin counts - #222
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Add an exact real-to-complex dyadic lift and transport nodal/Hessian error into the original field units. Reconstruct one deterministic K1 example and certify its finite-polynomial lifetime-bin counts without promoting it to a Gaussian draw. Preserve all frozen source objects.
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Exact-head adoption: 79d92ab, tree fcceb1460d3ba646d18e9d93b64cb01795e92077. The reviewer independently compared the committed blobs with the frozen inventories; all reviewed bytes are identical. The preserved report below records its original review timing and scope. Report SHA-256: e8515d2b0226da37830de9c955b4a12e5c83e6a091bfdbc2d90659333ba994d6.
C42 word-polynomial bridge: fresh mathematical and implementation review
Disposition: PASS_TECHNICAL. No substantive or blocking findings.
Dylan Roy — delegated AI review, performed by OpenAI / Codex, agent /root/c42_math_review, as a fresh nonauthor review of the additive word-polynomial bridge. The reviewer first encountered these authored files as review targets and did not implement or edit them. This is same-provider Codex lineage with zero organizational-independence credit. Dylan’s personal reading is PENDING. This is neither formal evidence nor scientific acceptance.
The reviewed candidate is commit 79d92ab2af199ade4740350ef25b6257180c15c8, tree fcceb1460d3ba646d18e9d93b64cb01795e92077, on codex/word-polynomial-bridge-20261001, based on 8ed4e73bafb3c9fd97a439b2e30e4a795a567a4f. The fresh source review began on the frozen uncommitted candidate; after publication, git show <head>:<path> independently confirmed exact SHA-256 and byte-count equality for all 12 candidate members and all 43 source-inventory members, with zero mismatches. Ownership claim: a808ae9d-f56a-4c7f-934e-178652927acb. The review is bound to the 12-file candidate inventory SHA-256 865a6fc7fd9cccd32476737aa69fba5178a3383ed3b1bfd56fba96d4744b8ed5 and the 43-file source inventory SHA-256 a76cd9e727b9fd4b2e42f13f48a86fe71d0105b9452a29f93bec6fbec43b04f5. Every inventory member matched actual bytes before and after the controls.
The five additive sources reviewed are:
| Source | SHA-256 |
|---|---|
experiments/periodic_h0/WORD_POLYNOMIAL.md |
7fee78ff0c23466fcbdf20fdb3b7dd400316a85f41bae3592dc23d350751a4d3 |
experiments/periodic_h0/test_word_polynomial.py |
142ea2c601c1e676c14d346e206313f6833282f0407e635961def375e8f1ffa4 |
experiments/periodic_h0/test_word_polynomial_certificate.py |
96d7fc07667cd4901cb410df6636aefa99a332c273d0799b9f99ad06439ad0e5 |
experiments/periodic_h0/word_polynomial.py |
98616041d9fbb9a5d2ec44225a7deaaf11d8fa4c459d7bf7e220144779596bce |
experiments/periodic_h0/word_polynomial_certificate.py |
6299087deca2a7aa2c4e8e5e6a58dc1f5325364a36c8fcbbd7bcd2a28af9724a |
The produced CERTIFICATE.json is SHA-256 0c6157788b2dfbb2480a1b7deae5d6287feebf39b951b70487cafa64b77d7e95; RUN.json is SHA-256 dba004cd54233c47f4a5fccbef5382bd0d049ec9b87979020f018b3658e2c581. Exact identities for consumed interfaces and all four new outputs are retained in REVIEW.json. The frozen consumed core/proof slice is unchanged from the base.
The conversion is correct: (C-iS)/2 under the positive-exponent unnormalized transform restores C cos(theta)+S sin(theta) after conjugate doubling. DC is unchanged. The row-major coordinates and both axes, including negative ky, agree with the physical side-24 polynomial. The maximal converted denominator is 2^193; the smallest permitted nonnegative power-of-two lift is 2^max(0,r-96), bounded by 2^97. No real coefficient is rounded.
Unit transport is correct. Samples are the integer real centers divided by lambda*2^96; their original-polynomial nodal error is E_lift/lambda. The sample spacing is 24/n, so the spatial term is (24/n)^2*c_lift/lambda. Division of the lifted certified Hessian bound is valid without supposing that outward rounding is homogeneous. The total matching budget is their sum.
The ordinary periodic H0 interface keeps one essential class, omits zero pairs, and interprets each finite lifetime as (birth-death)/(lambda*2^96). The bridge executes its separate connectivity checker on the full new integer array. Its exact bin-transfer call uses the correct strict clean-upper condition a>2epsilon and half-open endpoints.
The identified replay supports exactly one finite smooth-polynomial bar in [1/125,2/125), and zero in each of the other seven displayed bins, whose union spans [1/1000,32/125). The upward decimal bound is epsilon <= 0.000375408633135674692468, with lambda=2^97 and sample scale 2^193. The exact sample diagram has three positive finite bars; the two smaller bars lie at or below 2epsilon and do not establish smooth-field detections. This review makes no total-bar assertion outside the displayed bins.
The nine-word cutoff-one vector is a deliberately selected new arithmetic fixture. It is not an observation of the Gaussian ensemble. The earlier K24/K32 coupling and tail budgets do not certify this K1 field. The output correctly leaves the IID law unproved, new Gaussian barcodes zero, historical coupling and infinite-field diagram bounds unknown, and the lifetime law uncertified.
The fresh controls completed with 232 positive checks and 44 rejecting controls, using the standard-library command below. The file CHECKS.json retains each outcome and the exact rational result data.
python3 -B -S /Users/dylanroy/Documents/Codex/2026-09-19/connect-research-repository/work/continuation42/evidence/math-review/review_checks.pyThe nontrivial arithmetic controls used a direct Q(sqrt(2)) oracle at all 64 eighth-root grid nodes for a mixed K2 polynomial, independent of the evaluator’s trigonometric implementation. A positive-rescaling test made P and 8P have exactly the same lifted record, then verified the original-unit errors and sample scales. A literal two-peak periodic polynomial verified the finite and essential endpoints. Lift controls covered 24 DC/cosine/sine cases across the precision boundary and rejected smaller lifts. Strict invalid inputs, zero-mode aliasing, half-open boundaries and diagonal-threshold equality were exercised.
The produced directory was semantically replayed once. Raw encoded endpoint differences were separately counted in the expanded and contracted bins. A forged exact bin upper count failed an actual replay after digest rebinding. Rebound receipt scope, identity and role corruption, as well as altered result prose, were rejected. No candidate source edits or remote writes occurred. The full historical/master suite was not rerun.
The reviewer’s first control attempt encountered a harness-only TypeError: it supplied encoded endpoint strings to the two-argument Fraction constructor. The harness was corrected to convert those strings to integers; the complete corrected run passed. This was not a candidate defect, and no failure was silently attributed to the reviewed code.
This review imports the frozen sampler’s deterministic dyadic-coefficient interface, the nodal error proof, the Hessian interpolation cover, the integer H0 proof, and APPROXIMATION.md’s minimum-corner PL/stability argument. It checked their relevant interfaces and unit compatibility, without claiming to re-prove all upstream mathematics or the cited external q-tame matching theorem. The remaining transitive inventory is custody evidence, not a fresh review of every historical proof. The claims remain conditional on those identified interfaces and separate from required human review, independence and formal acceptance.
Source changes require a successor review of the affected bytes and dependency slice. The published commit has been bound to this exact review by unchanged identities; its publication does not enlarge the mathematical scope. The parent’s separately reported 21-stage master run was not performed by this reviewer and is not claimed as fresh review execution.
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Exact-head adoption: 79d92ab, tree fcceb1460d3ba646d18e9d93b64cb01795e92077. The reviewer independently compared the committed blobs with the frozen inventories; all reviewed bytes are identical. The preserved report below records its original review timing and scope. Report SHA-256: 5167f7e65df848a47f147dc76156362bef25a13f59b62f7e146552d05d736085.
C42 integration and custody review
PASS_TECHNICAL — no unresolved finding in the reviewed integration scope.
Dylan Roy — delegated AI review. Actual performer: OpenAI / Codex,
/root/c42_integration_review; fresh nonauthor source review and focused
execution audit. The reviewer read the author implementation and outputs but
made no candidate source edits. This is same-provider review with zero
organizational-independence credit. Dylan's personal reading is PENDING.
The coordination claim is a808ae9d-f56a-4c7f-934e-178652927acb.
The review binds the 12 candidate files and 43 source identities frozen before
commit creation, against base 8ed4e73bafb3c9fd97a439b2e30e4a795a567a4f:
| Inventory | SHA-256 |
|---|---|
| candidate-files.json | 865a6fc7fd9cccd32476737aa69fba5178a3383ed3b1bfd56fba96d4744b8ed5 |
| source-inventory.json | a76cd9e727b9fd4b2e42f13f48a86fe71d0105b9452a29f93bec6fbec43b04f5 |
All source bytes matched before and after the controls. All 31 inherited
source objects equal the base Git blobs. A later commit adoption must compare
its actual blobs to these identities; this report does not invent a reviewed
commit before one exists.
Verified behavior
The unchanged stored word_polynomial1 directory passed a real exact replay.
The independent review script then completed 40 control groups, comprising
three positive groups and 37 rejected mutations, in normal Python. The
full results, script, and retained
stdout specify what executed. The script uses temporary
copies for mutations and leaves repository source untouched.
The receipt enforces exact top-level roles, integer schema and byte counts
without accepting Boolean substitutes, canonical UTC, bounded nonempty
Python/platform strings, exact source/output roles, lowercase SHA-256 values,
and the receipt checksum. Duplicate JSON keys and nonfinite constants fail.
The source closure contains all twelve local modules imported transitively by
the bridge and wrapper; the produced receipt binds 34 source objects, including
the linked derivations and original upstream receipt objects.
The eight semantic controls altered an input word, a real coefficient, the
sample digest, one positive interval, the spatial bound, a bin bound, the
Gaussian-source status, or the infinite-field/historical coupling status.
For each, the changed input/polynomial digests, output identity, receipt hash,
and generated report were fully rebound. Metadata validation passed, then
actual arithmetic replay rejected the false result. Thus the evidence does
not depend on hashes alone for numerical correctness. Separate controls also
rejected false report prose, a modified inherited sampler source, and a
modified source with its parent receipt and checksum rebound. The hardcoded
parent receipt identity preserves the inherited source boundary.
The reader documents consistently restrict the result to the one deterministic
cutoff-one polynomial. The three grid bars, exact one-bar bin conclusion,
uncertified two smaller detections, 96-bit twiddle accuracy, and missing
Gaussian/IID, historical coupling, infinite-field, and lifetime-law predicates
remain distinguishable. All 62 relative reader links in the four affected
entry/result documents resolved. The mathematical derivation itself belongs
to the separate substantive review scope.
The periodic workflow differs from base by exactly the intended new verifier
step. Both normal/optimized experiment discovery, all frozen predecessor
certificate replays, and the complete pilot remain present. The coordinator's
planned 21-stage local master is appropriate; this reviewer did not duplicate
it. Hosted checks must still bind the actual tested commit and original formal
artifacts. A local PASS does not certify the future hosted run or merge.
Publication custody boundary
The current workflow, release operations, and amended C41 assembler,
formal-artifact reader, and registration preparer were read. The
C42 adaptation notes preserve their successful guards
and identify C42-specific substitutions. This is a preparation review;
acceptance of actual C42 publication scripts, archive, native review bindings,
pre/post hosted artifacts, owner-only upload/readback, register append/readback,
and claim release remains pending those artifacts. No remote write or whole
master run was performed by this reviewer.
Codex Review SummaryThis comment shows the latest Codex review activity on this pull request.
ℹ️ About Codex in GitHubYour team has set up Codex to review pull requests in this repo. Reviews are triggered when you
Codex reacts with 👀 while any review is running, comments if it has suggestions, and reacts with 👍 once all reviews finish with no findings. |
|
C43 pickup from the completed C42 handoff: prospective uniform finite-Fourier-polynomial H0 count bound and exact clipping-loss expectation bridge, under the explicit IID input model. New additive proof/certificate/tests and reader links only, based on main |
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C43 pickup delivered through merged PR223: uniform |
The finite-word sampler previously stopped at real Fourier coefficients. This change carries those exact coefficients through the existing complex Fourier evaluator, Hessian bound and periodic H₀ computation. A positive dyadic lift makes the coefficients exactly representable; dividing samples and error bounds back by the same factor preserves the original smooth polynomial.
The new deterministic nine-word, cutoff-one example certifies exactly one smooth-polynomial bar in [0.008, 0.016) and zero in the other seven displayed bins spanning [0.001, 0.256). Its diagram error is at most 0.000375408633135674692468. The two smaller grid bars lie below twice that bound and are not certified smooth-field detections. No conclusion is made outside the displayed bins or about a Gaussian ensemble, historical samples, expected intensity or a lifetime law.
The source-bound verifier regenerates words → coefficients → lifted nodes → Hessian enclosure → exact integer barcode → bin bounds. Frozen sampler, nodal, Hessian and H₀ sources/receipts are unchanged. Reader links, proof and a compact exact certificate are included; CI replays the successor.
Validation: focused exact controls and strict malformed/rebound-receipt mutants pass; the complete 21-stage master ran once on the stable tree, including 95 experiment tests in normal and optimized Python, all existing certificate replays, repository/navigation/formal-source and browser checks, and a fresh complete pilot. Fresh source-bound mathematical review and integration review pass with no findings at
79d92ab2. The mathematics reviewer ran 232 positive and 44 rejecting controls; the integration reviewer rejected 37 malformed or fully rebound mutations. All eight premerge check executions and GitHub's automatic code review completed without findings. PR #222 merged as3b242e76fd1c1e33b1461b9d135f42413d93dec1; all eight landed checks also pass. The original formal ZIPs for runs 36803654177 and 36804369199 bind the exact tested/landed commits, identical candidate tree, 31 axiom targets and ten rejected negative controls. Formal alignment and scientific acceptance remain separate.Review attribution: Dylan Roy — delegated AI review; actual OpenAI/Codex performers and source exposure are recorded. Personal reading remains PENDING and same-provider reviews add zero organizational independence. R17 claim a808ae9d-f56a-4c7f-934e-178652927acb; one owner-only C42 delivery now carries this result and the separate PR205/197 reporting follow-ups. Original upload bytes were downloaded and verified identically, registration was read back, and all three claims were released. Archive SHA-256:
f35093c03014b6714a1870bd1d1531baae2db101ac17d6a0344c656f4e8707e8.