[commdata2338]

Change and mathematical scope

Adds research/staged-fourth-dual-binomial-audit/ only: an independent exact audit of the pooled-dual proof in Ross Richardson's PR #4091, with a written proof, source pins, a deterministic certificate and 21 arithmetic/control tests. No discrepancy was found. No new κ.

• All 841 whole-helper bin patterns (5 / 21 / 158 / 657 at capacities 8 / 16 / 32 / 48), enumerated by a Cartesian product and cross-checked against a generating-function recurrence.
• All three duals (left / middle / right), re-solved from exact equality systems rather than executed from the published formulas, and checked on every pattern for every real 0.00173 <= a <= 0.00174: 11 / 8 / 9 identically tight patterns, every other reduced cost strictly positive.
• The central balance system (determinant −10) forces the unique global optimum x = 1080; affine primal witnesses cover every real mixture 0 <= x <= 3240.

The numerical method is new: a positive binomial series for (r^(1-a) - r)/a with integer outward rounding and an explicit geometric tail. It imports no predecessor Python, no logarithm/exponential engine and no optimizer. Dual coefficients and primal multiplicities are solved from exact equality systems. The theorem and its witnesses remain Ross Richardson's contribution; there is no new κ, root-digit refinement, construction or supplier admission.

The complete bill keeps 3240 L3 + x(L4-L3) and every residual, including empty bins, against (896*360)H73. The local paid-profile reports and the stock coefficient are inherited premises. This audit does not establish local-transition or native correctness, chronology, family exclusion, offsets, physical stock, compiler, all-size, precision/tape, BIT or outer admission. The selected record on main (#5413, κ = 0.003027288235) is a changed mechanism far above this fixed-model cap (about 0.0017310), so the cap does not constrain it.

A checked consequence answers PR #5280's pooled-feasibility question. At x = 1090, the right-face witness uses five capacity-48 bins (6,6,9,9,9,9) and leaves idle {3:22783, 5:1625, 48:138492}. The central eight-pattern support forces x = 1080, and its idle multiset has ten too few residual rank units for 1090 (the total residual rank grows by exactly one per fourth-entry call). So PR #4170's question about reaching that same multiset at 1090 needs a changed mixture or invoice. These consequences already follow from PR #4091's formulas; no discovery priority is claimed.

Task and overlap

Task card independent-moment (track independent-verification), priority revision 7; catalogue scope key 9f27273fb945e4fc4274; refined scope pr4091-binomial-dual-audit. The card was retrieved after reading the handoffs, with python3 -B scripts/choose_current_research_task.py --task independent-moment (seed c96fb7aca3d981b76569c66ee826ff3a). Full scope, stop rule and remaining obligations are in TASK.json.

Head dc5f8e94c8764d2048af85c0cc07a290d5ee153b on base main 32890bc5de034ab8fc93ac88ead7279a6a416d04. Canonical main was unchanged at 05:36 and 05:46 UTC on 2026-10-25. certificates/selected-result.json on main selects #5413 (κ = 0.003027288235); the priority page's 23 October selection refresh still names #4610 (audited in #5241). This package changes neither.

Exact dependency heads:

• PR #4091, audited: a29bdd8f56cd18b805917775cdec09ff935eddc4; its package is unchanged on main.
• PR #4088, frozen source of PR Prove the sharp pooled optimum for fourth staging #4091's inputs: 76088d49e3ed4142a958659aa996b0ca4da5e88e. The later head 72d7efc181edb21806a878f174847c96dd410ab0 is context only.
• Context only, not computational dependencies: PR #4170 6fc2f387b13157778aff09b1611425b5cdd2f406; PR #5280 a0dab1be029af75ef635a10eb9c905014dc54a31.

All eight live input files have Git-blob/SHA256 pins in SOURCES.json and replay declarations in VERIFY.json.

Scope status: refined to the card's explicit pooled-dual obligation for PR #4091 ("validate dual, exhaustive pattern scope"). Domain: widths 5..9, capacities 8/16/32/48, one paid residual per bin, all 841 patterns, real mixtures and the stated exponent interval. Milestone: scoped independent audit. Stop at a discrepancy or a completed certificate; no expansion or packing search.

Nearest overlap: PR #4091 supplies the original theorem and its own primary and secondary checks. PR #4170 and PR #5280 check source transfer and explicitly exclude the dual proof. PR #4181 reports chronological attainment of the cap; its bounded independent finding also excludes the ceiling proof. This account's PR #3237 concerned a different PR #2979 final-fill/control model.

The claim is in Discussion #3606, following the PR #4091 result. Overlap searches used the task plus staged-fourth-mixture-ceiling, pooled dual, 841 patterns and binomial, alongside the ledger and retained packages. The final offline report covered all 7,404 tracked local notes, 371 open PRs (snapshot 2026-10-25 05:31 UTC) and 4,779 Discussion records (05:33 UTC), with complete pagination as reported. A cached snapshot is not a live novelty guarantee or a search of arbitrary diffs.

Validation

Python 3.12.3 on Linux, standard library only:

python3 -B scripts/research_package.py --package research/staged-fourth-dual-binomial-audit --check-only
python3 -B scripts/research_package.py --package research/staged-fourth-dual-binomial-audit
python3 -B scripts/check_research_commit.py --package research/staged-fourth-dual-binomial-audit --check-source-history

All pass. The registered replay runs the checker and 21 tests in both normal and optimized modes. Expected checker line:

PASS_STAGED_FOURTH_DUAL_BINOMIAL_AUDIT patterns=841 duals=3 tight=11/8/9 unique_x=1080

The 2,523 dual inequalities split into 28 symbolic equalities and 2,495 strictly positive interval inequalities. Conservative lower bounds on nonzero slack divided by a are 4.38806, 0.69698 and 2.37161 for left/middle/right. Exact rational linear solves recover all three dual systems, the central determinant −10 and both affine primal segments. The signed bill has opposite signs at the coarse endpoints 0.00173 and 0.00174; the published 24-place root bracket is not freshly verified here.

Controls include exact rational powers and square identities for every rank 1..73; a Fraction partial-sum oracle; omitted-tail discrimination; omitted empty/nonempty patterns and count-preserving duplicates; a zero-rank-mass dual perturbation with genuinely negative slack; inadequate numerical precision; wrong mixture, width or historical replica factor; an omitted local charge; and corrupt source bytes. No floating-point decision supplies acceptance.

Committed-byte/source-history validation covers 16 payloads. The official export and its offline integrity check pass. All four direct commands then pass from that export with read-only files and directories, an empty PATH, no Git use, and all 16 payload hashes unchanged. A direct checker run takes about one second; timing is host-dependent.

The written proof plus finite exhaustive arithmetic supports the declared LP claim. It is not a formal-kernel proof or an independent human review. No full source-native, ordinary/outer, selected-frontier, Lean or Docker replay was run.

Credit

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.

Ross Richardson / PR #4091 retains the pooled theorem, the three duals and the witnesses. Cinna Davis / PR #4088 retains the fixed words and the original fill. Nick Petalas / PR #4170 and unpaidmoon / PR #5280 retain the transfer checks and the follow-up questions answered here. Rohan Arun / PR #4181 retains chronological attainment. Credit also Rohan Arun / PR #4076, David Ganz (Dugongue) / PR #3945, zxv / PR #4085, dataisfire / PR #3734, Erwan Secousses (ChaloupeMusic) / PR #3949 and #3815, and the geometry, native, owner, compiler and transfer contributors preserved in PR #4091's notices. New work is Apache-2.0; predecessor files are read in place and unchanged.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

See the package README and TASK.json. No further search is justified inside this fixed invoice: PR #4181 already reports attaining the cap. A consequential next construction must name a changed word, inventory or consumer boundary, identify the balance or cost equation that changes, and supply the complete signed invoice before borrowing any ceiling. Source-transition and physical-admission obligations remain separate; the inherited local paid profiles have not been reconstructed from every source transition.

Workflow feedback (optional)

The selected-record helper correctly resolved #5413 while the priority page's selection prose still named #4610. Literal overlap searches were dominated by copies of original_card inside unrelated TASK files; deweighting that inherited text in favour of actual_scope and open questions would make the same search easier to review.

🤖 Generated with Claude Code

https://claude.ai/code/session_01Jur4aiU6BBSmBiaRjkbECf