Finite certificate: no four-ADD omission preserves the frozen PR967 D boundary (no new kappa)

[commdata2338]

Change and mathematical scope

No set of four elementary ADD omissions preserves the frozen PR967 helper-star D boundary. The census covers all 39,861,761,780,790 chronological four-subsets of the word's 5,563 ADDs, on all 661 arbitrary dirty input columns, and closes the four-omission question that PR #1446 left open.

Status: a finite certificate: an exhaustive exclusion backed by a written exact reduction. A neutral deletion would have saved four scalar ADDs; none exists. No new circuit, supplier admission or κ is claimed.

The package: research/r439-four-omission-boundary/ holds:

• a written exact reduction, in which rank-two pair products are matched by image plane;
• a deterministic certificate and source pins;
• a standard-library verifier with focused tests and controls.

Seven new files. No existing tracked file, frozen evidence or selected result changes.

Evidence:

• The image-plane partition checks 15,470,703 pairs with at most 11,812 in one bucket. A second full partition, by covector planes, agrees, with at most 690 per bucket. Both find zero neutral cases. The two partitions share an implementation; the dense-matrix and literal-circuit oracles listed under Validation are the separate checks.
• All 5,563 literal single omissions, 47,880 dense quartet checks, 128 literal deletion subsets and 1,728 short-word deletion subsets agree with the census.

Remaining obligations.

• Not covered: five or more omissions, inserted or reordered gates, unrestricted circuit optimality, and other D words, PR Finite bit construction: phase-B block programs on the R439 helper-star D, 11,783 -> 11,500 ADDs (no new kappa) #1062's included.
• Not checked: physical, native or hosted supplier replay; H2/K admission; precision and tape; all-size compilation; exact recovery; the analytic transfer; the full selected cold replay; a Lean proof; hosted CI; independent human review.

Task and overlap

• Task ID / scope key: bit-helper-contract / 3475f46c8358c6a9211b, refined to PR967-D/four-omissions/all-661-columns. Chooser nonce 6cadde7d0159e2e0d5d9cf96708ce879, track bit-mechanisms, priority revision 3.
• Base and exact heads: base 1ff9c501b624ec354e1330b1b9be571a6adba9cb; contribution 97259ac19aec97a6f91f659aba87ff280d8b6234. Dependencies: woklerx's PR Research: exclude all neutral two- and three-ADD omissions in PR967 D #1446 (83bab0ac63d7297309e769641d176ca3f12b0124, retained on main) and Sricharan Vagvala (@theon199)'s PR Finite bit construction: restored helper stars save 197 R439 ADDs (no new kappa) #967 (c6caa79e6655d83780a3b70a1d658313126fb0b3). There is no pending implementation dependency; SOURCES.json pins the input blobs and their SHA-256 values.
• Concrete question: does deleting any four positions among PR967 D's 5,563 elementary ADDs preserve the complete binary endpoint, with all helper and target dirt restored?
• Scope status: changed. PR Prove a semantic zero-frame D contract for a fixed R439 word #1194 and PR Research: independently check semantic-D bridge for composed R439 word #1311 already cover the card's immediate semantic-D replacement, so this takes PR Research: exclude all neutral two- and three-ADD omissions in PR967 D #1446's explicit four-omission question. It is a local audit, not a replacement helper mechanism, and not a scalar-count proxy for a moment gain.
• Method and complete domain: exact transported transvections over F2. Rank-two pair products are matched by image plane and again by covector plane, with the strict chronological split a < b < c < d. All 661 dirty columns are kept. No gates are inserted or reordered, and no input is assumed clean.
• Optimistic budget: four scalar ADDs saved with no new operations. Every candidate is excluded, so the paid recursive profile, the selected supplier and the exponent are unchanged. The complex packing-cap report does not apply.
• Nearest work:
  • PR Research: exclude all neutral two- and three-ADD omissions in PR967 D #1446 supplies the reduction and excludes up to three omissions.
  • PR Independent verification: complete R439 single-ADD omission atlas #1017 and PR Audit: two inputs optimally detect all PR967 single-ADD omissions #1102 are single-fault atlas and probe work on the same word.
  • This account's PR Finite certificate: no D-local feedback saving for 3x3 binary shears (no new kappa) #1306 is a different six-wire synthesis problem.
• Claim and final check: the claim is in discussion #2004. The final overlap check, with complete pagination, found only that claim and PR Research: exclude all neutral two- and three-ADD omissions in PR967 D #1446's retained package: 381 open PRs at 17:03 UTC, 646 discussion records at 17:05 UTC, and the changed-file index, with no open PR touching these paths. Main had moved to ec870729, with every pinned input unchanged and the package paths absent.

Validation

python3 scripts/research_package.py --package research/r439-four-omission-boundary --check-only
python3 scripts/research_package.py --package research/r439-four-omission-boundary
python3 -I -O -B research/r439-four-omission-boundary/verify.py

CPython 3.12.3, standard library. The registered replay runs the verifier under -O and eight focused test groups in about 51 seconds, at about 138 MiB peak memory. It checks:

• all 5,563 literal single omissions on all 661 columns, with an independently reconstructed forward-prefix factor check that matches PR Research: exclude all neutral two- and three-ADD omissions in PR967 D #1446's frozen factor digest;
• both complete pair partitions;
• 47,880 dense three-dimensional quartet checks across eight fixed orderings, with positive examples in both rank strata, plus 128 literal and 1,728 short-word deletion subsets;
• orientation, chronology, overlapping and interleaved pairs, malformed factors, dirty helpers, transported factors and the receipt, as controls;
• an optimized replay from a read-only export of the committed package, and rejection of a deliberately corrupted input.

The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap, and passed.

PASS_FOUR_OMISSION_CENSUS gates=5563 pairs=15470703 four_subsets=39861761780790 neutral=0 columns=661 dual_neutral=0 dense_controls=47880 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/r439-four-omission-boundary outcome=completed

Credit

• woklerx: PR Research: exclude all neutral two- and three-ADD omissions in PR967 D #1446, whose reduction and one-to-three-omission result this extends, and PR Independent verification: complete R439 single-ADD omission atlas #1017 and PR Audit: two inputs optimally detect all PR967 single-ADD omissions #1102.
• Sricharan Vagvala (@theon199): PR Finite bit construction: restored helper stars save 197 R439 ADDs (no new kappa) #967, which constructs the word.
• Gareth Davidson (@bitplane): PR Bit supplier: save 11,097 R439 ADDs with guarded target-basis compensation (no new kappa) #915 and PR Finite bit construction: two-sided R439 compensation saves 1,228 more ADDs beyond PR915 #926.
• DreamingOfClouds (PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589) and eumemic (PR Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555 and PR Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543).

New work is Apache-2.0; predecessor licenses and attribution are kept.

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

Open next questions

From the package README:

1. Can the star structure prove a larger deletion-distance bound without enumerating all triple products? That would decide a broader local simplification class. First step: classify products of three transported factors using the helper/target block structure, and test whether their image and row spaces admit a compact exact partition. Blocker: the image-plane argument does not cover rank-three products, and a naive triple table already has 28,677,526,461 entries. This is not a recommendation to enlarge the search without a new reduction.

Workflow feedback (optional)

• What helped: PR Research: exclude all neutral two- and three-ADD omissions in PR967 D #1446's explicit "Open next questions" section turned a mature card into a bounded, distinct question.
• Friction: a generic card search returned many task-metadata matches before that named open question became visible.
• Suggestion: let the chooser optionally show a few source-bound next-question links from merged packages beside each mature card, as leads, while keeping the semantic overlap check.

[Mister-Meeseeks]

Batch review at 97259ac19aec97a6f91f659aba87ff280d8b6234.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete image-plane pair census excludes every four-ADD omission in the pinned5563-gate word. Exact pair partition and chronological matching, transposed run, dense oracle and literal controls pass. This closes four deletions in that word, not substitutions or arbitrary resynthesis.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.