[petalas]

Summary

Finite certificate and scoped written proof: the fixed binary Clebsch full-parity matrix has a sharp induced-subset threshold of 13 vertices for a strict query-dimension saving over integral edge/private-coordinate sharing.

13 unit payloads on the shipped triangle-free mask
  edge/private-coordinate sharing: 7 coordinates
  explicit full-parity factorization: 6 coordinates

Every 13-vertex restriction has that saving. No restriction with at most 12 vertices gives a strict saving using this fixed matrix. A quadratic-code argument separately proves rank six after deleting any five vertices. This answers a bounded form of Open next question 1 of JoyboyBrian #5023.

The classical full 16-vertex Clebsch rank-six storage code is credited to its existing literature, documented by Alexander Barg and Gilles Zemor in High-rate storage codes on triangle-free graphs, Section 2. That full construction and its separation from clique sharing are not claimed as new. The contribution is the exact induced-subset threshold and explicit smaller query witness. This contribution was produced by Nick Petalas.

Evidence

• Before: integral coordinate sharing on vertices 0,...,12 needs seven coordinates. After: the committed six-query factorization checks every ordered matrix entry and all 8192 binary inputs, giving 106,496 individual recoveries.
• Every one of the 65,536 induced subsets is checked by pivot elimination and complete row-span enumeration. Matching recurrence and independently computed odd-component upper bounds agree throughout. Exactly 697 subsets improve: all 560 size-13 subsets, 120 size-14 subsets, 16 size-15 subsets and the full graph.
• Ten malformed-input/factorization controls reject. The generated receipt binds all six local source/metadata files and records the full joint histogram and ordered census digest.
• Local CPython 3.12.15 normal and optimized direct checks each take 0.64 seconds. Both registered checks pass. Metadata/pins, Python syntax/function bounds, committed-byte/source-history preflight, official export integrity and read-only Git-free optimized replay pass. A separate internal proof/verifier review found no mathematical blocker; independent human and formal proof-kernel review remain unclaimed.

Exact commands from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/clebsch-puncture-query-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/clebsch-puncture-query-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/clebsch-puncture-query-gap
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/clebsch-puncture-query-gap --check-source-history

Expected scientific output:

PASS_CLEBSCH_PUNCTURE_QUERY_GAP subsets=65536 first_saving=13 strict_saving_subsets=697 literal_recoveries=106496 controls=10

Limitations

The threshold concerns principal restrictions of one fixed full-parity matrix over F2. It is not a minimum over all fitting matrices, a global smallest triangle-free example, a nonbinary or nonlinear theorem, a native supplier, or a complete multiplication theorem. The unrestricted fitting rank of the 13-vertex witness remains five or six. Physical queries, neighbor reads, delivery, coefficient preparation, dirty restoration, ownership, frames, stock, precision, tapes, compiler and outer costs are not supplied. No selected cold replay, Lean or Docker run was needed or performed. No new kappa, supplier admission or selected-frontier change is claimed.

Only research/clebsch-puncture-query-gap changes. The canonical source base is de5ec2e, priority revision 8.

Open next questions

1. Does the shipped 13-vertex graph admit an unrestricted directed binary fitting matrix of rank five? Start from its independent-set lower bound and seek a factorization or exclusion; the fixed full-parity rank does not decide this.
2. Can one actual recursive caller expose this mask and retain a gain after the complete physical bill? Start with a source-bound ordered mask and all query, target-delivery and restoration charges. No such caller is supplied.

Related open work

• Classify binary C5 optima at total payload seven (no new κ) #5873 classifies all rank-optimal binary C5 fitting matrices at positive total payload seven. This package studies fixed full-parity restrictions on Clebsch masks; the domains differ and no common numerical value is compared.
• Construct minimum-cost binary query bases with exchange certificates #5791 constructs a minimum-cost basis of a given binary query span; this package fixes a particular mask and compares its supplied rank with edge sharing. Acquisition/reconstruction prices are not compared.
• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies fitting-matrix floors for square root-code decoders on E8/P112/P104/D4 masks. The Clebsch induced masks and query-dimension contract here are different; no common numerical result is asserted.
• Checkpoint eight masked-decoder basis searches (exploratory; no new K) #5064 searches rational masked E8 completions and reprices decoder bases. This package gives an exhaustive finite binary result on a different graph and does not compare those decoder costs.

No same-Clebsch open PR was found in the live overlap search. The merged predecessor #5023 has a different, single-cycle mask class. Main-tree and merged-term searches supplement the sparse local helper; absence of a match does not certify novelty or cover unpublished work.

CI notes

At head 92a3d52f337c513d20bef62e9bf5125b2d20efee, run 40486801948 has plan failed; replay skipped. The plan job's only failed step is Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending with FAILED (failures=8, errors=2, skipped=22), exit code 1.

The two errors are missing follow_up_steps and catalog_first_step keys. The eight assertion failures concern frozen-card command resolution, missing current/historical references and sparse omission lists. This is the shared main de5ec2e7a failure reported on #5854 through #5863 and #5865 through #5870. Maintainers will repair it; no shared tests, implementation or files outside the package were changed.

Hosted replay has not certified the package; only local package checks have run. All scientific checks listed above are local evidence. The completion refresh inspected this account's open PRs created since the CLAIM and added related #5873; it computes no common Clebsch threshold and no disagreement was found in the stated scopes. No extra computations were run to validate other contributions.

Merge danger

Door: two-way. Blast radius: isolated. This adds seven self-contained research files and an opt-in bounded replay manifest, with no shared-code or selected-record change.