[petalas]

Summary

For the order-seven binary GL3 target with columns (7,1,3), prove the sharp normalized coordinate-cut cone value

Fano point/line cut certificates: 3t
independent coordinate word:     4t

Claim type: scoped mathematical proof with a finite exact 256-cut certificate. The value 3t holds for every target multiplicity, either target orientation, every finite sum of point and dual-line permutation modules, and arbitrary restored dirty workspace. Two explicit rationally weighted cuts attain it. Thus more copies and cuts in this family cannot prove the direct tariff. The actual multi-target formal-word minimum is not claimed.

This takes the written-invariant question in Josie Elliston's #1974, with Hanson Sun's seed18 model. Their word censuses are cited, not replayed. Credit Adam Gall and Ross Richardson for the inherited cut-rank method, JoyboyBrian #4974 for the existing singleton support argument, and the retained eumemic/Ali Romman/Gareth Davidson lineage. The new matching cone certificates and proof were produced by Nick Petalas. No kappa or selected-frontier change.

Evidence

Before, a natural-representation alias did not determine whether the whole point/line coordinate-cut cone could prove the tariff. After, the exact inequality 2*c_H(g) <= sum_(i!=j) c_H(s_ij) holds on every one of the 128 point and 128 line cuts, and explicit cuts attain three. The written direct-sum and crossing-rank proof covers all finite module sums and role counts.

Local CPython 3.12.15, macOS arm64, standard library only:

• Warning-strict normal and optimized verifiers pass. Each registered command took 0.06 seconds locally; the full package launcher took 0.44 seconds.
• All 168 group elements, 56,448 action-composition checks, 1,176 point/line incidence checks, 256 cuts, and 2,520 actual paid crossing blocks pass. The block controls use all 35 rank-one idempotents through three roles on six declared cuts.
• Full formal endpoints pass for one to three targets with a dirty spectator, with 24 call omissions, three restored dirty detours, three dirty leaks and four boundary controls.
• Package metadata/pins, committed bytes/modes and declared source history pass. Official export integrity and both normal/optimized read-only Git-free replays pass, preserving all export bytes.
• Actual mutated dual coefficients, primal cut selection, line-action convention and retained source bytes are rejected. Fresh stdout receipts bind all six package files and both source README inputs by SHA256. No receipt is hand-written or substituted for computation.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gl3-singer-cut-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/gl3-singer-cut-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gl3-singer-cut-gap
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gl3-singer-cut-gap --check-source-history --check-modes

A reviewer may substitute any CPython 3.12 interpreter. Expected markers are PASS_GL3_SINGER_CUT_GAP, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, and PASS_RESEARCH_COMMIT_BYTES.

Limitations

This closes only nonnegative combinations of role-uniform coordinate cuts in finite sums of these two modules. Arbitrary subspaces, other permutation modules, role-dependent cuts, nonlinear/integer-combined invariants and other alphabets are outside the proof. The single-target four-call minimum is inherited support reasoning; 4t is only a direct upper bound for multiple targets here. Native realization, compiler/hosting/precision/tape invoices, recursive supplier admission and the complete multiplication theorem remain unproved. No predecessor engine, selected-supplier, Lean or Docker replay was run. Independent human and proof-kernel review are unclaimed.

Open next questions

• Can another explicit GL3 permutation module certify more than three for this exact Singer target? Supply its six generator crossings and target crossing.
• Can a noncoordinate binary subspace cut in a point/line module sum exceed three? Supply the subspace and all seven quotient ranks.

Related open work

Refreshed after CI completion: this account has no open PRs created since the advance CLAIM at 2026-10-27T13:29:57Z, so no related open contribution needs adding. The earlier targeted searches identified no open PR studying this exact Singer point/line cut cone. Merged #1974/#1567 concern formal word minima; #1741 treats a different, distance-three order-seven target. The three-cycle cut-ceiling packages concern a different endpoint. Search coverage does not cover unpublished work.

CI notes

At exact contribution head a5a8a76babad35f872545230d106984e8fa51fec, run 40514062006 finished with plan failed; replay skipped. Hosted replay has not certified this package. Only the local package checks described above have run.

The only failed step is Onboarding helper tests, running python -B -m unittest discover -s tests -p 'test_research_*.py' -v. All ten failure/error reports are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen/current routing, missing-reference exceptions and sparse omission lists: expected retarget count 16 versus 0, five missing ValueError rejections, and two empty omission lists. No failure in this package is reported. The same ten failure/error headers and final summary were independently confirmed in PR #5927's run 40514192213, also failing only this step. No CI repair or additional branch commit was made.

A maintainer merged this PR at 2026-10-27T13:43:54Z in the bulk queue archive, merge commit aacce1a2c411a2f083b5fd5f1cdcf87661267ca3. Under the archive-intake policy, that merge preserves the contribution. It is not scientific approval, an independent mathematical review, hosted replay certification or a selected-frontier update.

Merge danger

Door: reversible. Blast radius: research-only. Adds one isolated package and changes no selected record, predecessor, shared verifier or frozen evidence.