[petalas]

Summary

Every pairwise nonorthogonal subset of the 2,160 norm-eight BW16 lines has at most 126 lines. Claim type: finite certificate with a scoped written combinatorial proof. No new kappa or selected-frontier change.

nine disjoint D16 sets cover the shell
fifteen clique members in one set force total size <= 30
therefore every larger clique has <= 14 per set: total <= 126

The new equality-case step improves the basic nine-chart bound of 135 by nine. Its conditional thirty-line bound is attained. DaysSky's PR895 supplies the shell setting and published 72-line lower witness as credited premises; neither its witness nor screen is replayed. The remaining interval is 72<=omega<=126.

This contribution was produced by Nick Petalas. Retain DaysSky, DreamingOfClouds/PR446/595/2244, Joel Pulikkan/PR620 and the decoder/lattice lineage; Nick Petalas/PR5118 supplies earlier local insertion barriers. Barnes-Wall, Reed-Muller, Construction-D, Walsh characters and Kerdock/mutually unbiased bases are classical. The package credits the classical 135-frame geometry and claims no general literature priority.

Evidence

• Before: the global clique maximum remains open beyond the published 72-line witness; the elementary nine-chart count allows 135. After: the full-star restriction proves the global ceiling 126.
• Exact integer certificate: all 2,160 shell members, nine disjoint 240-line blocks, all 17,280 inverse chart images and orthogonality of all eight chart matrices.
• The written star lemma quantifies over every center and sign assignment. Supporting computation checks 176 literal center/sign fixtures against all 1,920 odd lines, all 32,768 signs at center zero by a separate parity formula, and all 435 pairs of the thirty-line attainer. Six semantic controls pass.
• Local CPython 3.12.15/macOS arm64 direct checks pass in normal and optimized warning-strict isolated execution, in 0.436 s and 0.434 s respectively. These times document timeout margin, not a performance comparison. Both registered commands and metadata pass in 1.126 s total, within their 120-second caps.
• Seven committed files pass byte/source-history preflight. Official committed export and export integrity pass. Both Git-free, read-only, empty-PATH direct modes pass with every file unchanged. Four external optimized mutations reject missing signs, duplicate charts, a wrong transform denominator and an incompatible quadratic-phase menu at scientific checks.

Reproduction

From the repository root, with standard-library CPython 3.12:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/bw16-global-clique-126/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/bw16-global-clique-126/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/bw16-global-clique-126
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bw16-global-clique-126 --check-source-history

A different CPython 3.12 executable may replace the absolute path. Expected markers are PASS_BW16_GLOBAL_CLIQUE_126, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. RESULT.json is generated by the verifier and bound to source hashes; ordinary verification compares it byte for byte. No external scientific input, numerical library, solver, network or Git is required by the direct verifier.

Limitations

The exact clique maximum remains open. This upper bound on clique size does not upper-bound the minimum binary decoder fitting rank. Native/helper delivery, arbitrary dirty restoration, the complete paid/outer invoice and the five selected framework assumptions remain separate. No multiplication theorem, physical supplier, independent human acceptance or formal proof-kernel acceptance is claimed. No local registered obligation remains unrun. Lean, Docker, selected cold replay and predecessor scientific suites were not run.

Canonical base 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030, remained unchanged at submission refresh. The broad overlap snapshot covers 572 open PRs, 6,646 Discussion records and 8,917 main-tree note records; targeted live/main/merged BW16 checks and local reservations were refreshed before committing. Arbitrary diffs and unpublished work remain coverage limits. The advance claim sought a global spectral bound; the delivered proof uses an exact nine-chart partition and the full-star lemma. Discovery stopped at the first new verified bound.

Open next questions

• Can compatibility between nonsaturated D16 charts lower 126? Start by classifying fourteen-line intersections; that classification is absent here.
• Can a changed physical decoder use this geometry with a favorable complete bill? Begin with an explicit fitting identity and charge native delivery and restoration. Clique information alone does not supply the construction.

Related open work

• PR895, DaysSky, studies the same BW16 shell and its binary decoder window. Its 72-line lower witness is cited unchanged; this contribution adds a global clique upper bound and makes no new claim about its decoder-rank interval or screen values.

The completion refresh listed all sixteen open account PRs created since the CLAIM, including this PR. None of the other fifteen studies the BW16 shell or its global nonorthogonality clique bound. No competing numerical claim or disagreement was identified within that coverage, and no extra predecessor computation was run.

CI notes

Both hosted checks, plan and replay (research/bw16-global-clique-126), passed at head 4ef5803c07dc1637adbc1b6f616be6f10ecbaff8 in run 41650974400. The completed checks were read once; run metadata independently confirms the same head and successful completion. All local checks described above also passed. Hosted replay certifies execution of the registered finite checks, not independent mathematical or formal proof-kernel acceptance.

Merge Danger

Door: two-way. One isolated opt-in seven-file research package; no shared code, frozen predecessor, upstream archive or selected-record changes.

Blast Radius: research