[petalas]

Summary

Prove exact local insertion barriers around DaysSky's published BW16 72-line clique. This is an exact finite certificate and scoped combinatorial proof, with no new kappa.

one outside line              -> at least 20 old deletions, attained
two compatible outside lines -> at least 30 old deletions, attained
retain at least 43 old lines  -> no different clique of size 72 or larger

For exactly d old deletions and at least one outside insertion, no clique exists below d=20; the exact largest size is 73-d for 20<=d<=29. Thirty is a sharp two-line insertion threshold, not a proved sharp radius for an improving exchange. The attaining cliques have 53 and 44 lines. The predecessor's window floor remains 72.

Only research/bw16-clique-exchange-barrier/ is added. The CLAIM initially sought a larger clique. Two bounded heuristics found none; the scope update records the delivered local obstruction. The construction search stopped at this first new verified result. The result guides the next search beyond small replacements and makes no global maximum claim.

Evidence

• Before: No new κ: BW16 as a unit — window c ≥ 72 puts its ceiling at 0.649× E8's (checked) #895 supplies a 72-line witness and leaves larger cliques open.
  After: the new certificate excludes every improving replacement deleting at most 29 lines of that exact witness.
• All 2,088 outside lines and 2,178,828 unordered outside pairs are classified. The single-deletion histogram is {20:378,29:1080,36:630}. There are 1,316,988 compatible pairs; 11,340 attain deletion union 30. The complete pair histogram, attaining vectors and package hashes are generated on stdout.
• Standard-library CPython 3.12.15/macOS local replay passes with both signed-mask and literal integer-dot pair computations, each in normal and optimized Python. The four registered checks took 0.425, 0.426, 1.288 and 1.298 seconds in the recorded replay, below their 120-second limits.
• Committed-byte/source-history preflight, official committed export and export integrity pass. All four modes also pass in a Git-free export with read-only files and PATH=/nonexistent; all eight exported files, including the export receipt, retain identical hashes. Three deliberate semantic mutations are rejected: intersection in place of union, omitted compatibility filtering and omitted inner-product scale.
• New code is bounded, read-only and has no imported predecessor executable, solver or recursion. Its fixed-domain Python allocation deviation is documented in the README. No mathematical result from another contributor is audited as the deliverable.

From the repository root:

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

Expected markers are PASS_BW16_CLIQUE_EXCHANGE_BARRIER one=20 two=30 outside=2088 compatible_pairs=1316988, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

This is a finite neighborhood theorem about the exact fixed clique. It does not prove maximum clique, a stronger decoder-window floor, a complete paid supplier, physical/helper/restoration costs, compiler/outer admission or a multiplication theorem. Independent human and formal proof-kernel review are unclaimed. No registered local obligation remains unrun; no native, Lean, Docker, selected cold replay or historical suite was run.

The original card's historical orientation command research/complex-local-word-ceiling/verify.py raises KeyError: 'complex_profile' on the changed selected schema. That unrelated reporter is unchanged and is not evidence for this result. The current selected record and family guidance were read; this package changes neither.

Scientific base is canonical main 81e9bd69e4a5b00ba9c713a1620e36d576d5522e, priority revision 7, unchanged at the submission refresh. Overlap coverage includes the complete cached snapshot of 321 open PRs and 4,236 Discussion records, refreshed live handoffs and object-specific all-state/merged PR searches, and main-tree README searches. Sparse local notes, arbitrary diffs and unpublished work are not exhaustively covered. No overlapping successor was identified within that coverage.

Open next questions

1. Find a larger clique through a move discarding at least 30 old lines or a different starting construction. Begin with at most 42 retained old lines. The necessary threshold is not asserted sufficient.
2. Explain the deletion spectrum structurally, beginning with the 378 minimum-deletion lines under the actual stabilizer, or test a different inequivalent clique. No orbit/equivalence classification is supplied.

Related open work

• #895, DaysSky, head 3bf6fccaf17521ffdb9b3bec2740ba4886478556, studies the same BW16 shell and decoder window. Its 72-line clique is an unchanged attributed input; this package adds the insertion thresholds 20 and 30, without re-pricing or endorsing its screen. No competing numerical claim about these thresholds appears in that PR.

Credit

This contribution was produced by Nick Petalas. DaysSky supplies the shell setting and published clique. Its exact JSON bytes are retained unchanged with source commit, path, SHA256 and notice; the pending source head is not a local Git object, so the package contains the authenticated exported input. DreamingOfClouds #446/#595/#2244, Joel Pulikkan #620, #598/#606 and the classical Barnes-Wall, Reed-Muller and Construction-D lineage retain their credit. New material is Apache-2.0.

CI notes

CI blocked upstream at package head bede750d1aaa0e47930a398d4779078f8107370d. The plan job failed in Onboarding helper tests: 408 tests ran in 140.809 seconds, with eight failures and seventeen skips. Package selection did not run and replay was skipped, so hosted CI has not certified this package. The previously recorded local package checks remain the available replay evidence.

The checkout log identifies synthetic merge cfcc2e0130a605f694144e35a2fb5333353f055f, merging this head into main base 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba. The eight failures are:

• In tests/test_research_descendant_source_roles.py, test_historical_blob_and_hash_corruption_still_rejects fails for both git_blob and sha256. It expects respectively source blob provenance mismatch: scripts/research_package.py and source SHA256 provenance mismatch: scripts/research_package.py.
• In the same module, test_historical_history_test_pin_corruption_rejects fails for both fields, expecting the corresponding source blob provenance mismatch: tests/test_research_source_history.py and source SHA256 provenance mismatch: tests/test_research_source_history.py.
• test_live_history_guard_is_not_a_historical_source_substitution expects return code zero and receives one.
• test_original_gate_rejects_but_correct_role_accepts_current_helper expects working source drift: scripts/research_package.py.
• test_original_history_input_role_rejects_live_guard_successor expects working source drift: tests/test_research_source_history.py.

All seven assertions above instead receive exactly working source drift: scripts/check_research_commit.py. The shared fixture uses research/isolated-descendant-bytecode/SOURCES.json, whose live pin for that script is SHA256 ecf206a82bf1f4e748efb6cc639919582050d173b3537ec12f7265a4f7b55ead; the actual script at the CI main base hashes to 28fed1e156fe8601633ed453f682a158cb15e94981e6c89ed96ae72ffc9d0c6f. The unrelated live-input mismatch stops the fixture before its intended historical-source checks.

The eighth failure is tests/test_research_onboarding.py::OverlapTests.test_report_survives_a_cp1252_stdout. Its temporary repository copies only scripts/find_overlap.py; that script now executes from research_package import git_source_env at line 24 and fails with ModuleNotFoundError: No module named 'research_package', returning one instead of zero.

These are shared upstream fixture/dependency failures. Commit 862ac2a0f, confirmed as an ancestor of the CI main base, changes scripts/check_research_commit.py and introduces the find_overlap.py import. The two failing test modules are byte-identical between this contribution's head and that CI base. This contribution changes only its seven new package files; it changes no shared script, test or pinned historical package. These conclusions were checked against source bytes and CI logs; the shared test suite was not rerun locally in this follow-up.

The independently inspected failed logs show exactly the same eight failure names, field variants and assertion/error texts on:

• PR #5117, head eb0c12811253174a6a7834c118e2cf827cc9b706, run 39933925464: 408 tests, eight failures, seventeen skips; replay skipped.
• PR #5116, “Implement complete owned query streams on selected signed records”, head ee4bebf4bb1d441bbbcd43ff4bc2bcb2734a679e, run 39933826517: 408 tests, eight failures, seventeen skips; replay skipped.

No package or shared-file repair is included, no new commit was made, and DONE is withheld pending successful hosted CI. Scratch evidence is retained.

Merge Danger

Door: two-way

Blast Radius: isolated

Seven files in one new opt-in research package; no selected-frontier, shared-code or workflow change.