[petalas]

Summary

Scoped written proof with exact finite corroboration. This answers Open next question 1 of SHIN09-git's #6708: cancellation can produce a three-block, two-ports-per-block signature absent from every generic support pattern.

Six source masks: ABC, ABC, ABC, ABC, ABC, ABC
Special reader masks: BC, BC, AC, AC, AB, AB
Generic full-reader floor: at least 3; special witness: 0

The integer basis and inverse are unimodular, with entries in {0,1,-1}, so the same signature works over every field, including F2. The new generic theorem is sharp: on b >= 2 nonempty blocks, full source masks force at least b full reader masks. A determinant-matching graph and a source strongly connected component prove the exclusion for every support, not just this witness's support.

Only research/three-block-cancellation-signature/ is added. The new proof, construction and verifier were produced by Nick Petalas. Credit SHIN09-git #6708 for the question and two-block classification, Gonzalo Mier #3990/#3997 and Nick Petalas #6691 for the preceding source/reader model and screens. Structural determinant/cofactor matchings and the inverse-path description are classical. The predecessor's scientific verifier is not imported or replayed.

Evidence

Before: the cited package leaves three-block cancellation outside the generic census open. After: an explicit six-port integer inverse pair and a universal generic exclusion settle that question.

Local CPython 3.12.15, macOS arm64, standard library only:

• Both integer inverse products and separate rational elimination pass. The signature is checked over F2, F3, F5, F7 and F11.
• All 59,131 full-source supports on (1,1,1), (2,1,1), (2,2,1) are enumerated. The 58,081 structurally invertible supports pass separate cofactor-matching and graph-reachability checks. Two six-port patterns check sharpness and specialization.
• All 27 nonzero-entry omissions and five invalid-input controls are rejected.
• Both registered commands pass and produce identical generated receipts in direct normal/optimized runs. Direct times were 1.60 s and 1.47 s; registered replay took 3.33 s total. Every check is below its 120 s timeout.
• Metadata/source pins, committed bytes/modes, source-history ancestry and whitespace checks pass. The selected source-binding orientation passes; no selected scientific suite is replayed.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/three-block-cancellation-signature
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/three-block-cancellation-signature/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/three-block-cancellation-signature/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/three-block-cancellation-signature --check-source-history --check-modes

A different CPython 3.12 path may replace the local path. Expected markers are PASS_REGISTERED_RESEARCH_PACKAGE, PASS_THREE_BLOCK_CANCELLATION_SIGNATURE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The all-field witness and all-size generic bound use the written proof; the checks are finite corroborations, not an exhaustive six-port census. No complete three-block classification, physical decoder price, native schedule, supplier admission, new kappa or multiplication theorem is claimed. Preparation, delivery, dirty restoration, routing, precision, stock, compiler closure and the full signed invoice remain obligations. No Lean, Docker, independent human or formal proof-kernel review is claimed. The selected #7000 record and its five assumptions are unchanged.

Open next questions

1. Classify further three-block signatures absent from the generic census. Start with inverse bases having zero diagonal blocks.
2. Test whether a complete physical source/reader schedule for these matrices wins after every preparation, delivery and restoration charge. No favorable invoice is supplied here.

Related open work

No open PR studying this three-block square-basis signature quantity was found in the live title/body and claim checks. Merged #6708 is the direct predecessor. #7053's three internal paired-star blocks are a different object and quantity.

Advance claim and scope. Canonical main was refreshed before publication at e54523517d0f03e58352e647c403a5af3e525a57; priority revision 8. Explicit main-tree and merged-PR searches supplement the sparse working-note overlap inventory; arbitrary historical diffs and unpublished work remain coverage limits.

The completion check examined this account's eight other open PRs created since the CLAIM: #7179, #7180, #7181, #7182, #7183, #7184, #7186 and #7187. They study formal A2 tariffs, Gaussian inverse norms, Clifford response sectors, local-algebra lifts, radical directed-cycle tariffs, rational Hurwitz orbits and sampler sharing. None studies three-block square-basis source/reader signatures or computes a competing value. No additional scientific comparisons were run.

CI notes

CI did not certify this package. At exact head ede571e831278ee9752070f8e301350ddb57d9ee, run 42993082934, plan job 130430418440, failed in Onboarding helper tests on exactly one of 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

The job log includes the stale current section of docs/ci-verification.md, which still names #6030 and make frontier6030-verify. Direct inspection of base main e54523517d0f03e58352e647c403a5af3e525a57 confirms that certificates/selected-result.json already names make frontier7000-verify. The test, document and selected record are byte-identical between that base and this PR. The mismatch predates this package.

The same exact failing test, missing command, 760-test total, one failure and 24 skips were independently confirmed in the failed-step logs of all nine comparison runs:

• Other contributors: 42905094314 and 42903313790.
• Prove sharp multiplicative-shell growth for every rank alphabet (no κ) #7178: 42992088329.
• Prove five A2 targets need five rank-three calls (no κ) #7179: 42992340683.
• Prove the sharp half-shift Gaussian inverse lower envelope (no κ) #7180: 42992579989.
• Exclude mixed cubic and quintic Clifford response sectors (no κ) #7181: 42992679183.
• Classify exact two-versus-three costs for diagonal cube-zero lifts (no κ) #7182: 42992720936.
• Prove exact independent-radical directed-cycle tariff 206 (no κ) #7183: 42992841885.
• Classify rational saturated-header Hurwitz orbits (no κ) #7184: 42993026387.

The package replay job 130432261749 was skipped because planning failed. The local normal/optimized, registered-package and committed-byte/source-history checks above remain the available automated package evidence. No fix, retrigger commit, push or workflow rerun was made for this repository-level failure. Maintainers will repair main. Mathematical acceptance remains separate from replay.

Merge Danger

Door: two-way. Blast radius: isolated. Seven new package files; no shared implementation or selected-frontier edits.