[petalas]

Summary

Scoped algebraic proof with exact finite corroboration, produced by Nick Petalas.

same complete central class Z=(S+T)^2
    does not imply the same ordered Schur state
first collision: pivot 4, residual 14, ambient 18

In the sharp nilpotent identity-pivot family over every characteristic-two field, the same original profile (4,4,4,6) admits compressed products J3(0)+J1(0) and J2(0)+J2(0). Both central operators have Jordan form J2(0)^4+J1(0)^6, but rank((S+T)^3) is two versus zero. A complete central-class key therefore cannot deduplicate these ordered states. The dimensions are minimal within the stated same-pivot, same-profile nilpotent family.

Credit Malik Safwan / ssafwann #5267 for the nilpotent model and whole-pair rigidity, Ali Romman #5287 for the complete characteristic-two central law, and Mihir Srivastava #5360 for the general ordered gluing boundary. The package retains the wider Schur lineage. These are cited premises; their scientific suites were not replayed as this deliverable. The new contribution is the minimal collision, explicit original headers and ordered-state separator.

CLAIM. Six new files in research/schur-central-alias/; no selected-frontier file changes.

Evidence

• Before: the central law and whole-pair normal forms were available, without this minimal same-profile alias.
• After: a written proof, two literal 18-dimensional original headers and 14-dimensional residual pairs, an explicit invertible central intertwiner, all eleven partitions through pivot size four and all fourteen same-size unordered comparisons. Exactly the displayed partition pair collides. Eight rejection/separation controls pass.
• Warning-strict ordinary and optimized direct checks return PASS_SCHUR_CENTRAL_ALIAS, with identical computed hash-bound receipts. Local durations were 0.0292 and 0.0291 seconds. Both registered commands pass; the package runner took 0.313 seconds and returned PASS_REGISTERED_RESEARCH_PACKAGE.
• Metadata/pins and committed-byte, executable-mode and declared-source-history checks pass at 02f0ffab916042785f0a61ebfc4dacb9495a0ac2. Source compilation and whitespace checks pass. CPython 3.12.15, macOS arm64, standard library only. These durations are check runtimes, not performance claims.

Exact reproduction from the repository root:

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

Other machines may substitute their CPython 3.12 path. The direct verifier writes no files and computes its receipt from the literal evidence and source hashes.

Limitations and unrun obligations

The universal minimum relies on the written proof and cited normal-form rigidity. It is scoped to sharp nilpotent identity pivots at the same pivot size and ordered original profile. It is not a universal minimum for arbitrary idempotent pairs. Independent human and formal proof-kernel review are unclaimed.

No new kappa, physical saving, supplier admission or complete multiplication theorem is claimed. Physical access, coefficient preparation, movement, restoration, precision, stock, dispatch and a complete recursive invoice remain outside scope. All five selected framework assumptions remain. No predecessor scientific suite, selected cold replay, native replay, Lean or Docker was run.

Open next questions

• Specify an actual ordered-state registry and price acquisition of the full power-rank profile of S+T; the central class alone is insufficient.
• Under a separately named nonsharp or mixed-primary contract, find the first analogous alias. This PR holds no follow-on scope.

Related open work

No open PR found on the same sharp nilpotent ordered-state equivalence question. Nearby Schur-rank work uses a different paired-star contract:

• Record a partial F4 census of two thin maps on the three-leaf paired star #7006: partial F4 two-thin-map paired-star census; no ordered-state central equivalence claim.
• Prove one F8 nullity-7 paired-star class has rank sum at least 8. #7028: one nullity-seven paired-star class has rank sum at least eight; different matrices and invariant.
• Prove the next F8 nullity-7 paired-star class has rank sum at least 8. #7049: a second nullity-seven paired-star class; different matrices and invariant.
• Prove one nullity-32 paired-star class has Schur rank at least 4 #7091: a nullity-32 paired-star Schur-rank lower bound; different matrices and invariant.
• Prove the exact odd-rectangle residual-rank floor for any stage size and hub count: 72ρ at u = h = ρ (no κ) #7134: odd-characteristic rectangular residual-rank floors; no nilpotent state-equivalence claim.

Their bodies were read; no extra computations were run to validate their results or compare unrelated numbers.

The closeout check read this account's eight other open PRs created since the CLAIM: #7203, #7204, #7206, #7207, #7208, #7210, #7211 and #7212. None studies sharp nilpotent ordered Schur-state equivalence or computes the central-alias minimum here. No new related-work line or numerical comparison is required.

CI notes

CI did not certify this package. At unchanged head 02f0ffab916042785f0a61ebfc4dacb9495a0ac2, run 43012595145 failed in plan job 130495652283, step "Onboarding helper tests". I read the failed job log. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found

The assertion is at tests/test_research_current_status.py:58. At canonical base e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. I read both base files and confirmed that this PR changes neither file nor the test. This is an existing repository-level documentation mismatch.

The dependent package replay job was skipped because plan failed. The execution evidence for this package remains the local checks above. No fix, retrigger commit, push, job rerun or selected-frontier change was made.

I also read the failed logs of all six comparison runs below. Each has the same sole failing test, missing command and totals of 760 tests, one failure and 24 skipped:

• Run 42905094314.
• Run 42903313790.
• Run 43012497455.
• Run 43012524632.
• Run 43012862401.
• Run 43012902583.

Merge Danger

Door: two-way. Blast Radius: isolated.

Only a new research package and its opt-in replay are added. Existing scientific sources and the selected frontier are unchanged.