[petalas]

Summary

Scoped written algebraic proof: the historical actual 112-port doubled centre Gram admits one monic rational polynomial P with P(A_f)=0 at every positive tensor width and degree(P)<=2^93. Its spectral union is finite and eventually stabilizes.

- PR5168 leaves a uniform higher-width upper-degree bound open
+ exact signed Bell generators lie in a group of order 2^90
+ an eight-row faithful regular action gives one all-width annihilator

The new step is the quadratic signed-translation normal form and its application to this Gram. The bound is coarse, and the huge polynomial is defined by a finite determinant rather than materialized. This is an existence proof with finite corroboration, not a finite certificate of that determinant, a complete multiplication theorem or a new kappa.

The contribution was produced by Nick Petalas. Credit Krzysztof Wojt / #5168 for the one-slot theorem and open question; Ali Romman / #4958, #4720 and #5382 for the query, exact Bell action and two-slot spectrum; Nick Petalas / #5120 for the earlier lower certificate; #1719 and the retained Gauge44/native/compiler lineage. David Ganz / #4885 gives a related finite group-algebra recovery precedent for a different resonant operator. Boolean normal forms, regular representations and Cayley-Hamilton are classical. Predecessor spectral certificates are cited, not replayed as the deliverable.

Evidence

• The complete enclosing normal forms at n=1,2,3 give 4,64,4096 distinct signed permutations. Every one of the 4112 ordered products at n=1,2 agrees with literal signed composition.
• All 12544 ordered actual-port generator pairs agree under independent Boolean interpolation and monomial expansion; 62720 additional product-state checks pass.
• A separate eight-dimensional regular-action example agrees with nine exact determinant evaluations. Its same polynomial annihilates tensor representations of dimensions 8,32,128. Twelve rejection controls pass.
• Warning-strict normal and optimized direct checks pass in 3.188 and 3.057 seconds, respectively, below the 120-second registered limits. Both registered commands pass. Environment: CPython 3.12.15, macOS 27.0.1 arm64, standard library only.
• Metadata/pins, all nine committed package/input bytes and declared source history pass. The official export passes integrity checking. Both read-only Git-free replays pass with PATH=/nonexistent; all ten exported file hashes and modes remain unchanged. Six separate normal/optimized mutation processes reject source-byte corruption, a Boolean-for-integer receipt mutation and omitted composition phases.
• In-memory compilation and the function-size inspection pass. No shared scripts, predecessor files, frozen evidence or selected record change.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/uniform-centre-gram-annihilator
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/uniform-centre-gram-annihilator/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/uniform-centre-gram-annihilator --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/export_research_package.py export --package research/uniform-centre-gram-annihilator --ref HEAD --output /tmp/gram-annihilator.tar
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/export_research_package.py check /tmp/gram-annihilator.tar

Other machines may substitute their CPython 3.12 path. The direct marker is PASS_UNIFORM_CENTRE_GRAM_ANNIHILATOR; the launcher also prints PASS_REGISTERED_RESEARCH_PACKAGE. RESULT.json is generated and bound to all seven other package files and the exact historical query. Export SHA256: 9b87c0d78e90c445f0acac093c246f03b2d87e0372f82df1caf4c292e3a74bc9.

Limitations

The written proof supplies the universal quantifier. Finite checks do not enumerate the 2^90-element group or generate the huge regular characteristic polynomial. The actual generated subgroup, least annihilator, exact stabilized spectrum and stabilization width are undetermined. No practical decoder, physical supplier, native access, coefficient preparation, rounding/precision, ownership/restoration, tape/storage or complete invoice is supplied. The historical 112-port theorem discharges none of the current selected framework assumptions.

Independent human and formal proof-kernel review are unclaimed. No predecessor scientific suite, selected cold replay, Lean or Docker run was performed. Selected conditional kappa remains .003125804.

Open next questions

• Determine the actual generated subgroup and its irreducible structure, starting with the translation and central-phase subspaces.
• Determine which spectral components occur at each width, including unequal Bell labels.
• Price one concrete decoder with its complete source/target, preparation, access, restoration, precision and storage contract.

Related open work

The account's open PRs created since the advance CLAIM were refreshed before closeout. None studies this historical actual 112-port doubled centre Gram or its scalar minimal-polynomial degree, so there is no newly computed shared number to compare. The earlier live open-PR searches also found no other open PR on that object. Merged #5168 and #5382 remain the direct predecessors; their exact degrees are cited context and are not independently rechecked here. Unpublished work and arbitrary diffs are outside this coverage.

CI notes

Hosted CI did not certify this package. At head d09031c5c65f2e654a9fa4c91bfd2f8712b9ec5e, run 41106617094 failed its plan job. The only error was:

test_research_posix_sigterm.TerminationTests.test_registered_replay_terminates_owned_tree
subprocess.TimeoutExpired: ... /tmp/posix-term-c3qjhpm2/driver.py package signal ... timed out after 10 seconds
Ran 710 tests in 153.871s
FAILED (errors=1, skipped=24)

The traceback points to tests/test_research_posix_sigterm.py:202, process.wait(timeout=10), called by self.exercise('package') at line 234. I confirmed the cause from the failed log and inspected the test: it constructs its own temporary driver, synthetic test module and solver using the shared runner scripts. It does not load this research package. This unrelated harness timeout is outside the eight-file package addition; no test, runner or package change was made to work around it, and no push or rerun was requested. Neighboring completed workflow runs 41106545148 and 41106463337 succeeded; no matching failure appeared among the completed recent runs inspected.

The dependent package replay job was skipped as a consequence. All declared commands passed locally, including normal/optimized direct and registered replay, source-history/committed-byte checks, export integrity, read-only Git-free replay and mutation controls. Those local checks remain the available execution evidence. No hosted package pass or scientific acceptance is claimed.

Merge danger

Door: two-way. Blast radius: one isolated research package. Eight new files; no shared implementation or frontier changes.