[petalas]

Summary

A new finite exact certificate and scoped all-positive-tensor-size proof answer Open next question 1 of Ali Romman's #4958 negatively: the full historical doubled centre Gram cannot have minimal-polynomial degree exactly 409.

- scalar minimal-polynomial degree >= 409
+ scalar minimal-polynomial degree >= 410
- necessary scalar-polynomial inverse degree >= 408
+ necessary scalar-polynomial inverse degree >= 409

The new certificate uses the invariant direct sum of Bell blocks c=1 and c=11. It uses new seeds and does not import or rerun the predecessor's 409 certificate. The inverse statement assumes alpha I+beta A is invertible and beta is nonzero. Only research/centre-gram-degree410/ is added.

Produced by Nick Petalas. Credit Ali Romman #4958 at c8c48ef for the operator, Bell identities, invariant tensor embedding and open question; #4720 for the retained actual query; #1719 for the doubled representation. The unchanged historical notices retain icekylinx, Gauge44 and the native/helper/compiler/upstream lineage. Krylov/Hankel minors and modular nonvanishing are standard methods.

Evidence

The new 410-by-410 moment minor has determinant 58805 modulo 65537. Both operator implementations agree on all 1,638 block Krylov vectors, and modular Gaussian elimination and integer fraction-free condensation agree on the determinant. The receipt regenerates all 819 direct-sum moments, both component lists and seven source hashes. Eight dependence, pivoting, source and truncation controls pass.

On CPython 3.12.15 / NumPy 2.5.4 / macOS arm64, both warning-strict registered modes pass. Direct normal/optimized checks took 6.70/7.04 seconds, below each 180-second limit. Metadata, committed-byte/full-source-history preflight, official seven-file export integrity and read-only Git-free normal/optimized replays pass. All seven exported payload files retain the source-bound receipt hashes. The selected source-binding orientation also passes; it is metadata orientation only.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/centre-gram-degree410/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/centre-gram-degree410/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/centre-gram-degree410
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/centre-gram-degree410 --check-source-history

Any equivalent Python 3.12 installation with the declared NumPy pin can replace that local path. Direct output is PASS_CENTRE_GRAM_DEGREE410 degree>=410 inverse_degree>=409 det=58805 mod=65537. Add --emit-result to the direct command for the generated JSON receipt; write it outside the checkout.

Limitations

The exact full degree and an upper annihilator remain unknown. The all-size deduction uses #4958's cited Bell identities and common fixed vector. General ADD circuits, rational expressions, matrix-valued encodings, native access, rounding, dirty restoration, physical movement, recursive prices and complete supplier admission are outside this scalar-polynomial result. No new kappa or selected-frontier change is claimed.

All registered checks ran locally. Lean, Docker, predecessor scientific replays and the selected cold replay were not run. Independent human and formal proof-kernel review are unclaimed. The finite certificate and written scope are distinct from a complete multiplication theorem.

Open next questions

• Derive a characteristic-zero annihilator from the full Bell-block decomposition to determine the exact degree. The modular minor gives only a lower bound.
• Specify a decoder outside scalar polynomials with its actual source, output, dirty-restoration and complete paid contract. Those implementation costs remain unknown.

Related open work

No other open PR studying this doubled Gram's scalar minimal-polynomial degree was found in the pre-commit refresh. Merged #4958 is the direct predecessor; #4720's source-span bounds and #1719's lift have different claims. The complete remote overlap snapshot covers 323 open PRs and 4,243 Discussion records; sparse local coverage was supplemented with all-main README and merged-PR searches. Arbitrary diffs and unpublished work are not exhaustively covered.

Canonical advance claim. Scientific base 81e9bd6, priority revision 7; canonical main was refetched unchanged before submission. TASK.json declares the changed formal-interface-lemma scope.

CI notes

CI blocked upstream for head 34f5c3aa8c9df2f9f1c587c6bca0c4ea7af0c36e. The plan job failed in Onboarding helper tests; package selection and the hosted package replay were skipped. The log reports 408 tests, eight failures and 17 skips. The checked-out merge was b616414606fc3c6cb811b4cc955b9bf4e12b143f, merging this head into upstream 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba.

The eight failure entries have two shared causes, confirmed from the failed logs and exact merge-tree files:

• Seven entries in test_research_descendant_source_roles.DescendantSourceRoleTests stop on working source drift: scripts/check_research_commit.py. The historical research/isolated-descendant-bytecode/SOURCES.json still marks that checker as a live input with SHA256 ecf206a82bf1f4e748efb6cc639919582050d173b3537ec12f7265a4f7b55ead; the merge-tree checker has SHA256 28fed1e156fe8601633ed453f682a158cb15e94981e6c89ed96ae72ffc9d0c6f. This earlier rejection prevents the intended helper/history corruption or acceptance checks. The seven entries comprise both git_blob/sha256 subtests for each of the two historical corruption tests, plus test_live_history_guard_is_not_a_historical_source_substitution, test_original_gate_rejects_but_correct_role_accepts_current_helper and test_original_history_input_role_rejects_live_guard_successor.
• test_research_onboarding.OverlapTests.test_report_survives_a_cp1252_stdout fails with ModuleNotFoundError: No module named 'research_package'. Its temporary fixture copies only scripts/find_overlap.py at test line 39; that script now imports git_source_env from research_package at line 24, and the copied fixture lacks the module.

I compared all eight failure names/subtest labels and terminal error signatures with these other failed plan logs; all eight signatures agree exactly:

• PR #5116: plan job.
• PR #5117: plan job.
• PR #5118: plan job.

Their changed-file lists, like this PR's six-file package addition, contain no shared script or test changes. This is a shared upstream fixture breakage, not a failure of the new Gram certificate. No shared scripts/tests were modified, no new commit was pushed, and no DONE was posted. The previously passed local registered normal/optimized, source-history and export checks remain the available package execution evidence; hosted replay has not certified the package. Scratch logs and exports are retained for the upstream repair and rerun.

Merge danger

Door: two-way. Blast radius: isolated research package. Shared code, predecessor sources, frozen evidence and the selected frontier are unchanged.