[petalas]

Summary

Scoped written proof with exact finite corroboration for the pinned historical eight-by-112 doubled centre Gram.

- #6302: uniform degree bound 2^93; eventual stabilization without a width
+ actual generated Bell group has order 2^44
+ uniform rational annihilator has degree at most 2^47
+ the minimal polynomial is constant at every tensor width at least 10

The root coefficient occurs once, outside the tensor power. The stable polynomial is the minimal polynomial of the eight-row regular-action operator. The exact stable degree and width-two polynomial remain unknown. This is an algebraic side result, with no new kappa, supplier admission, physical decoder or selected-frontier change.

The original CLAIM included uniform existence. The scope update acknowledges merged #6302 and narrows novelty to the quantitative results above. Scientific base and refreshed canonical main are e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8.

This contribution was produced by Nick Petalas. Credit Krzysztof Wojt #5168 for the operator, one-slot degree 881 and original question; Ali Romman #4958/#4720/#5382 for the query, Bell identity and two-slot lower bound 1093; Nick Petalas #6302/#5120 for the earlier uniform and lower bounds; #1719 for the doubled representation; and Ali Romman's Clifford tensor Gram work for the finite-group and Burnside-Brauer viewpoint. The representation-theory ingredients are classical and cited in the proof. Predecessor spectral certificates are cited premises, not replayed as this deliverable.

Evidence

• Before: the enclosing group in Prove a uniform all-width annihilator for the doubled Gram (no κ) #6302 has order 2^90 and supplies no stabilization width. After: the actual group's label image has binary dimension 36 and its sign kernel has dimension 8. A faithful regular action bounds the degree by 8 times 2^44. At most eleven character values, the cited Burnside-Brauer theorem and the fixed Bell vector give stabilization by width ten.
• The finite verifier authenticates the query and computes both dimensions. Literal integer 512-by-512 Bell actions agree on 1,792 sampled label pairs over all 112 generators, covering both sign cases. The written identity handles all label pairs.
• Full enumeration of the two toy groups gives orders 64 and 16. Their kernels agree with the normal-closure algorithm, and every element satisfies the trace lemma. Four rejection controls pass. This does not enumerate the actual 2^44-element group or construct its huge regular matrix.
• Warning-strict normal and optimized direct runs pass with identical output. Each registered check runs well below its 120-second limit. Metadata/pins and the selected source-binding orientation pass. Functions are at most 42 lines, and in-memory compilation and diff whitespace checks pass.
• Committed-byte and declared source-history checks pass for all eleven package/input files at head 362a05a889ffc1ae923812f298a4186972be731e. The official export passes integrity checking. Both verifier modes also pass in a read-only Git-free export with PATH=/nonexistent, with all twelve exported file hashes unchanged. The verifier uses only the standard library and writes nothing. Independent human and formal proof-kernel review are unclaimed.

Local environment: 3.12.15 (main, Oct 14 2026, 00:55:37) [Clang 23.1.3 ], macOS-27.0.1-arm64-arm-64bit, standard library only.

Exact reproduction from the repository root:

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

Other machines may substitute their CPython 3.12 path. Direct marker: PASS_CENTRE_GRAM_WIDTH_STABILIZATION. Registered marker: PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations

The bound ten is sufficient, not claimed minimal. The stable polynomial and its degree are not computed; the universal quantifiers rest on the written proof and the cited classical theorem. The lower bounds 881 and 1093 are inherited, not new counts.

Open next questions

• Determine which irreducibles occur at widths two through nine and compute the corresponding minimal polynomials. Start with the sign kernel and label-image extension; projective representation blocks and their exact spectra remain unresolved.
• Lower the stabilization-width bound by determining which trace values actually occur. The 36-dimensional binary matrix span makes direct enumeration too large.
• Bound the stable degree more sharply, or exhibit eigenvalues beyond the inherited lower bound. One small representation block could provide a bounded first step.

No predecessor scientific replay, selected cold replay, Lean or Docker run was performed. Physical access, coefficient preparation, dirty restoration, storage, routing, precision, compiler closure, complete supplier cost and outer multiplication remain outside this result.

Related open work

The live titles and bodies were read. These PRs study subspaces of the same historical one-slot Gram; their numbers were not independently recomputed here.

• Cut one moved 48-eigenspace by a 12-dimensional A_1 subspace #7082 cuts a moved 48-dimensional eigenspace by a 12-dimensional invariant subspace; this PR treats all tensor widths through the generated group.
• Prove odd signs on u keep the X_3 trace -48 #7069 studies odd-sign invariance of an X_3 trace, rather than tensor-width stabilization.
• Record the characteristic polynomial of the 84-dimensional X_3=+1 subspace #7068 computes a characteristic polynomial on an 84-dimensional X_3 subspace.
• Record geometric multiplicity 24 for eigenvalue 0 on the other three 216 joint sign spaces #7066 computes zero-eigenvalue multiplicities on three other 216-dimensional joint spaces.
• Record the characteristic polynomial of A_1 on the 128-dimensional dihedral isotypic component #7065 computes a characteristic polynomial on a 128-dimensional dihedral component.
• Record geometric multiplicity 12 for eigenvalue 2 on the pinned 216-by-216 matrix #7064 computes the eigenvalue-two multiplicity on a pinned 216-dimensional joint space.
• Record the characteristic polynomial of A_1 on the 80-dimensional isotypic summand #7061 computes the characteristic polynomial on an 80-dimensional isotypic summand.
• Record that c=1 movers miss the 8-dimensional dihedral summand. #7059 studies how c=1 movers act on an eight-dimensional dihedral summand.
• Record geometric multiplicity 10 for eigenvalue 0 on the pinned 48-by-48 matrix #7057 computes the zero-eigenvalue multiplicity on a pinned 48-dimensional matrix.
• Record the characteristic polynomial of one 48-dimensional eigenspace of s #7036 computes a characteristic polynomial on a 48-dimensional moved eigenspace.
• Record that the 8-dimensional c=1 isotypic summand is reducible under… #7023 proves reducibility of an eight-dimensional c=1 isotypic summand.
• Record the c=1 stabilizer action outside the {g,t} normalizer. #7022 studies the setwise stabilizer of one 216-dimensional space outside its Klein-group normalizer.
• Prove degree-881 payload involutions split a sector-3 joint space #7020 studies payload-involution traces and dimensions in sector c=3.
• Record the characteristic polynomial of one 8-dimensional c=1 joint eigenspace #7019 computes characteristic polynomials on eight-dimensional c=1 joint eigenspaces.
• Record geometric multiplicity 24 for eigenvalue 0 on the pinned 216-by-216 matrix #7018 computes the zero-eigenvalue multiplicity on a pinned 216-dimensional matrix.
• Split the c=1 sign characters that g and t move #7009 splits c=1 sign characters moved by two predecessor involutions.
• Prove the c=1 ++ 216-space is reducible under its signed normalizer. #7008 proves reducibility of the c=1 joint 216-space under its signed normalizer.

The closeout check read this account's open PRs created since the CLAIM, including the following newer work. None computes this doubled Gram's generated group, tensor-width spectrum or minimal-polynomial stabilization; no shared numerical value is asserted.

• Lower P104 frame-0 hosting from 40 to 14 monomials (no κ) #7167 studies ANF polynomial hosting charges for P104 frame indicators.
• Prove the exact E8 Gram-class minimum through 24 deletions (no κ) #7168 minimizes a scalar E8 Gram-design support-pairing cost, a different operator and quantity.
• Exhibit a non-coordinate COPY71 completion at the first-failure boundary #7169 constructs masked and all-pairs COPY frames on one 93-line E8 unit.
• Bound scalar COPY savings at one 93-line E8 puncture (no κ) #7170 bounds scalar COPY perturbations on that 27-deletion boundary.
• Prove a uniform 13q lower bound for five commuting stages (no κ) #7173 proves a total-rank floor for five commuting stages in a shortest query root.
• Exclude the proposed 96-line E8 complement at inner dimension 8k (no κ) #7174 excludes a fitting block-mask realization of one 96-line E8 complement.
• Certify fixed-profile helper-split first-crossing ceilings for #7000 (no κ) #7175 bounds helper-split first crossings of the fixed Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 invoice.
• Prove sharp c=6 endpoint chain windows (no κ) #7176 proves endpoint windows for six-source characteristic-six diagonal chains.

The earlier #7132, #7133, #7134 and #7135 also study different objects or quantities. The overlap helper covered 466 open PR records and 7,089 Discussion records with complete remote pagination. Sparse local coverage was incomplete; separate Git-tree searches of main found #6302 and checked the sharper bound, group order and stabilization terms. Arbitrary diffs and unpublished work are outside this coverage.

CI notes

CI did not certify this package. At unchanged head 362a05a889ffc1ae923812f298a4186972be731e, run 42984145008 failed its plan job 130401216661 in "Onboarding helper tests". The log reports one failure among 760 tests, with 24 skipped. The exact failing test is:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

At tests/test_research_current_status.py:58, the assertion expects `make frontier7000-verify` from the selected record in the current CI documentation section. The AssertionError says that string was not found. The logged section still describes #6030 and make frontier6030-verify. I also read both files at base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json requires make frontier7000-verify, while docs/ci-verification.md describes #6030. Neither file is changed by this six-file package addition.

The dependent package replay job was skipped. No hosted package pass or scientific acceptance is claimed. The local registered, normal/optimized direct, metadata/pin, committed-byte/source-history, export-integrity and read-only Git-free checks remain the execution evidence at this head.

I independently read the failed logs of these comparison runs. Each reports the same sole failing test among 760 tests, with 24 skipped:

• Run 42905094314.
• Run 42903313790.
• #7167, run 42983769381.
• #7168, run 42983916692.
• #7169, run 42983930022.
• #7170, run 42984005087.
• #7173, run 42984180909.

The earlier failed logs of #7132 and #7135 had the same failure. No repository-level fix, retrigger commit or job rerun was made.

Merge danger

Door: two-way. Blast radius: one isolated research package, six new files. No shared implementation, frozen predecessor bytes or selected record changes.