[krzysztofwojt]

Change and mathematical scope

The full one-slot doubled centre Gram A_1 has minimal-polynomial degree exactly 881, in characteristic zero, for the pinned actual eight-by-112 query. Every invertible alpha I + beta A_1, with beta != 0, has a scalar-polynomial inverse of minimum degree 880. The common fixed Bell vector transfers the corresponding lower bounds to every positive tensor width; no higher-width upper bound is claimed.

This closes the one-slot annihilator question left by Ali Romman's #4958 (lower bound 409) and Nick Petalas's #5120 (lower bound 410). It adds a complete upper identity and matching lower certificate. The 2,097,152-dimensional operator is covered by six Bell-label orbits; two central involutions give twenty 1024-dimensional sectors plus the directly checked zero block. The new package provides a written proof, all 21 integer component polynomials, a verifier independent of the optional FLINT producer, and inspectable deterministic receipts.

Status: scoped computational proof, author-checked. No proof-kernel or independent human acceptance, general ADD-circuit lower bound, physical decoder, supplier admission, new kappa or selected-frontier change. Access, storage, routing, precision, restoration and outer admission remain separate obligations.

Task and overlap

• Canonical claim: #5142. Handoffs Handoffs and requests for independent checks #1166 were read first. Priority revision 7; formal-interface-lemma scope explicitly changed to the open finite exact decoder-algebra question, recorded with the original card in TASK.json.
• Fresh-clone scientific base and publication parent: a5df48c55ea806d7952b65acc9c76d878ff09b13. Contribution head: 51c5d48abd7de681265d009518ca0af59a0f6648.
• Exact predecessors: Prove centre Gram degree and least invariant scalar-cache bounds #4958 c8c48ef5a497ffd7f7e778434c1f074b7b777a03; Prove doubled centre Gram degree at least 410 (no new κ) #5120 34f5c3aa8c9df2f9f1c587c6bca0c4ea7af0c36e. Query SHA256 5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3. Source bytes, Git blobs and retained attribution inputs are pinned in SOURCES.json.
• The complete 11:25–11:26 UTC refresh covered 327 open PRs, 4,337 Discussion records, and 6,918 committed main-tree notes. All ten recent krzysztofwojt threads were separately checked; the A5, norm-six, Schur, reflection and earlier claimed topics are excluded. Canonical main remained unchanged at the final fetch; the newest 40 PRs, current claim replies and Handoffs and requests for independent checks #1166 were rechecked at 11:33–11:35 UTC immediately before submission. Arbitrary diffs and unpublished work are outside this coverage.
• JoyboyBrian's concurrent feasibility handoff independently reports the same six-label reduction. It explicitly releases the overlap with no polynomial, exact degree or PR, and defers to this claim. It was read after this derivation and receives concurrent credit; its exploratory scripts are not imported.

Merge danger: low in both directions. Only ten new files under research/exact-one-slot-gram/; no shared code, frozen inputs, dependency infrastructure or selected records change.

Validation

Clean Python 3.12.12 / macOS arm64 environment with numpy==2.4.1, sympy==1.14.0, and no FLINT:

python3.12 -B scripts/research_package.py --package research/exact-one-slot-gram
python3.12 -B scripts/check_research_commit.py --package research/exact-one-slot-gram --check-source-history
python3.12 -B scripts/export_research_package.py export --package research/exact-one-slot-gram --ref HEAD --output /tmp/exact-one-slot-gram.tar
python3.12 -B scripts/export_research_package.py check /tmp/exact-one-slot-gram.tar

• Registered normal and optimized (-O), isolated, warning-strict checks pass in 115.49 and 113.48 seconds, with identical receipts. The later completed metadata/pin check also passes.
• Every source-derived symmetry and the complete 512-label cover are checked. All 286,720 compressed port columns agree with literal two-term Bell embeddings. Arbitrary-precision Horner identities cover all twenty sectors through 324 complete basis-orbit representatives and every entry of all 512 zero-block matrices.
• 21 nonzero modular Hankel determinants, each computed by both Gaussian elimination and modular Bareiss condensation, match the exact upper degrees. The global rational and modular LCMs have degree 881. Ten failure controls pass, including deleted ports, changed phases, changed polynomials and determinant edge cases. Integer-overflow bounds are explicit in PROOF.md.
• Committed bytes and declared source ancestry pass for 19 files. Official export/integrity checks pass. Export SHA256: f928463a5a96d3fca638adcd7a3248d8b1766361ffddc1c1e355735296751358.
• Both direct verifier modes also pass from the fresh read-only, Git-free export, in 112.84 and 116.45 seconds. All twenty exported file hashes remain unchanged; both receipts equal the committed receipt. No Git or predecessor executable is available in PATH.
• git diff --check passes. No historical heavy suite or selected mathematical cold replay was run; selected source-binding orientation passed. Hosted CI run 39947253255 passes at the exact contribution head: both planning/shared tests and the registered package replay, including committed-input integrity, are successful. CI remains separate from mathematical review.

Credit

Ali Romman (#4958/#4720) supplies the query, Bell reduction and earlier obstruction; Nick Petalas (#5120) supplies the two-block strengthening. #1719 supplies the doubled representation. JoyboyBrian's concurrent reduction is credited above. The inherited source/compiler lineage, including icekylinx's Gauge44 work, remains credited in unchanged notices and license files included among export inputs.

Prepared by Krzysztof Wojt with substantial OpenAI Codex assistance. Bounded Luna support supplied read-only claim/symmetry inventories and consistency review; the primary agent regenerated the used maps/covers and owns the proof and integration. These are AI research roles, not independent human reviewers. New work is Apache-2.0.

Open next questions

The README and TASK.json record two unclaimed directions: derive a complete joint Bell-label classification at width two to decide new spectra and possible upper bounds; or supply a concrete decoder outside scalar polynomials with a full physical cost account. A higher-width complete similarity classification and an actual priced caller, respectively, are the blockers. Neither is settled by another small Krylov lower-bound increment.