[aliromman]

The historical centre Gram operator has new spectrum at two doubled slots. The complete equal-Bell subspace has exact minimal-polynomial degree 255, adding 212 degrees absent from the one-slot polynomial. Therefore every tensor width at least two has degree at least 1093, and every defined nonconstant affine scalar-polynomial inverse needs degree at least 1092. A concrete new eigenvalue is 8, absent at width one. This is a scoped algebraic result; the complete kappa calculation is unchanged.

The proof, exact polynomial receipt, verifier and source pins define the claim. A second Bell transform reduces every equal-label block to 8-by-8 signed root sums, with the root coefficient applied once. Source-checked symmetries cover all 512 equal outer labels through six representatives. Exact trace recurrences cover all 190239 compressed matrix pairs; independent literal port sums and symbolic determinants check all 886 distinct characteristic polynomials. Symmetry implies that each square-free characteristic polynomial is the exact block minimal polynomial. Their LCM is the degree-255 polynomial.

This answers the equal-label part of PR5168's released higher-width handoff. Its complete one-slot degree-881 theorem is inherited; its source model, certificate, polynomial LCM and hash are checked here. PR5120's earlier lower certificate is not replayed. The exact base is 1ec1119972c9aefe42525172d07b332fcff81cb4; the consumed one-slot contribution is 51c5d48abd7de681265d009518ca0af59a0f6648. The actual query digest remains 5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3.

With Python 3.12, numpy==2.4.1 and sympy==1.14.0, run:

python3.12 -B scripts/research_package.py --package research/equal-bell-two-slot-spectrum
python3.12 -I -W error -B research/equal-bell-two-slot-spectrum/verify.py
python3.12 -I -O -W error -B research/equal-bell-two-slot-spectrum/verify.py

Expected status: PASS_EQUAL_BELL_TWO_SLOT_SPECTRUM; the registered runner also prints PASS_REGISTERED_RESEARCH_PACKAGE. Normal and optimized registered checks pass on CPython 3.12.13, including 286720 literal Bell port/address columns and nine negative controls. Signed-integer overflow is excluded by a checked bound. The official 14-file export, committed-byte checks, Git-free optimized replay, source/typed-receipt corruption controls and a reversed-sign implementation control pass. The clean verifier environment contains no FLINT; the separate discovery calculation used python-flint 0.8.0. The selected source-binding baseline passes separately.

The exact degree 255 is an upper and lower result only on the equal-label subspace. The full width-two degree is unknown because unequal labels remain unclassified. The bound 1093 at larger widths is a lower bound, not an annihilator. The result does not bound general ADD circuits, rational or matrix-valued decoders. Native access, precision, storage, movement, dirty restoration and complete supplier prices remain open. Full predecessor scientific replay, full selected native replay, Lean, Docker and full-history verification were not run. No independent human or kernel acceptance is claimed.

Credit Krzysztof Wojt for the one-slot theorem and source symmetries; Nick Petalas for the preceding exact lower certificate; Ali Romman's PR4958 and PR4720 for the query and Bell/fixed-vector identities; and PR1719 for the doubled representation. JoyboyBrian's released symmetry findings and icekylinx's Gauge44/native/helper/compiler lineage retain credit in the proof and unchanged source notice. New reduction, classification and higher-width implication: Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.