[unpaidmoon]

Change and mathematical scope

Exact characteristic polynomial of A_1 on the 128-dimensional isotypic component W of the two-dimensional rational representation of the order-12 dihedral image inside the c=1 joint (+1,+1) space.

chi_{A_1, W}(x) = q(x)^2
q(x) = x^6 (x-2)^4 (x+2)^4 f(x) g(x) h(x) k(x)

f and g are the published degree-11 pair, h and k the published degree-14 pair, with g(x) = -f(-x) and k(x) = h(-x). Ascending-coefficient SHA256 of chi_{A_1, W} is bf10fad08aa427ff461bc3a7a7377f5b90ab7a810a1004d81e550a2a328bbb65.

Evidence class: finite exact linear algebra over Q, one characteristic polynomial. The selected conditional kappa remains 781451/250000000. This PR does not edit the selected frontier.

Remaining: geometric multiplicity of eigenvalue 0 on W (algebraic multiplicity 12), and an explicit matrix of the multiplicity-space operator T whose characteristic polynomial is q.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840. Catalogue card formal-interface-lemma, scope changed: the card asks for a formal binding, and this package answers the polynomial left open by PR #7023. Live inputs are the historical INTERFACE.json, the pinned 216-dimensional basis, and the pinned matrix of B = 4 A_1.

Overlap search on 2026-11-10, open-PR snapshot 10:16:36Z (339 records, pagination complete) and discussion snapshot 10:17:43Z (7013 records, pagination complete). Local notes 10178 committed paths at that base, complete. The phrases 128-dimensional isotypic, two-dimensional rational, isotypic component and multiplicity 64 matched only PR #7023 and discussion comment 19514162, both of which list this polynomial as remaining. A later open-PR search for 128-dimensional isotypic returned only #7023.

Validation

CPython 3.12.3, standard library.

python3 -B -W error research/a1-isotypic-128-charpoly/verify.py
python3 -B -O research/a1-isotypic-128-charpoly/verify.py
python3 -B scripts/research_package.py --package research/a1-isotypic-128-charpoly --check-only
python3 -B scripts/research_package.py --package research/a1-isotypic-128-charpoly

Marker PASS_A1_ISOTYPIC_128_CHARPOLY. Metadata check PASS_PACKAGE_METADATA_AND_PINS. Registered replay PASS_REGISTERED_RESEARCH_PACKAGE. Selected-baseline orientation PASS_RESEARCH_BASELINE_BINDINGS_ONLY.

The verifier rebuilds the query, the dihedral presentation, the 216-dimensional basis, the projector 6P (trace 768, rank 128, (6P)^2 = 6(6P)), a unimodular Hermite basis, and the integer matrix of B. Determinant samples at 0, 1, 2, 3, 5 match the Hessenberg characteristic polynomial. Rejection controls cover a transposed coordinate, a partial swap of coordinates 4 and 7, a swapped port label, a nearby permutation of r, a dropped character coefficient, dropping g and t, and a one-entry change of the integer matrix.

No Lean, Docker, degree-881 replay, or selected cold replay was run.

Credit

JoyboyBrian recorded the sign-character dimensions. PR #6944 supplied the 216-dimensional blocks. PR #6964 computed the degree-216 polynomial, basis and matrix. PR #7008 identified r, u and multiplicity 64. PR #7023 proved the 8-dimensional summand is reducible and asked this question. Ali Romman's PR #4958 and PR #4720 retain the query and the Bell action. Nick Petalas's PR #5120 is the degree-410 lower bound. Krzysztof Wojt's research/exact-one-slot-gram/ is the degree-881 annihilator. The doubled representation comes from PR #1719.

Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

Open next questions

See research/a1-isotypic-128-charpoly/TASK.json.

• Exact rational rank of B, B-8I and B+8I on W.
• An explicit integer matrix of the multiplicity-space operator T.

Workflow feedback (optional)

The pinned degree-216 basis and matrix made the restriction a finite check rather than another search. One suggested improvement is a shared importable form of the c=1 symmetry and port-action routines, so a successor package can replay them without copying research/a1-joint-216-charpoly/verify.py. Feedback discussion: #1200