[unpaidmoon]

Change and mathematical scope

Exhaustive finite census for the 8-dimensional dihedral isotypic summand inside the trivial-character (+1,+1) 216-space of the historical doubled centre block at c=1.

The c=1 stabilizer has order 3072. Exactly 768 elements preserve that 216-space, and 2304 move it. For every mover s, the image of the summand V is orthogonal to V, and s(V) meets the 216-space only at zero. In particular no mover preserves V, and none preserves V while acting non-scalarly on it.

One explicit mover is the signed permutation whose coordinate map k ↦ inv[k] is (0 3)(5 6), with every output sign positive. All eight basis images miss the 216-space, and they stay linearly independent modulo that space.

No new kappa. The selected conditional kappa stays 781451/250000000.

Evidence class: exhaustive finite census of a finite set of order 3072, plus an exact rank computation over Q for each mover. The completeness argument is the decomposition of the hyperoctahedral group into 30 left cosets of the line-preserving subgroup. This is not a heuristic search and not an unrestricted nonexistence claim.

Task and overlap

• Task: formal-interface-lemma, scope changed.
• Upstream base: 17469fff78c22941280f04d39dc5a1faee09c840 (priority revision 8).
• Assigned question: does any c=1 stabilizer element that moves the 216-space preserve the 8-dimensional dihedral isotypic summand and act non-scalarly on it?
• Predecessor: Record the c=1 stabilizer action outside the {g,t} normalizer. #7022
• Summand basis: Record that the 8-dimensional c=1 isotypic summand is reducible under… #7023
• Finding being answered: Handoffs and requests for independent checks #1166 (reply in thread)
• Package: research/c1-mover-8-summand/

The catalogue card asks for a formal caller binding. PR #7022 enumerated the stabilizer action on the 216-space and left this mover question open. Under the linear-character identification in PR #7008, elements that preserve the 216-space act by scalars on the summand; this census checks that prediction and then tests the movers. It does not redo the preserving-element census, the A_1 characteristic polynomial of PR #7023, the different 8-dimensional joint space of PR #7019, or the 80-dimensional and 128-dimensional components.

The 768 preservers of the 216-space all preserve the summand and act by ±I, 384 of each sign. That is a control. PR #7022 already places the outside-normalizer preservers among those 768 elements. This package does not recompute the normalizer.

Overlap on 2026-11-10 against 17469fff78c22941280f04d39dc5a1faee09c840: local notes 10180, HEAD paths 10178, complete; 339 open PRs, pagination complete, fetched 2026-11-10T10:22:32Z; 7013 discussion records, pagination complete, fetched 2026-11-10T10:23:39Z. The phrases 8-dimensional dihedral, moves the 216, and isotypic summand match #7008, #7022, #7023, and #7019. No open PR reports the action of a mover on this summand. Closed PRs were searched through local READMEs, task metadata, and the ledger only. No code diffs.

Validation

CPython 3.12.3, standard library only. Both modes passed. Metadata check returned PASS_PACKAGE_METADATA_AND_PINS. Registered replay returned PASS_REGISTERED_RESEARCH_PACKAGE. Selected baseline returned PASS_RESEARCH_BASELINE_BINDINGS_ONLY.

python3 -B -W error research/c1-mover-8-summand/verify.py
python3 -B -O research/c1-mover-8-summand/verify.py
python3 -B scripts/research_package.py --package research/c1-mover-8-summand --check-only

Marker: PASS_C1_MOVER_8_SUMMAND.

The line-preserving signed-permutation group has order 344064 and index 30 in the hyperoctahedral group. The summand is the image of the multiplicity-8 projector for ψ(r^k)=(-1)^k and ψ(r^k u)=(-1)^{k+1}. Its Gram matrix is 48 I_8. Rejection controls cover a coordinate transposition, the incomplete swap of coordinates 4 and 7, a swapped port label, a dropped projector support term, a repeated residual, and the PR #7022 witness, which is non-scalar on the 216-space and still acts by -I on the summand.

Not run: Lean, Docker, the degree-881 replay, any characteristic polynomial, and the selected cold replay.

Credit

Prepared by unpaidmoon with substantial Cursor Grok assistance.

JoyboyBrian recorded the sign-character dimensions. PR #6944 split the trivial c=1 character into 216-dimensional blocks. PR #6964 pinned one 216-basis. PR #7008 computed the dihedral image and the multiplicity-8 character. PR #7022 enumerated the stabilizer on the 216-space and left the mover question open. PR #7023 built this summand's basis and proved reducibility under A_1. Ali Romman's PR #4958 and PR #4720 retain the query and the Bell action. Krzysztof Wojt's research/exact-one-slot-gram/ supplies the 13 generators. The doubled representation comes from PR #1719. New work is Apache-2.0.

Open next questions

See research/c1-mover-8-summand/TASK.json.

• For the named mover (0 3)(5 6), does g or t act by a scalar on the image of the summand? The image misses the (+1,+1) space. This is not the basis-transport question of PR Record the characteristic polynomial of one 8-dimensional c=1 joint eigenspace #7019.
• Does any element of the line-preserving group outside the c=1 stabilizer preserve this summand and act non-scalarly?

Workflow feedback (optional)

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