[unpaidmoon]

Change and mathematical scope

The no-edge irreducible joint-image bound does not stay 3 in pattern dimension 4. For every ambient width W ≥ 4, the four projectors

N_i = e_i (e_i + e_{i+1} + e_{i+2})^T,    i in Z/4Z,

are the complete pool of their off-diagonal pattern span. That span has dimension 4 and contains no matrix unit. The joint image has dimension 4 and the action is irreducible.

This is an existence result under the same hypotheses as #6921: a complete noncoordinate rank-one idempotent pool over F2 with no matrix unit. It is not an upper bound at pattern dimension 4, and it is not a classification of every such pool. It is not a paid-profile improvement and no new κ is claimed. The selected value stays 781451/250000000.

The width-6 parity pool from #6838 has pattern dimension 4 and total image 5, but its image has a four-dimensional invariant kernel. That pool is reducible and does not answer the question. The dimension-at-most-3 theorem of #6921 is cited and not reproved.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a. This fork PR is based on the existing pin cursor/crocswap-main-7c49fe32-1dbb of that commit. Fork main is not upstream main.

Card bit-new-register-model, priority revision 8. Scope status: changed. The card asks for a larger or nonabelian register alphabet. The actual result answers open question 2 of #6921. No new alphabet or word is built. The original native acceptance gate remains unmet. Question 1 of #6921, paid-rank accounting after a relation split, is not this result.

Nearest work:

• Prove the no-edge irreducible joint-image bound is 3. #6921 proves the bound 3 for pattern dimension at most 3 and leaves dimension 4 open. Its Fano-plane case split is not reused.
• Prove sharp zero-edge image bound four and universal rank-three bound six (no new κ) #6838 supplies the total-image bound 4 for pattern dimension at most 3, cited not reproved, and the parity scope control rechecked here as reducible.
• Exclude irreducible five-dimensional one-edge binary pattern pools #6785 is a one-edge exclusion. Bound binary pattern cores and exclude the irreducible five-type saving (no new κ) #6782 is an edge-quotient bound. Prove sharp binary pattern-pool cardinality (no new κ) #5921 supplies the fiber lemma; this proof does not depend on it.

Overlap search on 2026-11-10 covered 885 open PRs and 6934 discussion records. The only records containing this question are #6921 and discussion comment 19474334. Both leave it open. No match is not a novelty certificate; the distinctions above are from reading those records.

Validation

Python 3.12, standard library only. From the repository root:

python3 -B research/pattern-dim4-irreducible-image/verify.py
python3 -I -O -B research/pattern-dim4-irreducible-image/verify.py
python3 -B scripts/research_package.py --package research/pattern-dim4-irreducible-image

Observed: PASS_PATTERN_DIM4_IRREDUCIBLE_IMAGE and PASS_REGISTERED_RESEARCH_PACKAGE. The verifier repeats the covector row reduction, all sixteen pattern weights, the six plane pairs, the complete pool at widths 4 through 8, and the full subspace lattice of F2^4 (67 subspaces, matching the Gaussian counts). Negative controls: deleting one cycle projector leaves an invariant line; the #6921 sharp triple stays irreducible; the normal-form triple stays reducible; the parity pool matches the #6838 width-6 witness and has a four-dimensional kernel. Those finite checks do not replace the all-width support casework in PROOF.md, and they do not enumerate every four-dimensional pattern space. No Lean, Docker, or historical replay was run. The selected baseline was not rerun.

Credit

Nick Petalas, #5921, for the fiber lemma and the joint-image question. CubeJerry, #6838, for the total-image bound and the parity pool. #6921 isolated this dimension-4 question and proved the bound for pattern dimension at most 3. Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

Open next questions

1. Can a complete no-edge pool of pattern dimension 4 act irreducibly on a joint image of dimension 5 or more? The four-cycle attains 4, so bound 3 fails, but it gives no replacement ceiling. The parity pool attains total dimension 5 and is reducible. First give one explicit pool with a written invariant-subspace check, or prove an upper bound under these same hypotheses. No support reduction here makes an exhaustive search finite, so a search that finds nothing would not be a proof.

Workflow feedback (optional)

The overlap cache separates this scalar question from unrelated four-cycle graph and centre-graph results. One link to the feedback discussion is enough: #1200