[unpaidmoon]

Change and mathematical scope

A complete no-edge pool of pattern dimension 4 can act irreducibly on a joint image of dimension 5. For every ambient width W ≥ 5, the five projectors

N_v = (e_0 + e_2)(e_1 + e_2)^T,
N_u = (e_1 + e_2)(e_0 + e_2)^T,
N_o = (e_0 + e_1 + e_2)(e_0 + e_1 + e_2)^T,
N_a = (e_2 + e_3)(e_0 + e_1 + e_3)^T,
N_b = (e_0 + e_2 + e_3 + e_4)(e_1 + e_4)^T

are the complete pool of their off-diagonal pattern span. That span has dimension 4 and contains no matrix unit. The joint image is the coordinate span of e_0 through e_4, and the action is irreducible.

This is an existence proof under the same hypotheses as #6921 and #6962: 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 first three projectors are the normal-form triple. On their own three-dimensional image that triple is reducible. N_a and N_b enlarge the pattern space to dimension 4 and make the five-dimensional action irreducible. The four-cycle of #6962 is not redone, and the dimension-at-most-3 theorem of #6921 is not reopened.

Package: research/no-edge-dim4-image-five/.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840.

Task card: bit-new-register-model, priority revision 8. Scope status: changed. The card asks for a new register alphabet. This package answers the open question of merged #6962, which is discussion finding 19507179. #6921 stops at pattern dimension 3. A search of open upstream pull requests for this question returned no match. #6785 is a one-edge exclusion. No match is not a novelty certificate.

The original native acceptance gate remains unmet. The selected frontier is unchanged.

Validation

Evidence class: a written existence proof for one explicit pool, valid at every width at least 5. The fifteen nonzero patterns of this one four-dimensional space are classified by hand. That list is not a census of all four-dimensional pattern spaces. The hit digraph on the five projectors contains a 4-cycle through a, v, b, o and a 2-cycle through a and u, so it is strongly connected, and every nonempty proper subset therefore has an incoming hit. Together with the image basis and the covector basis, that excludes every proper nonzero invariant subspace.

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

python3 -B research/no-edge-dim4-image-five/verify.py
python3 -I -O -B research/no-edge-dim4-image-five/verify.py
python3 -B scripts/research_package.py --package research/no-edge-dim4-image-five --check-only
python3 -B scripts/research_package.py --package research/no-edge-dim4-image-five

The verifier prints PASS_NO_EDGE_DIM4_IMAGE_FIVE. It repeats the fifteen-pattern census, scans every rank-one idempotent at widths 5, 6 and 7, checks the six digraph edges, and enumerates all 374 subspaces of F2^5. The normal-form triple is retained as a reducible control. These finite checks corroborate the written argument. They were run locally in this session. No selected-frontier replay was run.

Credit

#6921 for the normal-form triple and for the irreducible bound 3 in pattern dimension at most 3. #6962 for showing that bound 3 fails in pattern dimension 4 and for leaving joint image 5 open. CubeJerry, #6838, for the reducible parity pool of total dimension 5. Nick Petalas, #5921, for the fiber lemma; the row reconstruction here does not rely on it. Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

Open next questions

Can a complete no-edge pool of pattern dimension 4 act irreducibly on a joint image of dimension 6 or more? This pool attains 5, so any upper bound is at least 5. The smallest next step is one explicit pool with a written invariant-subspace check, or a scoped upper bound under the same hypotheses. No support reduction here makes an exhaustive search finite.

Workflow feedback (optional)

The open question on #6962 named the exact missing witness, which kept the session on one pool instead of a new alphabet. A link to the feedback discussion is enough: #1200