[unpaidmoon]

Change and mathematical scope

A complete noncoordinate rank-one idempotent pool over F2, in a pattern space of dimension at most three that contains no matrix unit, cannot act irreducibly on a four-dimensional joint image. Dimension 3 is sharp at every ambient width, by an explicit three-projector pool. This is a scalar pool theorem. It is not a paid-profile improvement and no new κ is claimed. The selected value stays 781451/250000000.

The total-image bound dim V ≤ 4 is cited from #6838 and is not reproved. The new statement is reducibility at dimension 4.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.

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 the scalar question left open by #6838: whether a no-edge pool of pattern dimension at most 3 can be irreducible in joint-image dimension 4. No new alphabet or word is built. The original native acceptance gate remains unmet. The same card was already used for scalar pattern-pool questions in #5921 and #6838.

Nearest open work:

• Prove sharp zero-edge image bound four and universal rank-three bound six (no new κ) #6838 proves the sharp total-image bound 4 and states the irreducible question as open. Its attaining example is reducible.
• Exclude irreducible five-dimensional one-edge binary pattern pools #6785 asks the zero-edge irreducible question and proves a different one-edge statement. Its argument assumes a nonzero edge space.
• Bound binary pattern cores and exclude the irreducible five-type saving (no new κ) #6782 gives the edge-quotient bound and an irreducible example only when edges are present.
• Prove sharp binary pattern-pool cardinality (no new κ) #5921 supplies the fiber lemma and leaves sharp joint-image dimension open.
• Prove the two-relation rank bound at arbitrary helper width (no new κ) #6777 is relation-word context, not a proof dependency.

Overlap search on 2026-11-08 covered 825 open PRs and 6920 discussion records. The phrase pair "no-edge" and "irreducible" matched only this package. "zero-edge image" matched #6838. "irreducible joint" matched #6785 and this package. No match is not a novelty certificate; the distinctions above are from reading those PRs.

Validation

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

python3 -B research/no-edge-irreducible-image/verify.py
python3 -I -O -B research/no-edge-irreducible-image/verify.py
python3 -B scripts/research_package.py --package research/no-edge-irreducible-image

Observed: PASS_NO_EDGE_IRREDUCIBLE_IMAGE and PASS_REGISTERED_RESEARCH_PACKAGE. check_research_commit.py --check-source-history passed at 6020400853abc110134804fad04b695adc360bcd.

The verifier enumerates GL(3, 2) (168 matrices, 18 survivors), the sharp pool on four coordinates, a one-extra-bit support check, the seven four-point caps, and negative controls: the #6838 parity pool is reducible, the normal-form triple is reducible, and dropping one sharp projector leaves an invariant line. Those finite checks do not replace the all-width case analysis in PROOF.md. No Lean, Docker, or historical replay was run. The selected baseline was not rerun.

Credit

Nick Petalas, #5921, for the fiber lemma. CubeJerry, #6838, for the total-image bound and the open irreducible question; #6782 for the edge-quotient context. Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

Open next questions

1. After a valid relation, cube-span, and coordinate-call split, does bound 3 change the paid-rank accounting for a three-relation word? The scalar theorem does not prove that split. First state the hypotheses from Prove the two-relation rank bound at arbitrary helper width (no new κ) #6777 and Bound binary pattern cores and exclude the irreducible five-type saving (no new κ) #6782.
2. Does the irreducible bound remain 3 in pattern dimension 4? The parity construction one dimension higher has total image 5, and the Fano-plane case split stops at dimension 3.