Research: q<=4 barrier for a noncentral UT4 root pair

[1337isnot1337]

Result

This package gives a complete q<=4 no-go for one fixed nonabelian four-register model. In the full formal algebra F2[UT4(F2)], with paid labels x=I+E12, y=I+E23, z=I+E34 and any binary rank-one idempotent N in M4(F2), no word of at most four calls D_N(g)=I+([g]+[1])N realizes diag([xyxy],[yzyz],[1],[1]). The direct word uses eight calls, so the minimum in this fixed model is between five and eight. Lengths five through seven are unsearched.

Exact evidence

The package enumerates 120 rank-one idempotents and 360 factors, constructs all 129,600 ordered pair products in the full group algebra, then exhausts all 360 length-three endpoints and 129,600 length-four suffixes. It also cross-checks all 4,096 UT4 group products by ordinary 4x4 matrix multiplication, verifies both target roots are noncentral, checks the direct eight-call word, and rejects an omitted-call and wrong-label corruption.

Reproduce:

python3 -I -B research/ut4-noncentral-commutator-seedbfe59/verify.py
python3 scripts/research_package.py --package research/ut4-noncentral-commutator-seedbfe59 --check-only

The exact result, stopping rule, source pins, overlap, and validation limitation are in research/ut4-noncentral-commutator-seedbfe59/README.md. Local mathematical verification passed under CPython 3.14.7. The registered generic replay requires Python 3.12, which was unavailable locally.

Cost and scope

This prices one formal call as one unit: the direct construction costs eight, and a q<=4 identity would have halved that count. Natural-representation displacement rank gives a floor of two. The selected baseline positive control says the complex primitive remains binding; it also reports a 5.827190801985042 bit multiplier under retained planning assumptions. This finite bit-model result does not change the selected record or kappa. There is no physical supplier, recursive schedule, or complete interface invoice here.

PR #1333 established a rank-sum minimum with the full rank-one affine-label alphabet. This search restricts to three fixed generators and a noncentral two-root endpoint. Open PRs #1662, #1680, and #1690 concern UT4 central targets. No predecessor certificate or selected frontier was modified.

Prepared by Cinna Davis (@1337isnot1337) with substantial OpenAI Codex assistance in mathematical framing, implementation, verification, and writing.

Coordination claim: #1759

[Mister-Meeseeks]

Batch review at 0f4f6d79b14573116e47d31d012f8e3f7a49cd87.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete formal F2[UT4] two-plus-two meet excludes at most4 calls for the specified four-role noncentral root-pair target; direct8-call word checked, leaving5..8. All4096 group products cross-checked, all360 idempotent-labelled factors included, unchanged baseline bindings pass.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.