Research: q=6/7 obstruction for four-label affine targets

[1337isnot1337]

Result

This package gives a scoped negative result for the four-register target diag(h,h^-1,tx,ty) over the fixed paid-label subset {a,b,tx,ty} of the binary affine group. Each call is I+([g]+[1])N for any rank-one idempotent N in M4(F2), with arbitrary dirty registers.

The direct predecessor, closed PR #1333, proves exact cost six when every rank-one affine displacement is available as a paid label. This contribution fixes four labels and one target tuple. In the homogeneous affine representation the target displacement rank is six. Exact enumeration excludes q=6 and q=7, so the minimum in this restricted alphabet lies in [8,10]; the direct ten-call word is explicit. The exact minimum is unresolved, and no kappa improvement is claimed.

The q=6 certificate recomputes all 480 factors and hashes the complete lowering frontiers. It verifies the direct positive control and rejects a dropped-call and a wrong-label control. The q=7 trajectory program and full captured output are hash-pinned in the certificate. The q=7 absence result uses a necessary representation condition and therefore excludes exact group-algebra words in this declared model.

Pins and distinction

• Base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea
• Selected-result SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32
• Certificate SHA-256: 83927854ee05f412319cb234de37c246f924c96e315c413b4819c74ae38ac111
• Direct predecessor: PR #1333, closed head 7f25b6d0e1d7e995d4d2840f231d98b1a1a9fec2
• q7 program and output hashes: in research/affine-mixed-four-targets-seed5638/CERTIFICATE.json

The live overlap review covered 419 open PRs and 375 Discussions with complete pagination and no warnings at 2026-10-14 08:28 UTC. Concurrent Discussion #1738 studies a three-register UT3 central pair with {x,y}; the target group, tuple, register count, and search method differ. PR #1711 and #1716 likewise use different affine target tuples/register counts.

Reproduction

python3 -I -B research/affine-mixed-four-targets-seed5638/verify.py
python3 -I -B research/affine-mixed-four-targets-seed5638/q7_trajectory_search.py
python3 scripts/research_package.py --package research/affine-mixed-four-targets-seed5638 --check-only

The recorded q7 replay took about 9m34s. The selected conditional kappa remains complex-bound. No physical, helper, recursive, frame, copying, hosting, precision, interface, or restoration saving is admitted. Prepared with substantial OpenAI Codex assistance; no independent human review is claimed.

[Mister-Meeseeks]

Scoped batch triage at 60f4c042da10aaf702a2f879b0f3cb11940741fc. This PR remains open.

The q=6 full replay passed; q=7 exceeded the extended recorded budget (952.67s total run) before complete output. Finish q=7 and compare the complete trajectory/census to the committed expected result. Timeout is not evidence against the mathematical claim.

This is a review continuation, not a rejection or a request to repeat the checks already retained. Credit and source pins are preserved in the batch ledger. The selected bound is unchanged. Thank you for the contribution.