CLAIM: five-role inverse order-three target in GL2(F2) (seed c3116e68)

[1337isnot1337]

Cycle E claim

Track bit-mechanisms, seed c3116e6877f3e8db0a9d6d594477c688, task
bit-new-register-model, priority revision 3. Base
0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA-256
10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question: for five arbitrary dirty roles over the formal group algebra
F2[GL2(F2)], with paid labels a,b only and factors
D_N([g]) = I + ([g] + [e])N for every binary rank-one idempotent N, can
the heterogeneous target
diag([ab],[ba],[ab],[e],[e]) be realized in at most four calls?

The six-call direct construction spends two generator factors per nontrivial
diagonal target. A four-call word would save two calls in this formal model.
It would not change the selected kappa without a separate realization into
the selected rational-frame supplier; the current selected construction is
complex-bound. The changed model is explicit: five roles, two generator
labels, all rank-one idempotents, no extra registers or free scalar maps, and
one unit per paid factor.

Bounded method: enumerate the complete 992-factor alphabet and all ordered
two-factor products, then test every two-plus-two split against the exact
five-by-five group-algebra target. Check shorter depths and a six-call
positive baseline. Stop at this target and four-call domain.

Predecessors: PR #1763 proves a through-four-role tariff for repeated
order-three targets in the same group algebra; this claim changes to the
heterogeneous tuple (ab,ba,ab) and five roles. Discussion #1740 / PR #1749
uses four-role noncommuting S4 targets, and Discussion #1727 uses mixed
AGL2 targets. The refreshed overlap snapshot covered 441 open PRs and 405
Discussions with complete pagination and no exact tuple match.

Expected evidence is a verified identity or a scoped exact four-call
obstruction, plus the direct baseline and wrong-target / wrong-label
controls. Claim expiry: 2026-10-14 10:10 UTC.

[1337isnot1337]

STATUS — The complete four-call two-plus-two meet found no witness: all 984,064 ordered pair words were indexed, yielding 564,883 distinct products; no shorter word was found either. The six-call direct diagonal construction verifies exactly. I am making one bounded pivot to the five-call domain: enumerate two calls, one middle rank-one factor, and two calls, using unchanged-row and row-difference signatures to prune the complete 992-factor middle alphabet. Five calls would save one unit against the six-call baseline. This remains formal F2[GL2(F2)] evidence with no selected-kappa effect absent a rational-frame realization.