Research: restricted three-register AGL2 target triple costs six calls

[1337isnot1337]

Result

For the exact three-register formal model M_3(F2[AGL_2(F2)]), the target
diag([a],[b],[h]), with h = a*b*a*t_x, has minimum paid-call cost six when the paid address labels are restricted to {a,b,t_x}. The target acts as h(x0,x1) = (x1,x0+1) on its third register. Arbitrary free scalar GL_3(F2) mixing is included and normalized by augmentation; all three register inputs are arbitrary dirty data.

The direct six-call word is exact. A complete radius-three group-algebra BFS and all 4,782 radius-two/radius-three joins found no word through five calls. The formal layers contain 1, 84, 4,697, and 232,720 distinct states. The homogeneous displacement-rank relaxation is four calls, so this closes the two-call gap for this restricted alphabet. This is a scoped negative result; it does not change the selected conditional bound or kappa.

Reproduce

python3.12 scripts/research_package.py --package research/bit-affine-mixed-target-triple-seed818dbf

Expected markers:

PASS_BIT_AFFINE_MIXED_TARGET_TRIPLE group=24 generators=84 layers=1,84,4697,232720 probes=4782 minimum=6 hit=False
PASS_REGISTERED_RESEARCH_PACKAGE research/bit-affine-mixed-target-triple-seed818dbf outcome=negative-result

The package includes the formal word model, exact layer digests, an independent dense multiplication control, all 84 generator inverse checks, direct-word omissions, wrong-label and wrong-target controls, source pins, and the stopping rule.

Assignment and overlap

Seed 818dbf90f2581f79a4e4150cdd77e415, track bit-mechanisms, task bit-new-register-model, priority revision 3. Exact base 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

PR #1696 treats the related linear GL_2(F2) target (a,b,aba) on three registers. PR #1711 treats two affine targets on two registers with the four-label alphabet {a,b,t_x,t_y}. This result uses the mixed three-target endpoint (a,b,h) on three registers and only {a,b,t_x}. The refreshed repository-scoped AGL2 Discussion search found only #1317, where this claim is tracked: #1317 (comment) .

PR #1333 supplies the broader affine displacement-rank theorem; its argument is credited and not claimed as new. Prepared by Cinna Davis (@1337isnot1337) with substantial OpenAI Codex assistance in experiment design, exact-search implementation, and package drafting.

[Mister-Meeseeks]

Batch review at fdbcd61eea435f9f8d497290a1db6513bb769d51.

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

Reviewed scope: Complete full AGL2 group-algebra radius-three census excludes five calls for the specified mixed three-target endpoint and three-label alphabet; the six-call word attains the minimum. No extra-workspace or alternative-label exclusion is claimed.

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.