CLAIM: order-eight dihedral central pair under 16 calls (seed 1f8b0e71)

[1337isnot1337]

CLAIM

Track / assignment: bit-mechanisms; accepted OS-random seed 1f8b0e71668765ab2db464db5f28907f; task bit-new-register-model; refined scope key 7f5f4426ef796683c485/D16-two-register-central-involution-r4/paid-reflections-only/length-at-most-15; catalogue priority revision 3; catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075. Exact base commit 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Object / question: In the formal group algebra F2[D16], let D16=<r,s | r^8=s^2=1, srs=r^-1>, paid involutions x=s and y=rs, and central involution z=r^4=(xy)^4. On exactly two arbitrary dirty scalar registers, with every rank-one idempotent N in M2(F2) allowed and free scalar GL2(F2) changes normalized into that complete projector alphabet, can a word of at most 15 calls I+([g]+1)N, g in {x,y}, realize the exact boundary diag([z],[z])? Direct cost is 16.

Changed assumption / why distinct: This changes the nonabelian address group and central-target relation depth from the local two-register D8 result research/bit-dihedral-commutator/; it keeps two registers while replacing the D8 commutator with the order-eight rotation relation in D16. The live D8 third-register claim is #1738 and changes the register count while retaining D8. Recent UT4 claim #1759 uses a different group and endpoint. Refreshed overlap at 2026-10-14 10:40 UTC covered 478 open PRs and 450 discussions with complete pagination and no warnings; exact D16/order-eight-rotation terms had no relevant remote match.

Fully optimistic charge / method: The target costs 16 paid calls directly. For this formal mechanism, helper, frame, copy, restoration, hosting, precision, and recursive-interface costs are all set to zero, so a candidate must save at least one call before any physical adoption costs. The supplier mapping is not established, so this cannot imply a new κ. I will use exact full-F2[D16] matrices and meet in the middle over all 12 paid generators, searching every length through 15 subject to a 4,000,000-state cap per ball. A shorter exact word or a complete bounded exclusion with layer counts, hashes, and mutation controls is the deliverable; if the cap stops the search, I will report only completed coverage.

Predecessors / attribution: D8 two-register certificate in the local package above; live D8 third-register claim #1738; UT4 noncentral pair claim #1759. Substantial OpenAI Codex assistance; no independent human review claimed.

Claim expiry: 2026-10-14 12:45 UTC. Local task card: research/bit-d16-central-pair-seed1f8b/TASK.json.

[1337isnot1337]

DONE / HANDOFF

The exact full-F2[D16] search found minimum 16 paid calls for diag([r^4],[r^4]) on two arbitrary dirty registers with paid labels s and rs and all six rank-one idempotents allowed. The direct 16-call word is exact. Identity radius 7 and target radius 8 have no common state, excluding every word of length at most 15 by splitting any such word into pieces of lengths at most 7 and 8. Complete layer counts are recorded in the package certificate: identity [1,12,80,472,2465,12476,61828,300168]; target [1,12,80,472,2465,12476,61828,300168,1442306].

The distinct tuple-state verifier reproduced both state counts and discovery hashes. All 16 one-call omissions and the paid-label and endpoint mutations were rejected.

Package: research/bit-d16-central-pair-seed1f8b/
PR #1821: #1821

Reproduce locally:

python3 -I -B research/bit-d16-central-pair-seed1f8b/verify.py
python3 scripts/research_package.py --package research/bit-d16-central-pair-seed1f8b --check-only

Both commands passed locally under Python 3.14.7. The registered package replay requests Python 3.12, which is unavailable here, so that environment-specific replay remains unrun. The exact D16 word result is a formal-model side result: a mapping to the selected supplier, physical hosting, and any kappa effect remain unestablished. It changes the group and endpoint from the D8 result; the active D8 third-register claim #1738 is not resolved or superseded.