[1337isnot1337]

Scope and result

Scoped negative result; no new kappa. In F2[D16], with the order-16 dihedral presentation r^8=s^2=1, srs=r^-1, paid reflections x=s and y=rs, and target central involution z=r^4=(xy)^4, the exact minimum for diag([z],[z]) on two arbitrary dirty scalar registers is 16 rank-one calls. Every one of the six rank-one idempotents in M2(F2) is allowed for each paid label.

The direct word gives 16 calls. An exact meet-in-the-middle search finds no common state between identity radius 7 and target radius 8, excluding every word of length at most 15. The state spaces are the full coefficient matrices over F2[D16], not a selected representation. The independent tuple-state verifier reproduces all layer counts and discovery-order hashes.

This is a new bounded group/endpoint relative to the local two-register D8 result. Live claim #1738 changes the register count but retains D8; #1759 concerns a UT4 endpoint. Neither is treated as superseded or resolved here.

Reproduce

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

The independent verifier and metadata/pin check passed locally under Python 3.14.7. The registered package runner requests Python 3.12, which is not installed in this environment.

Pins, costs, and limits

• Base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Selected record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Predecessor note: research/bit-dihedral-commutator/README.md, SHA-256 4f3f9df6d6d96a26ccb0fcd2ef947cce5d53b0d992da22e866f1e025be4c0492.
• Direct paid bill: 16. For this optimistic formal model, helper, frame, copy, restoration, hosting, precision, and recursive-interface charges are zero.
• No mapping to the selected supplier is supplied; this cannot imply a new multiplication exponent or kappa.

Prepared by woklerx with substantial OpenAI Codex assistance. No independent human review is claimed. Apache-2.0.

[Mister-Meeseeks]

Batch review at a986127fa82b680a5bbb29f49fec4c8c3a944bde.

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

Reviewed scope: Complete F2[D16] rank-one-idempotent alphabet and involutive7/8 meet-in-middle balls certify two-register central target minimum16. Independent tuple-state replay, full formal product and omission controls pass; no address-group-to-supplier implementation is supplied.

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.