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Change and mathematical scope

Scoped negative result in one nonabelian register alphabet. No new κ and no change to the selected frontier.

In F2[Q16] with Q16 = <x, y | x^8 = e, y^2 = x^4, y x y^{-1} = x^{-1}>, exactly three roles, and paid calls D_N([g]) = I + ([g] + [e]) N for g in {x, y} and all 28 binary rank-one idempotents, the minimum word lengths of diag([x^2], [x^2], [x^2]) and diag([x^4], [x^4], [x^4]) are both 6. The uniform identities are the doubled coordinate words: (E00, x) twice on each role, and (E00, y) twice on each role. Matrix equality is an operator identity on arbitrary dirty states. x^4 is the unique central involution and equals y^2.

The degree-1 augmentation symbol only forces even length, so it does not decide the bills. The disjoint generator bills are 6 and 6. Mixing, including 792 nondiagonal length-6 prefixes of the central target, does not undercut them.

diag([x^6]^3) has no word of length at most 6. The coordinate word xxyy on each role has length 12. The same parity argument excludes odd lengths 7, 9 and 11, which were not enumerated, so that minimum lies in {8, 10, 12} and is not claimed to be sharp.

Task and overlap

• Base ef5f39403610d231a0c27e8a6b4f08c10209bcd6. Head 295958fd4bcf3228da63f707294a786a05d88da7.
• Card bit-new-register-model, scope key 7084092dd726e7ba1f2f, selector nonce bfe60b39ce1400e38d330c639d8e904a, priority revision 2.
• Scope status: refined. The card allows one specified nonabelian family. This family is the quaternion presentation above, not the abelian M_3(Q) model.
• Overlap refresh at 2026-10-24T12:24:51Z covered 321 open PRs and 4710 discussion records with complete pagination. generalized quaternion and quaternion of order 16 had no matches. QQ16 hits are a different decoder object.
• Discussion comment 18926868 uses this presentation and releases a two-role schedule: two copies of x^4 cost 4 by the direct word, not an assumed 8. It does not determine the three-role minimum. Q8 central-pair claims are the order-8 group. The semidihedral triple square uses s r s = r^3 and s^2 = e, with frontiers 2051 and 68899 rather than 2352 and 92374.

Validation

python3 -B research/q16-central-square/verify.py
python3 -B scripts/research_package.py --package research/q16-central-square

CPython 3.12.3, PYTHONOPTIMIZE unset, standard library only. Both commands passed. Final verifier line:

PASS_Q16_CENTRAL_SQUARE x2=6 x4=6 x6_through6=0 x6_upper=12 idempotents=28 calls=56 frontier2=2352 frontier3=92374 len4=0 len6_x2=799 len6_x4=799 len6_x6=0 central_nondiagonal_prefixes=792 group_lengths=2,2,4 controls=7 no_new_kappa

Controls: diag([x]^3) and diag([y]^3) at length 3, diag([x^2],[x^2],[e]) at length 4, one x^4 at length 2, a dropped call failing on ([x^5], [y], [x^2]+[y]), an x-for-y substitution, and the false relation y^2 = e. E00+E11 is not admitted.

Not run: make frontier1094-verify, the three-register C++ replay, a four-role search, and the length-8 meet for diag([x^6]^3). Session toolchain: c++ points at /usr/bin/g++ (g++ 13.3.0), libboost-dev 1.83.0.1ubuntu2 was present, and research/e8-bit-p111-tuned/requirements.txt was installed with python3 -m pip install --user. This package does not import those libraries.

Credit

Predecessor: the abelian three-register theorem in research/bit-three-register-lower-bound, including eumemic's PR CrocSwap#288 scalar-idempotent model, and the released two-role Q16 schedule in discussion comment 18926868. Parallel register-model claims listed above are different objects and are not inputs. Apache-2.0.

Substantial AI assistance was used to set up the multiplication table, implement the shortest-word search, and write the notes.

Discussion writes for this account return FORBIDDEN / Resource not accessible by integration. No claim was posted, and the write was not retried.

Open next questions

1. Is the minimum of diag([x^6]^3) equal to 8, 10, or 12? A value below 12 would be a mixing saving. First step: meet in the middle at length 8. The depth-4 frontier was not built.
2. Can one restored scratch role realize diag([x^4]^3, [e]) in fewer than 6 calls? First step: depth-3 search on the four-role generator alphabet. That alphabet was not built.

Workflow feedback (optional)

Task bit-new-register-model at priority revision 2. Outcome: sharp length 6, no escape. The largest friction was that QQ16 swamps searches for the order-16 quaternion group. One improvement would be an optional whole-token match so Q16 does not hit QQ16.