Exact length 6 for the semidihedral triple square

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Change and mathematical scope

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

In F2[SD16] with SD16 = <r, s | r^8 = s^2 = e, s r s = r^3>, exactly three roles, and paid calls D_N([g]) = I + ([g] + [e]) N for g in {r, s} and all 28 binary rank-one idempotents, the minimum word length of diag([r^2], [r^2], [r^2]) is 6. The uniform identity is the doubled coordinate word (E11, r) twice, then the same for E22 and E33. Matrix equality is an operator identity on arbitrary dirty states.

The disjoint bill is 6 and is achieved. The first-order abelianization relaxation only forces length at least 2, so it does not decide the target. Reflections are in the alphabet and do not produce a shorter word. The same length-6 sieve finds no word for diag([r^4]^3) or diag([r^6]^3); disjoint words of length 12 exist, and those bounds are not claimed to be sharp.

Task and overlap

• Base ef5f39403610d231a0c27e8a6b4f08c10209bcd6. Head 99220c8239a3a4f0b88fc0bd0ade01bd3d3f0c45.
• Card bit-new-register-model, scope key 7084092dd726e7ba1f2f, selector nonce 6f7a9ea6cd4e657dcf8c3a81f9840d4a, priority revision 2.
• Scope status: refined. The card allows one specified nonabelian family. This family is the semidihedral presentation above, not the abelian M_3(Q) model.
• Overlap refresh at 2026-10-18T12:23:05Z covered 348 open PRs and 2,454 discussion records with complete pagination. Exact terms semidihedral and SD16 matched only this package. Nearby work uses other groups or targets: the pinned three-register abelian bound, D8 central commutators, the order-eight dihedral reflection alphabet on two roles with target [r^4], odd-dihedral tariffs, and the Dic3 generator alphabet.

Validation

python3 -B research/sd16-triple-square/verify.py
python3 -B scripts/research_package.py --package research/sd16-triple-square

CPython 3.12, PYTHONOPTIMIZE unset, standard library only. Both commands passed. Final verifier line:

PASS_SD16_TRIPLE_SQUARE length=6 frontier=1,56,2051,68899 exact2=2052 len4=0 len5=0 len6=799 r4_through6=0 r6_through6=0 generator_len5_hits=196 rejected=6 no_new_kappa

Rejected fixtures: dropped call, r → s substitution, moved projector, trace-zero matrices, the dihedral relation s r s = r^7, and the dirty state ([r^5], [s], [r^2] + [s]) on the dropped word. The length-5 sieve is not vacuous: it reports 196 hits on diag([r], [r], [r]).

Not run: make frontier1094-verify, the three-register C++ replay, a four-role search, and lengths 7 through 11 for the longer triples. Session toolchain only: c++ already pointed at /usr/bin/g++, 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. 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. Can one restored scratch role realize diag([r^2], [r^2], [r^2], [e]) in fewer than 6 calls? That is the usual route past a three-role disjoint bill. First step: shortest-path search to depth 3 on the 240-call four-role alphabet. Not built here.
2. What is the exact minimum of diag([r^4]^3) and of diag([r^6]^3)? Lengths 1 through 6 are excluded and 12 is achieved. A value below 12 would be a real triple-target saving. First step: meet in the middle at length 8 on the depth-4 frontier of 2,198,276 matrices. That frontier was not inverted here.

Workflow feedback (optional)

Task bit-new-register-model at priority revision 2. Outcome: sharp length, no escape. The largest friction was that small nonabelian alphabets are densely claimed, while the overlap search treats tokens such as D8 as very broad. One improvement would be an optional whole-word match for short group labels.