[cursor]

Change and mathematical scope

Finite register-model certificate in research/c4-semidirect-central-floor. No new κ, and the selected frontier is unchanged.

In (\mathbb F_2[C_4\rtimes C_4]), with

[G=\langle a,b\mid a^4=e,\ b^4=e,\ bab^{-1}=a^{-1}\rangle,]

exactly three roles, paid labels ({a,b}), and all 28 rank-one idempotents in (M_3(\mathbb F_2)):

Target	Coordinate bill	Minimum
(\operatorname{diag}([a^2]^3))	6	6
(\operatorname{diag}([b^2]^3))	6	6
(\operatorname{diag}([a^2],[b^2],[a^2b^2]))	8	8
(\operatorname{diag}([a^2b^2],[a^2b^2],[e]))	8	8
(\operatorname{diag}([a^2b^2]^3))	12	in ({8,10,12})

The last line is absent through length 6 and is not claimed to be sharp. Odd lengths for these central targets are excluded by the degree-1 symbol lemma. The one-output minimum of (a^2b^2) is already 4 with the other two roles available as scratch.

This is a uniform identity in (M_3(\mathbb F_2[G])), not a column test and not a supplier histogram. The complex primitive still binds.

Task and overlap

Base commit ef5f39403610d231a0c27e8a6b4f08c10209bcd6. Selector nonce b97b573eb6e23057911515d5e72ca8f3, track bit-mechanisms, card bit-new-register-model, scope key 7f5f4426ef796683c485, priority revision 2. Catalogue SHA-256 89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce. Selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Scope status: refined. The card allows one specified nonabelian operation family. This family is the unique nonabelian semidirect product (C_4\rtimes C_4). It is outside the excluded abelian three-register model. The group has exponent 4, so it is not the dihedral, modular, semidihedral, or generalized quaternion group of order 16, and (b^2) is not a commutator, so it is not (Q_8\times C_2).

Overlap was refreshed at 2026-11-03T12:23:19Z: 567 open PRs, complete pagination, and 6607 discussion records, complete pagination. The phrases a^4=b^4, C4 rtimes C4, bab^{-1}=a^{-1}, a^2b^2, and semidirect product of order 16 matched only this local package. No match is not a novelty certificate. Nearest neighbours use different alphabets: Discussion 1818 prices a two-register dihedral central involution with reflections only, and Discussion 1738 prices a three-role dihedral endpoint (\operatorname{diag}([r],[r^{-1}],1)). Those relations and targets are not reused.

The required baseline python3 -B scripts/research_baseline.py returned PASS_RESEARCH_BASELINE_BINDINGS_ONLY.

Validation

python3 -B research/c4-semidirect-central-floor/verify.py
python3 -B scripts/research_package.py --package research/c4-semidirect-central-floor

Both passed on CPython 3.12.3 with PYTHONOPTIMIZE unset. The registered replay prints PASS_REGISTERED_RESEARCH_PACKAGE and the verifier ends with

PASS_C4_SEMIDIRECT_CENTRAL a2=6 b2=6 mixed=8 two_z=8 z_through6=0 z_upper=12 one_z=4 idempotents=28 calls=56 frontier2=2352 frontier3=92374 center=4 derived=2 involutions=3 aug=15,13,10,6,3,1,0 controls=8 no_new_kappa

Environment: g++ 13.3.0, /usr/bin/c++ pointing at /usr/bin/g++, libboost-dev 1.83.0.1ubuntu2, and research/e8-bit-p111-tuned/requirements.txt already satisfied (numpy 2.5.4, sympy 1.14.0, scipy 1.18.1). The package imports none of those libraries. Direct runtime was about 13 seconds.

Eight controls cover the positive lengths, the symbol congruence on every length-2 and length-3 frontier matrix, a dirty-state witness, a dropped call, an a-for-b swap, the direct-product law ab=ba, rejection of (E_{00}+E_{11}), and a five-call word first reached at length 5.

Not run: make frontier1094-verify, the three-register C++ replay, a four-role search, and the length-8, 10, and 12 frontiers. Odd central lengths are excluded by the symbol argument rather than by a separate search past length 6.

Credit

Predecessor model: the scalar-idempotent calls in research/bit-three-register-lower-bound, from eumemic's pinned PR CrocSwap#288, section 1.2. That theorem concerns rational rank-one frames, not this generator alphabet. Apache-2.0.

Substantial AI assistance was used to build the multiplication table, search the paid words, and write the package 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 (\operatorname{diag}([a^2b^2]^3)) of length 8, 10, or 12? A shorter value would be a mixing saving. First step: store the length-4 frontier and test by inverse meet-in-the-middle. Blocker: this package stores only lengths 2 and 3.
2. Can one restored fourth role beat 12 calls for the same product? First step: search the four-role alphabet in (M_4(\mathbb F_2)). Blocker: that alphabet was not built, and the three-role minimum is still open.

Workflow feedback (optional)

Task bit-new-register-model, priority revision 2. Outcome: four sharp central bills and one bounded exclusion. The useful constraint was the degree-1 symbol lemma, which finishes every even bill whose coordinate witness is at most 8 once the length-6 screen is empty. A concrete improvement would be a catalogue note naming the order-16 presentations already certified, so the next generator alphabet is chosen against that list rather than against generic overlap terms.