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Change and mathematical scope

Finite register-model certificate in research/m16-generator-triple. No new κ. The selected record, both primitive suppliers, and upstream/ are unchanged.

In (\mathbb F_2[M_{16}]) with (M_{16}=\langle a,b\mid a^8=e,\ b^2=e,\ bab=a^5\rangle), exactly three roles, paid labels ({a,b}), and all 28 rank-one idempotents in (M_3(\mathbb F_2)):

• (\operatorname{diag}([a^2]^3)) has minimum length 6
• (\operatorname{diag}([a^3]^3)) has minimum length 9
• (\operatorname{diag}([a^5]^3)) has minimum length 9

Each minimum equals the disjoint generator bill, so mixing does not undercut it. (\operatorname{diag}([a^4]^3)) and (\operatorname{diag}([a^6]^3)) are only known to lie in ({8,10,12}).

The degree-1 symbol argument excludes every even length for the two odd triples. The odd lengths 1, 3, 5 and 7 are excluded by exhaustive frontiers of sizes 56, 2052, 68955 and 2199152. Equality is in (M_3(\mathbb F_2[M_{16}])), not a finite column sample.

Task and overlap

Base commit ef5f39403610d231a0c27e8a6b4f08c10209bcd6. Selector nonce a1330c55bf86b0ea1c2eb0ac57772471, card bit-new-register-model, scope key 7084092dd726e7ba1f2f, priority revision 2. Catalogue SHA-256 89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce. Selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Scope status: refined. The card allows one specified nonabelian family. This family is the modular group of order 16. The abelian (M_3(\mathbb Q)) model is not reused. The complex primitive still binds, so these sharp bills do not move κ.

Refreshed overlap on 2026-10-31, complete pagination, 320 open PRs and 6141 discussions. A search for this presentation matched only this package. Unrelated comments matched the digit string 16. Closed work is covered only through local notes and the ledger. Nearest different alphabets: research/bit-three-register-lower-bound (rational frames), open fork PRs #11 (quaternion order 16, center order 2) and #5 (semidihedral order 16). This group has center ({e,a^2,a^4,a^6}).

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

Validation

python3 -B research/m16-generator-triple/verify.py
python3 -B scripts/research_package.py --package research/m16-generator-triple

Both ended with

PASS_M16_GENERATOR_TRIPLE a2=6 a3=9 a5=9 a4_through7=0 a6_through7=0 a4_upper=12 a6_upper=12 idempotents=28 calls=56 frontier2=2052 frontier3=68955 frontier4=2199152 center=4 derived=2 controls=7 no_new_kappa

Environment: CPython 3.12.3, PYTHONOPTIMIZE unset, g++ 13.3.0 with /usr/bin/c++ pointing at /usr/bin/g++, libboost-dev 1.83.0.1ubuntu2, and research/e8-bit-p111-tuned/requirements.txt installed with python3 -m pip install --user. This package imports none of those libraries. Direct runtime was about 60 seconds. The registered replay also printed PASS_REGISTERED_RESEARCH_PACKAGE.

Seven controls cover the length-3 generator triples, shorter positive bills, a dropped call on a dirty state, an a-for-b substitution, the dihedral relation bab=a^7, rejection of (E_{00}+E_{11}), and one seven-call matrix first reached at length 7.

Not run: make frontier1094-verify, the three-register C++ replay, a four-role search, and lengths 8, 10 and 12 except the explicit length-12 upper words.

Credit

Predecessor model: scalar-idempotent calls in research/bit-three-register-lower-bound, from eumemic's pinned PR CrocSwap#288, section 1.2. Quaternion and semidihedral certificates on this fork use different relations. Apache-2.0.

Substantial AI assistance was used to build the multiplication table, search the paid words, and write the package notes.

Open next questions

See research/m16-generator-triple/README.md.

1. Is the minimum of (\operatorname{diag}([a^4]^3)) equal to 8, 10, or 12, and likewise for (\operatorname{diag}([a^6]^3))? A value below 12 would be a mixing saving. First step: store one word for each of the 2,199,152 depth-4 matrices. Blocker: this package builds that set and does not store the words.
2. Can one restored scratch role realize (\operatorname{diag}([a^5],[a^5],[a^5],[e])) in fewer than 9 calls? First step: a four-role search. Blocker: that alphabet was not built.

Workflow feedback (optional)

Task bit-new-register-model, priority revision 2, outcome completed for three named triples. The largest friction was the crowded nonabelian-register discussion list; a phrase search for the presentation was what separated this group from earlier order-16 alphabets. One improvement: the overlap helper's default excerpt cap hides a specific presentation unless the query is that presentation.