Research exact Q8 mixed-unit register minimum

[1337isnot1337]

Result

This adds an exact finite minimum for the target diag(i,j,k) over the abstract noncommutative register alphabet F2[Q8], with factors D_N(g)=I+(g+1)N, all 28 binary rank-one idempotents, and every nonidentity Q8 label. The one-factor census covers 196 words and the ordered two-factor census covers 38,416; neither contains the target. A direct three-factor word realizes it, so the minimum is three under the stated unit-per-factor convention. The nearest two-factor outputs have coefficient Hamming distance two, attained by six words.

This is scoped finite evidence only. It does not identify a physical supplier, charge an all-size interface, or change the selected conditional bound or kappa.

Relationship to open Q8 work

Open PR #2248 studies diag(-1,-1,i/j) and diag(-1,-1,-1) using paid labels {i,j}. This package studies the distinct target diag(i,j,k) and exhausts lengths one and two with all seven nonidentity Q8 labels.

Pins and reproduction

• Seed 189ad74f7eb7b99d43830c0ed0f15958, track bit-mechanisms, task bit-new-register-model, priority revision 4.
• Base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• python3 -I -O -B research/bit-q8-three-register-two-call-seed189ad74/search.py
• python3 -I -O -B research/bit-q8-three-register-two-call-seed189ad74/verify.py
• python3 scripts/research_package.py --package research/bit-q8-three-register-two-call-seed189ad74 --check-only

The exact search and independent set-based replay pass on CPython 3.14.7; the package metadata and source pins pass. The registered package replay targets Python 3.12, which is unavailable locally; hosted replay remains pending.

The original scope claim is Discussion #2300. Prepared with substantial OpenAI Codex assistance.