Research exact Q8 three-role target minima

[1337isnot1337]

Result

This adds exact three-register paid-call minima in the full group algebra F2[Q8] with x=i, y=j, z=-1=x^2=y^2, paid labels {x,y}, and all 28 rank-one idempotents in M3(F2):

• Each of the six coordinate permutations of diag([z],[z],[x]) or diag([z],[z],[y]) has exact minimum 5 calls.
• diag([z],[z],[z]) has exact minimum 6 calls.

All three registers may contain arbitrary dirty data. Equality is coefficientwise in the eight-element group algebra. The result adds no supplier or kappa change.

Method and evidence

The verifier performs a breadth-first search over exact 3-by-3 group-algebra matrices through depth four. The 56 paid factors produce new-state layer sizes [1, 56, 2324, 89574, 3286437]. It checks sorted-layer digests, Q8 associativity, the C2 x C2 abelianization, all 168 GL3(F2) conjugations of the 28 idempotents, and 952 packed-versus-generic matrix transitions. Direct coordinate words attain the stated upper bounds; all one-call deletion controls fail.

The first-order trace argument supplies the remaining parity step. The mixed targets require odd length, so the complete depth-four census and the five-call direct words give minimum 5. The all-central target requires even length; the census excludes lengths through four and parity excludes five, giving minimum 6. The depth-four census also reproduces the known diag(z,z,1) control at distance four.

Pins and reproduction

• Cycle AF accepted seed: 809fc39d62870d18f94dd48473f46972; track bit-mechanisms; task bit-new-register-model; priority revision 4.
• Exact base commit: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2.
• Catalogue SHA-256: d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7.
• Selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Package: research/q8-r3-two-central-one-generator-seed809fc39d/.

Run from the repository root:

python3 -I -B research/q8-r3-two-central-one-generator-seed809fc39d/verify.py
python3 -I -O -B research/q8-r3-two-central-one-generator-seed809fc39d/verify.py
python3 scripts/research_package.py --package research/q8-r3-two-central-one-generator-seed809fc39d --check-only

The direct verifier and metadata/source-pin check pass locally on CPython 3.14.7. The registered replay targets Python 3.12, which is unavailable locally; the PR workflow will provide that replay.

Scope and predecessor credit

The Q8 central-pair result in merged PR #1822 covers diag(z,z,1) and an anti-diagonal pair; the Q8 mixed-target result in merged PR #1925 covers a different target family. The three-central target was listed as an open question in the #1822 package. Open PR #2047 covers one central output with two order-four outputs for every r>=3. This package is restricted to the stated r=3 target profiles and depth-four search; it does not imply a bound for other targets, register counts, groups, labels, or longer words.

Each target's direct call bill is already optimal in this model. There is no physical mapping of these Q8 factors into the selected bit supplier, and the selected complex primitive still binds. No selected-bound change is claimed.

Prepared with substantial OpenAI Codex assistance. The result is a scoped exact finite obstruction, not an unconditional multiplication theorem or a full theorem audit. Apache-2.0.