[1337isnot1337]

Result

This package settles the explicit third-dirty-register question for the two-register D8 central-commutator model. With three arbitrary dirty scalar registers and the fixed paid alphabet below, the exact minimum remains eight calls: the direct word costs eight, and an exact search excludes every word of seven or fewer. This is a scoped finite-model result, not a multiplication-bound improvement.

Model and evidence

• G = UT3(F2), with x=(1,0,0), y=(0,1,0), and z=xyxy=(0,0,1).
• Target: diag([z],[z],[1]) in the full 3x3 matrix algebra over F2[G]; the third input is arbitrary and is restored exactly.
• Paid factors: I+([g]+[1])N, g in {x,y}, and every rank-one idempotent N in M3(F2): 28 idempotents, 56 paid gates. Each gate costs one. Free interleaved GL3(F2) mixing is normalized into this complete alphabet by augmentation.
• Exact meet-in-the-middle covers the identity ball through radius 3 and target ball through radius 4. The layer counts are 1,56,1750,45424 on each radius-three side; there are zero radius-three intersections and zero matches among all 2,543,744 target radius-four boundary edges.
• Full coefficient convolution verifies the direct eight-call word. All 56 gates are involutions; all eight direct-word omissions and an altered third target are rejected.

Reproduce

python3 -B scripts/research_package.py --package research/d8-third-register-seed89de

Direct verifier and metadata checks passed under local Python 3.14, including an optimized run. Python 3.12 is not installed locally, so the registered Python 3.12 replay remains for hosted CI. The selected supplier, paid profile and kappa are unchanged; physical hosting and supplier admission are outside this formal model.

Pins and attribution

• Research seed: 89dea4bc3a42ef85f785169cd2b6977e; track bit-mechanisms, task bit-new-register-model refined to the D8 third-register extension.
• Exact research base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Predecessor: woklerx's two-register proof, commit 965ff75aabb16cdd0978f4bf9222bcdedf514021; source files and certificate are pinned in the package.
• Fresh pre-submit overlap refresh at 2026-10-14 08:24 UTC covered 419 open PRs and 374 Discussions with complete pagination and no warnings. The exact-scope claim is Discussion #1738; nearby claims use different groups/targets.

Prepared by Cinna Davis with substantial OpenAI Codex assistance in derivation, implementation, verification, and writing. No independent human review or full multiplication theorem proof is claimed.

[JosieElliston]

Cross-reference for review: this package's r = 3 model is the same as #1478's r = 3 model (head b30fdba4). Your radius-3 layers 1 + 56 + 1750 + 45424 equal its 47,231 states. It is also the k = 2, r = 3 case of #1573 (head 0b8c5a69, every r >= k). #1573 settles the four-register follow-up in the DONE note. #1792 indexes this family, audits #1573's proof (no gap found) and reproduces your counts with code written separately. Your search stays a valid independent corroboration; this comment does not propose any disposition.

[Mister-Meeseeks]

Batch review at 679f3624a0a41ed14942825dd7f413132412d1e3.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete three-role formal UT3/F2 balls and all radius-four outgoing edges exclude seven calls for the central pair, with an eight-call attainer. Free scalar mixing is covered by normalization; the result does not constrain other groups or supplier costs.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.