Research: exact five-call minimum for binary tetrahedral mixed triple (no new kappa)

[1337isnot1337]

Result

Adds research/sl2f3-conjugate-triple-seed592f/, a scoped exact result in the full formal algebra F2[SL2(F3)]. For the three-role target diag([i],[w],[w^-1 i w]) and four address labels {i,i^-1,w,w^-1}, the exact paid-call minimum is five. The separate direct word costs five; exhaustive exact coverage excludes every four-call-or-shorter word. This is a negative result for one new nonabelian alphabet, not a new kappa or supplier construction.

Exact model and evidence

The group has order 24, generated by the pinned matrices i=[[0,1],[-1,0]] and w=[[0,-1],[1,-1]] over F3. The paid factor is D_N(g)=I+([g]+[e])N for each of the 28 binary rank-one idempotents in M3(F2), giving 112 paid factors. Free scalar GL3(F2) mixing is included by augmentation/conjugation closure. Every paid factor costs one.

The verifier covers all lengths 0 through 4. The 12,544 ordered two-call words collapse to 8,905 exact states; each distinct state is checked as a suffix against the complete prefix set for both 1+2 and 2+2 splits. This exhausts all 1,404,928 length-three and 157,351,936 length-four raw words, as well as shorter lengths. All 12,544 two-call products are cross-checked by independent full coefficient convolution. The direct five-call word, all five call omissions, a mutated generator, the factor inverse law, group-table associativity, and a known two-call target control pass. CERTIFICATE.json records the two-call state digest 7313957379a9c51963604fae5f6a69edda1771006526c9c2ef4ab068e76bed95.

The direct target costs five. A four-call factorization would reduce this finite model bill by 20%; the exact minimum remains five. The bit-model address labels have no priced implementation in the selected multiplication supplier, so this experiment yields no exponent change.

Pins and overlap

Track bit-mechanisms; seed 592f04c45f93ecfac35ea0ab971b12ba; task bit-new-register-model; priority revision 3; catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075; exact research base 1c47876ef742ba64bf9140b0527d63acf656a3b4; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32. The package pins the in-tree three-register abelian predecessor. The nearest nonabelian results are PR #1822 (head ee736b4b11bbd63b04d94c8edafb3790161ebdea, Q8 repeated central-commutator pair) and PR #1925 (head 009ed860f34ed5d7b0d626098ce93c4c2a7eb4df, Q8 mixed targets); this scope changes the group to SL2(F3) and uses the heterogeneous i,w,w^-1iw endpoint. It imports no predecessor code. The exact-term overlap search found no separate SL2(F3) or binary-tetrahedral PR/claim.

The selected record remains the maintainer-reviewed conditional finite construction with kappa = 836330098538459/500000000000000000; its certificate and reproduction obligations are unchanged. The experiment does not invoke the selected-source cold replay.

Verification and limits

• python3 -I -O -B research/sl2f3-conjugate-triple-seed592f/verify.py — passes on CPython 3.14.7.
• python3 -B scripts/research_package.py --package research/sl2f3-conjugate-triple-seed592f --check-only — passes source pins and metadata.
• git diff --check — passes.

The package runner requests Python 3.12; it is not installed locally, so I did not run that wrapper. The direct exact verifier passes under the available interpreter. No proof-assistant formalization or physical source/hosting/precision/restoration budget is claimed. The result is limited to the named finite address group, alphabet, endpoint, and three roles.

Attribution

The model is informed by PR #468's scoped three-register abelian result and the Q8 predecessors above. Prepared by Cinna Davis with substantial OpenAI Codex assistance; no independent human review is claimed. Apache-2.0.

[Mister-Meeseeks]

Batch review at 125ed77d114e0db0cb2c684494c7a5c90f559d60.

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

Reviewed scope: Complete formal SL2(F3) three-role alphabet and meet-in-the-middle search exclude all words through four calls for the chosen mixed target; five calls attain it. All inverse/suffix joins use full group-ring products, with convolution cross-checks. Fixed local alphabet and no extra roles; no implementation claim.

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.