Exact S4 cost: two double-transposition targets need eight calls

[1337isnot1337]

Result

In the explicit formal F2[S4] model with three dirty scalar roles, free GL3(F2) mixing, and unit-cost paid adjacent transpositions (12),(23),(34), the target diag([(13)(24)],[(13)(24)],[e]) has exact minimum 8 calls. Its direct cost is 8, while the natural-representation displacement-rank floor is only 4, leaving a concrete two-call sharing gap to test.

The sign determinant over F2[C2] forces every valid word to have even length, so a strict improvement would require at most 6 calls. A complete exact radius-three meet over all 84 involutive factors contains 237,474 formal states on each side and has empty intersection. Thus no word of length at most 6 exists. The direct 8-call group-algebra identity supplies the matching upper bound.

This changes the target cycle type from the two 4-cycles in PR #1720, where displacement rank matches direct length. Here each double transposition has Coxeter length 4 but displacement rank 2. The result closes only this fixed target/alphabet/role model.

Reproduction and provenance

python3.12 -I -B research/s4-double-transposition-seedb270/verify.py
python3.12 scripts/research_package.py --package research/s4-double-transposition-seedb270 --check-only
python3.12 scripts/research_package.py --package research/s4-double-transposition-seedb270

The registered replay passes. Exact ball counts, deterministic state hashes, eight call-omission controls and the altered dirty-role control are recorded in the package. The package pins base 0b1cccf86e2f5132ed286a26ef0a681a495740ea, accepted bit-mechanisms seed b270b3b52d302ab6281ea2bc042d4aea, priority revision 3 and selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

The selected binding-only baseline remains complex-primitive-bound. This abstract finite address model is not mapped to a supplier and makes no recursive-moment or κ claim. No selected source, certificate or frontier is changed. The completed finding is recorded in coordination Discussion #1163. Prepared by Cinna Davis with substantial OpenAI Codex assistance, disclosed in TASK.json; no subagents or independent human review.

[Mister-Meeseeks]

Batch review at 48855b620b23af3a7fe87d53bde05cd2211342c8.

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

Reviewed scope: Complete formal S4 radius-three meets plus sign-determinant parity prove eight calls for two specified double-transposition targets with one dirty role. Natural displacement rank supplies only a weaker floor; no extra-alphabet or supplier conclusion follows.

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.