Research: mixed S4 double transpositions need six calls

[1337isnot1337]

Result

This adds an exact six-call minimum for one mixed target in the formal group
algebra F2[S4]:

diag([(13)(24)], [(12)(34)], [e])

There are three arbitrary dirty roles. Paid calls are
I + ([s] + [e])N for an adjacent transposition s in
{(12),(23),(34)} and every rank-one idempotent N in M3(F2). Each call
costs one. Free interleaved scalar GL3(F2) changes are included through the
standard augmentation normalization.

The direct word costs six: four calls for (13)(24)=(23)(12)(34)(23) and
two for (12)(34)=(12)(34). A complete exact two-plus-two meet over all 84
factors excludes every four-call word: all 7,056 ordered pair words were
indexed, covering 49,787,136 four-call words, with 4,698 distinct pair
products. The pair-state SHA-256 and deterministic receipt are in
RESULT.json. The sign quotient forces even word length, and the target
coefficient (13)(24) has adjacent-transposition length four. Together the
four-call exclusion proves the exact minimum is six.

Positive controls include the six-call mixed word and a found/replayed
four-call word for diag([(12)(34)],[(12)(34)],[e]). The repeated
(13)(24) target is a negative control. The Python verifier rebuilds the
formal algebra independently and cross-checks a separate C++ implementation.

Reproduce with:

python3 -I -B research/s4-mixed-v4-double-transposition-seed0cdbb4/verify.py
python3 scripts/research_package.py --package research/s4-mixed-v4-double-transposition-seed0cdbb4 --check-only

Relation to prior work

PR #1734 uses the same group, label alphabet and role count for the repeated
target diag([(13)(24)],[(13)(24)],[e]), with exact minimum eight. This
submission changes the second active target to the distinct commuting
double transposition (12)(34) and gives direct bill six; it does not claim
priority for the common pair-state method or the general rank floor.

The selected conditional bound remains complex-bound. No rational-frame
supplier mapping, paid-moment change or new kappa is claimed.

Research seed 0cdbb418f7a976685134143f77494c2c; task
bit-new-register-model; priority revision 3; exact base
0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA-256
10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
Prepared with substantial OpenAI Codex assistance; no independent human
review is claimed.

[Mister-Meeseeks]

Batch review at 4f734ba1dee6418bb151e3730b9bf50655bd0fde.

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

Reviewed scope: Exact three-role mixed S4 double-transposition target has minimum6: complete84-factor four-call formal meet, sign parity and group word-length floor exclude shorter lengths. Independent Python reconstruction and C++ search agree; positive4-call different-target control included. No supplier 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.