Research: exact four-call floor for a D8 rotation/inverse pair

[1337isnot1337]

Result

In the exact two-role formal F2[D8] model with paid reflection labels s,t only and arbitrary free GL2(F2) mixing, the noncentral rotation/inverse target diag([r],[r^-1]), r=st, has exact paid-call minimum four. The direct four-call word passes; exhaustive full-coefficient enumeration rules out every word of length zero through three.

The regular representation gives only the lower bound three (target difference rank 12; each paid factor has rank 4). The exact word enumeration rules out that remaining three-call possibility.

Scope and relation to predecessors

This is a target-class extension of merged PR #1443, which uses the same group, two reflection labels, two roles and free scalar mixing but proves an eight-call minimum for the central target diag([r^2],[r^2]). This PR concerns the noncentral order-four inverse pair and its four-call bill. It does not restate the central-target result. Other nearby work includes the open quaternion central-pair result in PR #1822 and the D8 third-role central-pair claim in Discussion #1738.

Reproduction

python3 -I -B research/d8-rotation-pair-seedff568/search.py
python3 -I -B research/d8-rotation-pair-seedff568/verify.py
python3 scripts/research_package.py --package research/d8-rotation-pair-seedff568 --check-only

The search covers 1, 12, 144, 1728 words at lengths 0,1,2,3. The independent verifier checks the direct word, complete group-algebra boundary, altered and truncated controls, and the rank screen. The registered package replay requests Python 3.12; direct checks in this environment use Python 3.14.7 because Python 3.12 is unavailable.

This is a scoped abstract address-model result only. The selected supplier and selected moment are unchanged; there is no new kappa or physical/compiler admission. Substantial OpenAI Codex assistance; no independent human review is claimed.

[Mister-Meeseeks]

Batch review at 5a51deaaab2746d42eddb454bbb968bda3af02cc.

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

Reviewed scope: Both rank-one search and independent full group-algebra matrix replay exclude all1885 words through length3; direct D8 inverse-rotation pair attains4. Regular-representation rank bound3 is correctly distinguished from the exhaustive stronger result. Two-role reflection alphabet only.

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.