Research: four-role mixed D8 target has exact five-call minimum

[1337isnot1337]

Result

This adds an exact minimum for one four-role heterogeneous target in the formal group algebra F2[D8], where D8=<a,b | a^2=b^2=e,(ab)^4=e> has order eight. Paid factors are I+([g]+[e])N for g in {a,b} and every binary rank-one idempotent N in M4(F2). The target is diag([a],[ab],[ba],[e]).

The exact minimum is five. The package gives the direct word
D_(a,E00) D_(a,E11) D_(b,E11) D_(b,E22) D_(a,E22) and excludes every word of length at most four by exact coefficient arithmetic over all of F2[D8].

Verification

The replay enumerates all 120 idempotents and 240 paid factors, forms all 57,600 ordered pair products (32,651 distinct states), and checks the target-specific length 0–4 meetings. Matrix entries retain all eight group coefficients. It also checks the positive five-factor word and seven negative controls.

python3.12 -B scripts/research_package.py --package research/d8-mixed-target-seeda97c8e

Local result: PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result; git diff --check passes. Package metadata and exact base pins pass. The selected-result baseline orientation script was attempted but cannot complete in this sparse worktree because its selected certificate package inventory is not materialized; that limitation does not affect the D8 certificate.

Scope and attribution

This result extends the D8 line of PR #1742 (head 679f3624a0a41ed14942825dd7f413132412d1e3), which establishes an eight-call minimum for a three-role duplicated central-commutator target and leaves four-role models open. This contribution changes the role count and target to a reflection, both orientations of an order-four rotation, and one identity role. The initial S3/GL2(F2) candidate was dropped because PR #1763 (head 07910c7bee862dadbb3b3d30ea5dfcdbbbe23fda) uses the same four-role alphabet and pair-state method for repeated order-three targets. No predecessor executable is imported.

Cycle seed a97c8efa02e2955139574300f12b9c03, track bit-mechanisms, task bit-new-register-model, priority revision 3. Exact base 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

The fourth role is restored on arbitrary inputs. The full finite model is counted, but no source-word mapping, supplier adoption, paid-moment change, or kappa improvement is claimed. Prepared with substantial OpenAI Codex assistance; no independent human review is claimed.

[Mister-Meeseeks]

Batch review at 16b032604b73befc27c62b1ae6c471d263217fe7.

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

Reviewed scope: Full F2[D8] four-role census of240 factors and all57600 pairs excludes lengths up to4 for diag(a,ab,ba,1); explicit5-call word attains. Scope is the declared alphabet and roles, without supplier implementation or a broader free-operation 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.