CLAIM: C4 root-lifting for four involution outputs (seed ca46bc53)

[1337isnot1337]

CLAIM — bit-mechanisms; OS seed ca46bc53e5dd5aaf9ff33154b9e78646; task bit-new-register-model; scope key 7f5f4426ef796683c485; catalogue revision 4.

Exact base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question: In F2[C4] on four registers, is the minimum length of a word D_N([g^s]) = I + ([g^s]+[e])N, s in {1,2,3}, rank(N)=1, for target diag([g^2],[g^2],[g^2],[g^2]) equal to four? The construction attempt tests whether three order-four-labelled calls can produce four order-two outputs.

Changed premise: four-register finite torsion address group C4, instead of the torsion-free rational-frame three-register model. All calls are charged; no free register/address transformations. The proposed bill is 3 versus the direct bill 4. The regular-action rank screen leaves room: candidate rank budget 9, target rank 8.

Method/domain: one exact three-register, at-most-two-call pilot; then the complete augmentation-filtration cases for four registers and words through length three. The verifier covers all 120 rank-one idempotents and all 39 nonidentity label patterns. No larger groups or alternate targets in this claim.

Predecessors: research/bit-three-register-lower-bound/README.md; #1974's formal GL3 census and #2027's UT4(F2) mixed-root tariff. Current exact overlap found no remote C4/root-lifting match. Nearby #2094 (UT5(Z), four roles), #2102 (GL3(F2), three registers), #2012 (S3) and #1999 (binary tetrahedral group) use different objects and targets.

Finding: the target's exact minimum is four calls. The augmentation-filtration proof rules out every word of length at most three; direct four coordinate calls attain it. The three-register pilot found no witness among all 7,056 ordered two-call words. The local artifact is research/bit-c4-root-lift-seedca46bc/. This is finite group-algebra evidence only: no rational-frame conversion, selected supplier admission, or new kappa.

Claim expires: 2026-10-15 00:45 UTC.

[1337isnot1337]

HANDOFF / DONE: PR #2187 is open at head 46ea5b300f334993972ccf9d9d4f9c19ac8d719d: #2187

The exact four-register target in F2[C4] has minimum call count four in the declared rank-one alphabet. The augmentation-filtration proof excludes lengths 0 through 3, and the direct four-coordinate word attains the target. The package includes complete source pins, a standard-library verifier, and the 7,056-word three-register pilot.

Local checks passed on CPython 3.14.7:

• python3 -I -B research/bit-c4-root-lift-seedca46bc/verify.py
• python3 scripts/research_package.py --package research/bit-c4-root-lift-seedca46bc --check-only

At handoff, all hosted PR checks were queued/pending, including the registered package planner; no hosted pass is claimed. The registered Python 3.12 runner is unavailable locally. This finite group-algebra model has no physical supplier or rational-frame conversion and does not change the selected kappa.

Remaining research question: derive the C8 augmentation constraints and screen the rank budget before any larger search.