CLAIM: UT5 height-three root-pair bound (seed 2df87cb7)

[1337isnot1337]

CLAIM — UT5 height-three root-pair label bound

• Track: bit-mechanisms; seed: 2df87cb762758beffb181880cae1bf82; task: bit-new-register-model; catalogue revision 3.
• Exact base: 0b1cccf. Selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Object/question: In full F2[UT5(F2)], for four-register endpoint diag([I+E14],[I+E25],[1],[1]), what minimum call count is forced using paid labels x1=I+E12, x2=I+E23, x3=I+E34, x4=I+E45 and any rank-one idempotent N in M4(F2)?
• Changed assumption: Four registers and nilpotency-class-four UT5 replace the three-register abelian model; both targets are noncentral height-three roots.
• Method/domain: Exact finite profile enumeration of UT5 label words through the 20-call direct cap, combined with a coefficient-support relaxation and the first augmentation layer. No arbitrary idempotent-word enumeration.
• Cost screen: The direct nested-commutator construction costs 20 unit calls; natural rank gives floor two. A 14-call word would save 30% formally. All supplier, recursive, helper, frame, copy, hosting, restoration, precision, and interface expenses remain unbuilt, so no kappa claim follows.
• Predecessors/overlap: merged PR Research theorem: exact paid cost for binary affine register words #1333 (head 7f25b6d) uses the full affine-label alphabet; open PRs Research: UT4 central-target tariff bound with arbitrary dirty workspace #1662/Research: UT4 two-target unit count is independent of dirty workspace #1680/Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690 concern UT4 central roots; Discussions CLAIM: third dirty register for the D8 central commutator (seed 89dea4bc) #1738/CLAIM: four-register S4 sharing for distinct noncommuting 3-cycles (seed a27baaa7) #1740/CLAIM: UT4 noncentral root pair through four calls (seed bfe59e15) #1759 concern D8/UT3 central, S4 cycles, and the just-completed UT4 E13/E24 pair. Exact-term refresh at 09:13:37Z covered 437 open PRs and 399 Discussions, no warnings or UT5-root matches.
• Deliverable: A precise lower bound, complete profile certificate, exact 20-call positive control, missing-label controls, source pins, and reproducible verifier under research/ut5-root-pair-multiplicity-seed2df8/.
• Expiry: 2026-10-14T11:20:00Z.

[1337isnot1337]

COST UPDATE — improved positive control, same claimed scope

The initial 20-call cost screen used nested commutator expressions of length ten per root. A complete BFS in the 1024-element UT5 Cayley graph found shorter eight-letter words for both roots, so the direct coordinate-projector endpoint costs 16 calls. The task card and profile cap are updated to 16. The original 20-call expression remains a valid but nonminimal control; the target, alphabet, and claimed question are unchanged.

The exact words are:

• u=E14: x1 x2 x3 x2 x1 x2 x3 x2
• v=E25: x2 x3 x4 x3 x2 x3 x4 x3

The profile enumeration now covers all count vectors through that 16-call direct cap.