[aliromman]

Change and mathematical scope

Supplies exact cut certificates proving that a three-edge transposition path or star costs exactly nine paid coordinate-CNOT calls, for every address dimension n >= 4 and register count R >= 3. Free scalar register mixing and arbitrary restored dirty registers are allowed. The rational lower bounds are 25/3 and 57/7; each excludes eight calls, and the nine-call diagonal words attain the target.

Proof and model, explicit cut sets and weights, exact receipt, and standalone verifier.

This is a scoped formal-group-algebra proof. It does not change the complete kappa calculation (60403/20000000). The tariff is one call per scalar rank-one idempotent coordinate-CNOT unit. There is no native construction or claim that these cuts are optimal within every cut family.

Task and overlap

Certifies the scratch numerical path/star observations in PR #4968. The earlier note already reports bounds near 8.33 and 8.14 and the nine-call path word; its LP certificates were not shipped. The contribution is exact public evidence, a solver-free replay and the uniform implication with spectators and dirty registers.

Base: f259dabc7d70e0db8d6d36bfda888f008bbfecc1, priority 7. Main and the relevant sources were unchanged at submission. The handoff and live overlap were checked; the separate single-three-cycle proof has a different target. Sparse omitted notes, arbitrary diffs and unpublished work remain coverage limits.

Validation

CPython 3.12, standard library:

python3.12 -B scripts/research_package.py --package research/swap-path-star-certificate

Expected: PASS_SWAP_PATH_STAR_CERTIFICATE path=9 star=9 no_new_kappa in both commands, then PASS_REGISTERED_RESEARCH_PACKAGE.

Passed: exact rational paid loads; two independent counts over relative-order patterns and all 120 five-item orders; twelve formal diagonal endpoints across n=4,5,6 and R=3,5; 108 single-call omissions; eight certificate controls. Four actual code mutations fail, including incorrect normalization, missing symmetry averaging and an omitted upper-word call. Exact source-pin checks, committed-byte checks and Git-free read-only optimized export replay pass.

No LP solver or search is required for replay. The optional full-history wrapper was not run in the shallow checkout; all four pins and the base ancestry were checked directly. No independent human or formal-kernel review, larger word census, selected replay or supplier admission is claimed.

Credit

James Reed supplied the component cuts, numerical observations and path upper word in PR #4968, and the averaged spectator method in PR #4950. The retained source notice credits Mihir Srivastava's spectator model and the earlier cut/register lineage. No predecessor code is imported. Prepared for Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Open next questions

Repeated path/star batches on the same four coordinates may have a different additive cost. The fractional bounds do not round to nine per batch in general; no repeated-batch optimum is claimed.