[petalas]

Summary

A scoped proof, conditional on the cited spectral inequality in #5063, lowers the register-independent three-cycle permutation-cut ceiling to 17t/3. Final rounding therefore certifies at most 17 calls for three targets. This method cannot prove the direct 6t tariff already at t = 3.

- ceiling: ((27 + 9 sqrt(2))/7) t; obstruction starts at four targets
+ ceiling: (17/3) t; obstruction starts at three targets

The new step uses the 168-element regular action, the order-three crossing cap k <= 56, the even six-CNOT boundary, and a rational gap. Orbit lifting extends it to every finite action, address dimension, target triple/orientation and dirty-register count in the stated formal model. This contribution was produced by Nick Petalas. Krzysztof Wojt's spectral theorem and James Reed's cut formulas retain their credit; neither is replayed or claimed as new.

Evidence

• All 14,364 scalar pairs checked exactly. The maximum allowed by the retained scalar constraints is 17/3, at (k,d) = (17,18), (34,36), (51,54). These are arithmetic witnesses, not asserted realizable cuts.
• All 128 seven-point subsets and their regular lifts checked for fiber multiplicity, crossing transport, parity and the order-three cap. Three essential-hypothesis controls reject the false statements.
• Registered normal and optimized warning-as-error checks pass on CPython 3.12.15, Darwin 27.0.0 arm64. Each direct check took 0.03–0.04 seconds; the complete package replay took 0.29 seconds.
• PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES with source-history ancestry pass at 1ac2f221a987b2cd0a5154e762ecad03b854eb60. The generated stdout report binds the five package files by SHA-256.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/three-cycle-boolean-rounding
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/three-cycle-boolean-rounding/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/three-cycle-boolean-rounding/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/three-cycle-boolean-rounding --check-source-history

Any standard-library CPython 3.12 interpreter can replace the displayed local interpreter path.

Limitations and unrun obligations

The mathematical corollary depends on #5063 at 9ad4c47068f900fa1619191292539adcced0d638. Its spectral certificate is a cited external theorem, not a runtime input or independently verified premise here. Local PASS checks the new arithmetic and lifting step. The exact Boolean-cut optimum, attainability of 17/3, actual three-target word minimum, independent human/formal-kernel review, native realization, supplier/compiler/precision/storage admission and selected-construction cold replay remain outside this result. There is no new kappa or frontier change.

Open next questions

1. Determine the exact Boolean-cut optimum on the 168-element regular action using a rigorously certified primal/dual pair and complete optimality argument. No attaining cut or global Boolean certificate is provided.
2. Decide three-target word additivity with a nonlinear or integer label-count invariant beyond this cut cone. No new invariant or complete three-target word census is provided.

Related open work

• Prove a uniform ceiling for three-cycle permutation-cut certificates #5063 studies the same three-cycle permutation-cut quantity. Its spectral ceiling is the cited predecessor; this adds Boolean-integrality rounding and moves the obstruction threshold from four targets to three. It does not recompute its spectral numbers.

CI notes

CI pending for 1ac2f221a987b2cd0a5154e762ecad03b854eb60. Both registered commands and the committed-byte/source-history check passed locally. Hosted CI has not certified this contribution yet.

Advance claim.

Merge Danger

Door: two-way. Blast Radius: isolated research package. No shared code, selected frontier or predecessor package changes.