[petalas]

Summary

Scoped proof with exact finite corroboration, answering Cinna Davis's PR #2271 Open next question 2. For every n>=2 and positive toggle prices sorted c1>=...>=cn, the minimum closed borrowed-anchor tour through every pair mask is

minimum transitions = 2*c1 + 2*sum_(i=2)^n (i-1)*ci
visit {1,n},...,{1,2}; {2,n},...,{2,3}; ...; {n-1,n}
write each assigned dirty target at its mask; restore the anchor

The forest lower bound permits arbitrary intermediate masks and repeated visits. For four toggle sources and two targets per pair, the complete restored-source map costs 26 row ADDs instead of 36 direct ADDs, saving ten. Prices (7,3,2,1) have exact transition optimum 34. The new theorem, constructor, proof and independent finite oracles are produced by Nick Petalas. Cinna Davis retains the borrowed-source interface, transition-sharing example and open-question credit.

Evidence

• Before: direct independent source writes emit 36 ADDs for the four-source/twelve-target map. After: the emitted tour has 14 toggles and 12 writes, totaling 26. Both complete maps are checked on exact basis columns; the new circuit is also checked on all 131072 dirty states of that 17-row instance.
• Warning-strict normal and optimized verification pass: 121 permutation optima, 117 literal hypercube optima, 189 larger tours, 630 circuit basis columns, 132128 complete dirty states and 168 wide-row fixtures. Thirteen invalid inputs, four tour corruptions, three circuit corruptions and one legal expensive order exercise the boundaries.
• Each registered check takes about 1.2 seconds locally, far below its 120-second bound. The final registered hook completes in 2.78 seconds. Metadata/provenance pins, committed bytes and source history pass. Generated normal/optimized source-bound receipts agree; an isolated Git-free optimized replay passes. A separate read-only proof/code review found no blocker. No independent human or formal proof-kernel review is claimed.

Exact reproduction from the repository root, CPython 3.12 and standard library only:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/weighted-pair-mask-tour
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/weighted-pair-mask-tour/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/weighted-pair-mask-tour/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/weighted-pair-mask-tour --check-source-history

Expected direct status: PASS_WEIGHTED_PAIR_MASK_TOUR. Replace the interpreter prefix with a local CPython 3.12 installation elsewhere.

Limitations and unrun obligations

The alphabet permits only anchor toggles from unchanged sources and one anchor write per target. This is an optimum within that class, not an unrestricted CNOT minimum. The all-n real-price theorem is a written proof; the executable supports exact rational prices, 2–64 sources and a uniform 1–8 targets per pair. Native frames, actual caller occurrence, hosting, tape movement and recursive billing remain external. No predecessor executable, full selected cold replay, Lean or Docker run was performed. No supplier admission, complete multiplication theorem, new kappa or selected-frontier change is claimed.

The bit-helper-contract card explicitly changes scope from R439 H2/K to this local row-register side result. Canonical base is 624de9dc66858af0945e5f8a0942488d62280d43; the source README and catalogue are provenance pins. The advance CLAIM, live open PRs/claims, merged-term searches and main README object/formula searches were checked before publication. Sparse local notes and unpublished work remain overlap limits.

Related open work

• #3891 prepares and restores one fixed helper response; this result optimizes transitions across the complete varying pair family. Its R439 interface is not imported or rechecked.

• #1865 groups exactly commuting complex fanout intervals; this proof concerns binary source-mask tours and makes no physical profile comparison.

• #4891 lower-bounds dense dirty-output bank scans under monomial maps; this result prices one-anchor scalar row ADDs with specified pair demands. No predecessor numbers are recomputed for comparison.

• #4896 determines dirty-root stock for a fixed repeated source order serving every nonzero mask; this result fixes one borrowed anchor and optimizes weighted transitions through pair masks. Root stock and tour cost are different quantities, and no numerical comparison is inferred.

• #4894 studies within-class pair-sum excess on simple binary prefix walks; this result concerns the cost of visiting a prescribed complete pair-mask family and allows repeated walk states. Its census and bounds are not replayed here.

Open next questions

• Locate an actual aligned caller demanding these pair parities at one cut, then price its frames, references, restoration, storage and tape movement.
• For an actual caller needing only a subset of pairs, determine its weighted tour optimum. The complete-pair forest bound need not remain sharp.

CI notes

Both hosted checks passed for head 1957aba96cf0d0f09d89e0588bdde875885ed44c: plan and the registered package replay. Successful CI run. The registered replay executes the warning-strict normal and optimized verification commands. Local registered replay, direct normal/optimized checks, Git-free replay and committed-byte/source-history preflight also pass.

The post-CI refresh inspected this account's ten open PRs created since the CLAIM, including this PR. Newly relevant #4896 and adjacent #4894 are listed above; #4891 was already listed. These results optimize different quantities, with no computed numerical disagreement identified.

Merge Danger

Door: two-way. Blast radius: isolated research package. Seven new package files; shared code, predecessor sources and the selected record are unchanged.