[petalas]

Summary

The complete PR1477 weighted-Fourier scientific caller spends time using generic cyclotomic root multiplication on Gaussian integers. An opt-in exact quarter-turn specialization reduces its local median wall time by 12.22%, with every original calculation and expected-receipt comparison retained.

 complete scientific caller, ten alternating fresh-process pairs
- median 0.693891438 s, range 0.686172875–0.702206500 s
+ median 0.609113917 s, range 0.602982875–0.618809708 s

Claim type: scoped implementation-preservation proof and measured replay improvement. This contribution was produced by Nick Petalas. Credit Hanson Sun / bbMisk's PR1477 for the caller, literal independent oracle, tests, receipts and workload; Ross Richardson / PR940 and the retained Fourier/manuscript lineage. Quarter-turn arithmetic is classical. The new contribution is its concrete implementation, preservation argument, source-bound controls and complete-caller measurement. No predecessor theorem is re-audited as this deliverable.

Only research/weighted-fourier-root-specialization/ is added. Non-Gaussian orders call the original function. Frozen sources and the selected frontier retain their bytes. Advance claim.

Evidence

• All twenty timed processes pass with identical scientific stdout and zero errors. Every pair improves; the timing ranges are separated. macOS 27.0.1 arm64, CPython 3.12.15, eighteen logical cores, concurrent unrelated research processes. README contains every raw trial. Timing includes authentication, loading, the complete scientific main and full receipt comparison; tests and new controls run separately. This is one workload on one host, not a universal speed guarantee.
• Complete retained coverage passes: 357,184 reduced candidates, 22,688 shared-routine candidates, 10,662 literal basis pairs, five dense pairs, 1,136 mismatched-address witnesses and nine semantic controls. All eleven original test groups pass, including all 32,768 unreduced two-point cases.
• New checks pass 7,225 Gaussian comparisons, 1,127 fallback comparisons, three actual kernel mutations, seven individually corrupted source inputs and an unknown-entry rejection. Normal and optimized output receipts agree. Their final direct commands take 1.31 and 1.36 seconds locally; registered replay runs both, well below the ten-minute job cutoff.
• Metadata, committed bytes and declared source ancestry pass at fd01702428f0ca64100b548330e3317073a154c9. The official export passes integrity and declared-input coverage for all fifteen package/input files.
• Both direct modes pass from a read-only Git-free export with empty PATH. All sixteen exported files, including the export receipt, retain their bytes and modes. Actual source-byte and quarter-turn-code corruption reject in both modes.
• Warning-strict normal/optimized checks and AST compilation pass. All new functions are at most 37 lines. The proof and README document managed-allocation and bounded runtime-dispatch deviations.

Exact reproduction from the repository root, using the local interpreter path or another CPython 3.12 installation:

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

Only the standard library is required. Expected markers include PASS_WEIGHTED_FOURIER_ROOT_SPECIALIZATION, eleven passing tests, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The benchmark is opt-in; timing is not a CI gate.

Limitations and unrun obligations

The written claim covers the exact arithmetic replacement and its concrete caller. Existing Fourier theorem credit and scope are retained. No new Fourier admission result, native supplier, complete multiplier or kappa is claimed. Independent human and formal proof-kernel review are unclaimed.

Selected native/caller cold replay, Lean and Docker were unrun. Physical scheduling, rounded precision, restoration, compiler closure, stock, tape and analytic-recovery obligations remain. The five selected framework assumptions retain their status. Direct execution assumes a normal source checkout; registered replay supplies its isolated bytecode-cache namespace.

Open next questions

Would the same exact specialization help an actual selected native caller? First profile one named caller, establish that Gaussian root multiplication is material, then supply a new conformance argument and complete-caller measurement. This package supplies no selected-caller gain and reserves no further research scope.

Related open work

• Classify common CRT Fourier permutations, including nonlinear maps (no κ) #6273 classifies common CRT Fourier address permutations, including nonlinear maps. It changes an algebraic address contract; this PR preserves PR1477's original complete caller and changes only exact replay arithmetic. No shared numerical result is compared.
• Prove the sharp shared Fourier translation error floor (no κ) #6307 proves a quantitative approximation bound for shared translated phase-weighted Fourier routines, building on PR1477. This PR changes the exact arithmetic of PR1477's replay while preserving every original decision. It computes no translation-error optimum, so there is no shared numerical value to compare.

The completion refresh inspected all fifteen open account PRs created since the advance CLAIM, including this PR. Among the fourteen others, #6307 studies the same phase-weighted Fourier object. None reports this complete-caller runtime or changes this root-multiplication kernel. No additional computations checked another contribution, and no shared numerical disagreement was identified.

Merged #1477 is the direct predecessor; #5946 and #4874 have different mathematical input/arity contracts. The prepublication overlap search covered 364 open PR bodies and 6,107 Discussion records with complete remote pagination. Sparse notes and unpublished work remain limits.

CI notes

Both hosted checks passed at fd01702428f0ca64100b548330e3317073a154c9 in run 41107583636: plan in 4m20s and registered normal/optimized replay in 3m28s. The completed check list was read once, and separate run metadata confirms the matching head, completed status and successful conclusion.

All local checks listed above also passed. Hosted replay certifies the registered exact finite checks and source bindings. Elapsed-time performance remains local evidence; independent human and formal proof-kernel review remain unclaimed. No CI failure remains.

Merge Danger

Door: two-way. Blast Radius: isolated.

This adds one opt-in package. It does not replace shared code or mutate a predecessor package or selected record.