[petalas]

Summary

Scoped written proof with exact finite corroboration. For every finite product of cyclic groups and every nonzero order-two translation, a two-input shared phase-weighted Fourier routine has the sharp optimum

min relative coefficient error = min worst basis-pair relative output error
                               = sqrt(2 - 8/(3 sqrt(3)))

attaining weights: w(x) = 1/sqrt(3) - i sqrt(2/3) chi(b,x)

The same weights attain the optimum on every basis-input pair. Both forward passes share the routine, and the final pass conjugates its coefficients. Both Fourier signs use 1/N, so the desired output is (u*v)/N². No new kappa is claimed.

Hanson Sun / bbMisk's PR1477 already excludes exact translated shared convolution, extending Ross Richardson's PR940. The new step is the quantitative optimum, its two-moment proof and attaining phases, produced by Nick Petalas. Elementary character and moment methods retain their mathematical credit. No predecessor census is replayed as the deliverable.

Only research/shared-fourier-translation-floor/ is added. Advance CLAIM. The originally posed approximation question is a changed joint-query-controller side result concerning a precision/analytic boundary, not a selected paid-word improvement.

Evidence

Before, the exact classification established impossibility under this shared contract. The new proof gives a group-size-independent positive error floor and an equality construction.

• Exact arithmetic over Q(sqrt(2),sqrt(3),i) checks 16 group/offset fixtures and all 772 basis-input pairs, 1,332 rational phase assignments, 120 additional literal basis-pair comparisons, 512 field-basis triples and 560 scalar grid points.
• The polynomial identity proving the scalar maximum is checked coefficientwise. Literal three-pass circuits, coefficient norms and scalar moments agree. Eight malformed-domain rejections and six semantic controls distinguish missing conjugation, wrong global phase/sign and an exact escape with independent phases.
• Warning-strict normal and optimized direct checks pass with identical generated hash-bound receipts. The two registered commands were timed separately at 1.40 s and 1.43 s locally, below their 120 s limits. Registered metadata and both replay commands pass.
• Committed bytes, owner-execute modes and declared source ancestry pass at 53dbf9395c03f5253f3a810855bdca043360c86f. The official nine-file package/input export passes integrity checking. Both direct modes pass from its read-only, Git-free, empty-PATH tree with unchanged bytes; actual source-byte corruption rejects in both modes.

From the repository root, with CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/shared-fourier-translation-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/shared-fourier-translation-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/shared-fourier-translation-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/shared-fourier-translation-floor --check-source-history --check-modes

Use another CPython 3.12 path elsewhere. Expected markers are PASS_SHARED_FOURIER_TRANSLATION_FLOOR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Receipts are generated on stdout; none is hand-written.

Limitations

The universal claim rests on the written proof. The finite checks use the stated groups up to eight elements. The attaining algebraic phases are not a priced Gaussian-dyadic program. Other-order translations, different charts, arbitrary unit-norm-input operator error and rounded implementations remain open. No relative-error assertion is made for arbitrary inputs whose convolution may vanish.

Native realization, physical routing, restoration, precision/storage, the complete recursive invoice and all five selected framework assumptions retain their prior status. No supplier admission or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed. Selected cold replay, Lean, Docker and a complete physical caller were unrun.

Open next questions

1. Determine the sharp error for translations of higher order. Begin by deriving the constrained character moments on Z/3; the nonbinary moment optimization is unresolved.
2. Determine the sharp bilinear error on arbitrary unit-norm input pairs. Begin with the exhibited minimizer's two-output formula, then prove a matching lower bound over all phases. The different-norm optimizer and physical costs are unresolved.

Related open work

• PR6311 specializes exact arithmetic in PR1477's complete weighted-Fourier replay while preserving its original decisions. This PR proves a new quantitative approximation optimum for a shared translated routine. PR6311 computes no translation-error optimum, so no shared numerical value is compared.
• PR6273 classifies common CRT Fourier tensorization permutations. It studies a different algebraic address contract and computes no shared-translation error optimum.

The completion refresh inspected the titles and bodies of all fifteen open account PRs created since the advance CLAIM, including this PR. Among the fourteen others, PR6311 studies the same phase-weighted Fourier object. None computes this error optimum. No additional computation checked another contribution, and no shared numerical disagreement was identified.

Merged PR4874 concerns exact cancellation at higher input arity. Merged PR5946 concerns exact convolution after restricting the second input with unequal reads and fixed first/final charts. Neither is the approximation contract proved here. Searches do not certify absence of unpublished work.

CI notes

Both hosted checks passed at 53dbf9395c03f5253f3a810855bdca043360c86f in run 41107337654: plan in 4m31s and replay (research/shared-fourier-translation-floor) in 1m24s. The completed checks were read once. Separate run metadata confirms the matching head, completed status and successful conclusion.

All local checks listed above also passed. Hosted replay certifies the registered finite checks and source bindings, including normal and optimized execution. The universal theorem rests on the written proof; independent human and formal proof-kernel review remain unclaimed. No CI failure remains.

Merge danger

Door: two-way. Blast radius: isolated. This adds a proof package and its opt-in replay; predecessor sources and the selected frontier are unchanged.