[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. For a fixed phase-weighted permutation of the second Fourier spectrum, while the first input is unrestricted and the first/final Fourier charts are standard:

maximal second-input dimension = number of permutation cycles with phase product 1
sharp input codimension       = N minus that count

An explicit cycle basis attains the bound. Translation by an element of order d with constant phase lambda admits exactly the time inputs supported on the character level chi(b,x)=lambda. The dimension is N/d when lambda^d=1, and zero otherwise. On (Z/4)^2, translation by (1,0) and phase i admit exactly the four-dimensional space supported on x_0=1; no five-dimensional linear input space works for this wrapper.

This changes the arbitrary-second-input contract of Hanson Sun / bbMisk's #1477, with Ross Richardson's #940 as its unweighted predecessor. Their results are cited, not audited or replayed. Weighted-permutation cycle decomposition, character orthogonality and rank-nullity are classical. The new maximal-input contract, construction and verifier were produced by Nick Petalas.

Only research/fourier-input-defect/ is added. Advance CLAIM.

Evidence

• Before, the cited full-input contract excludes unequal forward read maps. After restricting only the second input, the proof gives its exact maximal dimension and an attaining basis.
• Every one of 6,564 weighted permutations at dimensions one through four agrees with an independent exact rational realification-rank calculation. The census has 13,556 cycles and 3,389 balanced cycle vectors.
• Literal Gaussian-integer Fourier products and direct group-addition convolution pass 25,625 accepted basis pairs and 6,413 dense first operands across 12,324 layouts. All 46,139 enumerated forbidden-coordinate witnesses fail the identity as required.
• The translation corollary passes 104 layouts and 1,168 time-coordinate columns, plus all 64 basis pairs of the explicit sixteen-coordinate sparse construction. Seven malformed domains and three semantic controls pass.
• Local CPython 3.12.15 warning-strict normal and optimized executions pass with identical generated hash-bound JSON receipts. The registered commands take 3.49 and 3.26 seconds locally; the complete registered runner takes 7.05 seconds. AST inspection confirms bounded loops, no recursion or acceptance assertions, and all 22 functions below sixty lines.
• Source metadata, exact committed bytes and declared source history pass at 04a5da5ce867cc965103094b8e52d51368291fac. The orientation-only selected binding check passes. No predecessor scientific suite was run.

From the repository root:

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

Substitute another CPython 3.12 executable elsewhere. Expected markers are PASS_FOURIER_INPUT_DEFECT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier generates receipts on stdout bound to all six package files. There is no hand-authored receipt.

Limitations

The written claim covers finite abelian groups with a standard nondegenerate symmetric pairing, arbitrary permutations and unit phases. The finite corroboration uses fourth roots, rank dimensions up to four and the explicitly listed groups up to sixteen coordinates. No dimension-three Fourier computation is claimed.

The first and final maps are fixed. Coupled restrictions on both operands, arbitrary final-map changes and approximation remain outside scope. Physical sparse-input preparation, movement, restoration, native execution, precision/storage, stock and a complete supplier invoice remain unresolved. No selected assumption, supplier admission, new kappa or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected cold replay were not run.

Open next questions

Can an actual convolution caller provide one of the character-level sparse operands and absorb the shifted read and phase at a favorable complete cost? First identify a literal operand and translation, prove its support contract, then charge preparation, movement, restoration, precision and the complete wrapper. This PR supplies no physical caller or invoice.

Related open work

The completion refresh read the titles and bodies of all ten other open account PRs created since the advance CLAIM. None studies this restricted-input dimension or computes the sparse-wrapper maximum; no conflicting shared value was identified. No additional computations were run to check those contributions.

• Prove the unique optimal Gaussian chirp origin (no new κ) #5949 optimizes the origin of a Gaussian chirp matrix. It does not compute the input subspace of a mismatched Fourier convolution wrapper.
• Reduce signed Hadamard lowering from 222 to 208 primitives (no new κ) #5945 lowers a signed Hadamard connector with extra source queries. Its target and primitive invoice differ from this Fourier-input dimension contract.

The directly related predecessors #1477 and #940 are merged. These publication searches do not establish absence of unpublished work.

CI notes

Both hosted checks passed for 04a5da5ce867cc965103094b8e52d51368291fac in run 40530455596: plan in 4m30s and replay (research/fourier-input-defect) in 2m59s. The completed check list was read once, and the run head equals the PR head. The registered replay executes both normal and optimized commands and confirms committed inputs remain unchanged.

This certifies the bounded registered checks, not independent human or formal proof-kernel acceptance of the written theorem. All local checks listed above also passed. No CI failure remains.

Merge danger

Door: two-way. Blast radius: isolated. The change adds a scoped proof package and its opted-in replay; the selected frontier and predecessor source bytes are unchanged.