[petalas]

Summary

Proves the exact shared-routine criterion for a product of k independent Fourier inputs, for every k>=2 and every finite product of cyclic groups. Claim type: scoped algebraic proof with finite corroboration.

pi(x) = L(x)+b                  L self-adjoint and invertible
(k-1)b = 0                     permitted translation
w(x) = c*chi(-x,b)              c^(k-1) = 1

These conditions are necessary and sufficient for k copies of D_w P_pi F^-, followed by its coefficientwise conjugate on their pointwise product, to return the k-fold convolution divided by N^k. On Z/2, the translated layout with weights (1,-1) works at three inputs and fails at two. The exact finite-phase routine count is S(G)*|G[k-1]|*gcd(M,k-1) when M contains the group exponent.

This extends Hanson Sun's merged #1477 phase-weighted two-input theorem and Ross Richardson's merged #940 unweighted adjoint criterion. Their method and two-input result retain predecessor credit. The new step is the higher-arity torsion/root condition and count. No literature-priority claim for elementary character algebra is intended. This contribution was produced by Nick Petalas.

Evidence

• Before: the shared-routine phase prohibition was established for two inputs. After: the written proof classifies all k>=2 and supplies a three-input translated example.
• The exact census compares coefficient equations with affine normal-form reconstruction on all 90,752 declared candidates over Z/2, Z/3, Z/4 and (Z/2)^2 at arities 2..5. Exactly 194 are admitted; every row agrees with the count formula.
• A separate evaluator imports no primary or predecessor code. Literal exact Fourier passes and cyclic convolution agree on all 106,332 admitted basis tuples and 194 dense tuples. Seven semantic fixtures exhaust their own basis domains. The primary has four semantic and twelve input/capacity controls. Generated PRIMARY.json and ORACLE.json bind source hashes and ordered output digests.
• All four registered warning-strict normal/optimized commands pass locally. Measured elapsed times were 0.299s, 0.297s, 2.828s and 2.824s. Every command is below five minutes.
• Registered metadata/replay, AST validation, committed-byte/source-history preflight for ten files, committed export integrity, optimized isolated Git-free export replay and eight offline/integrity/occupied-output controls pass. Source and receipt corruption reject without overwriting them. No dedicated linter was installed.

Exact reproduction from the repository root:

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

Substitute the host's CPython 3.12 path when needed. The registered command runs both direct checkers normally and optimized. Expected output includes PASS_MULTIINPUT_SHARED_FOURIER 90752 candidates; 194 accepted, PASS_MULTIINPUT_FOURIER_ORACLE with 106332 basis tuples, and PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations

The universal claim is the written proof; finite replay is separate corroboration. Independent human and formal proof-kernel review are unclaimed. No physical schedule, free phases, rounding, tape routing, ownership/restoration implementation, supplier admission, new kappa or complete multiplication theorem is claimed. The selected frontier is unchanged. The full selected supplier and predecessor verifiers were not replayed. The executable census is deliberately bounded; those limits do not narrow the written theorem.

Canonical base and refreshed main: 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Only the new isolated package is changed. Exact retained predecessor READMEs are live hash-bound inputs; no pending Git object is required. The scope change from the current audit card to an original interface extension is explicit in TASK.json. The advance claim matches the delivered scope.

Related open work

No open PR on this same shared monomial k-input convolution classification or its routine count was found. Coverage includes the full cached 395-open-PR and Discussion snapshot, refreshed recent open PR bodies/claims, targeted merged searches and main README searches. The local overlap helper omits sparse notes, so its lack of additional matches is not a novelty guarantee. Merged #1477 and #940 are the direct predecessors. No competing result was recomputed.

Completion refresh inspected all 61 open PRs on this account, including the nine other submissions created since the CLAIM: #4873, #4875, #4876, #4878, #4879, #4880, #4881, #4882 and #4883. None computes this shared-routine cancellation criterion or its routine count; no same-quantity disagreement was found in that coverage.

• Prove optimal dyadic rotation switch tariffs (no new κ) #4876 proves exact logical dyadic rotation switch tariffs. Its rotations are adjacent to the permitted affine read shifts here, but this package classifies algebraic cancellation and claims no physical read-layout cost. No common numerical result is compared or recomputed.

Open next questions

• Price one literal three-input caller that uses the permitted two-torsion shift and character weights, including all read maps, phases, intermediate products, restoration and output conversion. A physical caller and favorable complete invoice are missing.
• Classify separately weighted and addressed forward routines at general k. Begin with independent multilinear coefficients. This follow-on was left unpursued after the first verified result; no physical benefit is established.

CI notes

Both hosted checks passed for dd12c73794fa1941b14222f7d16eba408c922351 in run 39867084655: planning in 3m40s and registered package replay in 1m32s. Run metadata confirms the matching head, completed status and success. The checks were read once at completion. Registered replay executes all four warning-strict normal/optimized commands. Local checks and publication preflight also passed. Hosted execution is finite corroboration; independent human and formal proof-kernel acceptance remain unclaimed.

Merge Danger

Door: two-way. The addition is an isolated research package.

Blast Radius: package. No shared implementation, pinned predecessor or selected-frontier file changes.