Metya–Mukherjee's exact independent-convolution L1 minimum

Nilava Metya and Satyaki Mukherjee, Approximate Uniformity in Finite Convolution Models, arXiv:2609.38243v1, §4.3, Conjecture 4.7, states:

The optimal value of (21) is (2n−1)⁻¹ for n≥3.

The source declares CC BY-SA 4.0. The quotation and mathematical source correspondence here are attributed to that version under the same license. The proof below is a repository formalization of the sharp bound.

Equation (21) minimizes the full L1 norm of the deviation from uniform over ordinary convolutions of two independent probability vectors. For every natural n≥3 and real closed-simplex vectors p,q on Fin n, put

The exact target is that t is the least member of the set of all \sum_{k=0}^{2n-2}|r_k-t| obtained from these legal pairs. Both factors range over the whole real closed simplex. They may be unequal, asymmetric, irrational or have zero entries. Addition is ordinary natural addition.

The source separates this L1 objective from its squared-L2, relative-entropy and line-Wasserstein objectives. Equation (22) gives an attaining family, but its matching universal L1 lower bound is stated as Conjecture 4.7. The attaining family is a published supplier; its formal verification does not by itself establish the minimum.

Preregistration #11735 records the original complete target, tier-one classification and bounded source and literature readings before the mathematical probe. The recorded readings include the primary v1 and Asgarli–Hartglass–Ostrov–Walden, Kovačević and Klich at the versions identified there. Their inspected conclusions were not the full sharp L1 result. The source's Theorem 4.6 is about line-Wasserstein distance. No exact L1 resolution was identified in those inspected surfaces; this does not assert worldwide absence or publication priority. Inaccessible surfaces and uninspected cited dice papers remain within the boundaries stated in the preregistration.

The concrete Lean supplier search found no matching sharp theorem in D5 or the pinned Mathlib. The proof uses Mathlib's real simplex, finite sums, trigonometric identities, power-order estimates and intermediate value theorem. Its model does not use rational finite laws or descending-factorial convolution coefficients.

Write m=n−1≥2 and θ=π/m. A zero actual output coefficient immediately gives one deviation equal to t. In the remaining branch every r_k is positive. For the actual generating product R=PQ, pairing indices j and m+j in F(s)=Im R(s exp(iθ)) gives positive constants A,B and the bounds

The choices a=min(1,A/(2B)) and b=max(1,2A/B) give F(a)>0>F(b). Continuity and the intermediate value theorem supply a positive crossing. Nonnegative input coefficients give nonnegative imaginary parts of both factors on this ray. Positivity of the actual r_1 forces a strictly positive imaginary part of at least one factor. The complex product identity then gives Re R≤0 at Im R=0. A crossing need not be a root.

At a crossing radius 0<ρ≤1 use

The proof establishes |b_k|≤1, Σb_k≥1 and Σb_kr_k=Re R≤0. These imply Σ|r_k−t|≥t. When ρ>1, reverse only the output coefficient vector at width 2m+1. Its evaluation at ρ⁻¹ exp(iθ) is ρ⁻²ᵐ conjugate(R(ρ exp(iθ))), and the full L1 sum is preserved by the fixed-width permutation. No degree-based reversal or replacement of the product optimization is used.

For attainment take the parameter-one member of the source's equation (22): p_0=p_m=1/2, all other p_j=0, q_0=t and q_j=2t for j>0. Both vectors are legal simplex members. Their ordinary output is r_0=t/2, r_m=3t/2 and r_k=t at all other output indices. Thus their full L1 value is t. This membership and the universal bound are the two parts of the least-element statement.

A source-model mismatch, a legal pair whose exact full L1 error is less than t, or a failure of simplex membership or the stated output of the attaining pair would invalidate the claimed source-level resolution. An arbitrary coefficient vector or arbitrary coupling with smaller error is not a counterexample to this product-law target.

The single public resolution is D5/S3/TotalVariation/IndependentConvolutionL1Minimum.result. ordinaryConvolution and fullL1 define its exact objective and indices. Two local substantive proof blocks supply the actual-product crossing and the nonpositive-real coefficient certificate; reversal and attainment are local parts of the result proof. The canonical Lean sources use only the accepted axioms propext, Classical.choice and Quot.sound.

The resolution's admission basis is open-problem-resolution under the preregistered external named question. Proof shape, utility and literature eligibility are separate obligations. The general real-parameter argument is not a positive finite enumeration or a numerical reduction. The published attainment family is credited to the source, and no equality classification or global novelty claim is made.

The full quantified least-element statement includes attainment and the universal lower bound. The decisive lower-bound structure is a nonpositive real crossing of the actual independent product followed by a finite dual certificate. The certificate also applies conditionally to arbitrary real coefficient vectors at the fixed width; that auxiliary observation does not change the optimization domain of the resolution.

The theorem does not classify all minimizing pairs, cover n=2, treat unequal input widths or minimize a different distance. Those are separate questions. No source result depending on the conjecture is promoted beyond the stated least-element conclusion.

The bounded literature reading is not an exhaustive known-result exclusion. Source fidelity and per-declaration admission require independent review. Freeze, final admission checks and publication are separate from the Lean proof. This dossier asserts no completed repository publication lifecycle.