slug: bilyk-et-al-2022-averaged-kernel-characterization-refutation
bibkey: bilykferizovicglazyrinmatzkeparkvlasiuk2022multivariate
doi: 10.1007/s00209-022-03000-z
url: https://arxiv.org/abs/2104.03410
triage: theorem
motivation_gids:
  - D5/S3/Analytic/MultivariateEnergy/Result.result

The averaged-kernel characterization fails

Problem

D. Bilyk, D. Ferizović, A. Glazyrin, R. Matzke, J. Park and O. Vlasiuk, Potential theory with multivariate kernels, Math. Z. 301 (2022), Section 5, paragraph before Theorem 5.3, verbatim:

We suspect that the property that σ minimizes I_K is equivalent to the fact that U_K^{σ^{n−2}} is conditionally positive definite, i.e. the two-input energy I_{U_K^{σ^{n−2}}} is minimized by σ. This conjecture is supported by all the examples known to us.

The setting is the unit sphere in real Euclidean space, normalized surface measure σ, and continuous symmetric rotationally invariant kernels with n ≥ 3. Definitions 2.1–2.2 give the measure and energy conditions; (5.1) gives invariance.

Motivation

Averaging detects a quadratic variation at the uniform measure. A global minimization criterion must also control higher-order terms in the energy. A cubic centered-moment term separates those two requirements.

Gap

Conditional positive definiteness of the averaged kernel does not imply that normalized surface measure globally minimizes the original multivariate energy. The counterexample has positive quadratic variation and negative energy at a Dirac probability measure.

Route

For x on the unit sphere in dimension d, define

$$A(x)=x{x}^{\mathsf{T}}-I/d,L(x,y)=\mathrm{tr}(A(x)A(y)),T(x,y,z)=\mathrm{tr}(A(x)A(y)A(z)).$$

The three-input family is

$$K_{d,\beta }(x,y,z)=\frac{\beta }{3}(L(x,y)+L(x,z)+L(y,z))-T(x,y,z).$$

The pair trace is $(x\cdot y{)}^2-1/d$. The triple trace is

$$(x\cdot y)(y\cdot z)(z\cdot x)-\frac{(x\cdot y{)}^2+(y\cdot z{)}^2+(z\cdot x{)}^2}{d}+\frac{2}{d^2}.$$

These formulas prove continuity, symmetry and orthogonal invariance. Coordinate reflections annihilate off-diagonal second moments of σ; coordinate permutations make diagonal moments equal, and their sum is one. Thus $\int x{x}^{\mathsf{T}}d\sigma =I/d$ and $\int Ad\sigma =0$. For any Borel probability measure μ, put $Q=\int Ad\mu$. Then

$$I_K(\mu )=\beta \mathrm{tr}(Q^2)-\mathrm{tr}(Q^3),U_K^{\sigma }(x,y)=\frac{\beta }{3}L(x,y).$$

For finite positive measures μ₁,μ₂ the quadratic difference form of the averaged kernel is $\frac{\beta }{3}\mathrm{tr}((Q_1-Q_2{)}^2)\ge 0$ when β≥0, since the Q matrices are real symmetric. This holds without a mass constraint. On the other hand, for any unit vector e,

$$I_K({\delta }_e)=\frac{d-1}{d}\left(\beta -\frac{d-2}{d}\right)<0=I_K(\sigma )\left(d\ge 3,0<\beta <\frac{d-2}{d}\right).$$

Falsifier

The universal equivalence would imply minimization for the admissible kernel at d=n=3 and β=1/10 because its averaged pair kernel is conditionally positive definite. The competitor δ_e has energy −7/45, whereas σ has energy zero.

Evidence

The unconditional theorem D5/S3/Analytic/MultivariateEnergy/Result.result has type ¬ Defs.claim. Its principal family theorem is D5/S3/Analytic/MultivariateEnergy/Result.family_refutes, for every d≥3 and 0<β<(d−2)/d. All three conjuncts, admissibility, conditional positive definiteness and failure of minimization, are used in the refutation.

Principal dependencies are Moments.integral_coord_mul_coord, Moments.integral_A_eq_zero, Moments.L_eq, Moments.T_eq, Moments.isAdmissible_familyKernel_three, Energy.energy_family_eq, Energy.avgKernel_family_eq and Energy.isCondPosDef_family, all in D5/S3/Analytic/MultivariateEnergy/. The result’s axiom closure is exactly propext, Classical.choice, Quot.sound.

The proof route was derived with the assistance of the ChatGPT Pro model. The source proves necessity and a local converse in Theorem 5.3 and leaves the global converse as a conjecture. No settlement was found in the examined scope: the source title and identifier, author names with counterexample and conditional positive definiteness terms, Matzke’s dissertation, two follow-ups from 2023, arXiv:2410.01489, arXiv:2410.22104, and the formal-conjectures collection. Citation indexes were not examined exhaustively. This scope does not establish an exhaustive priority claim.

Triage

What the settlement shows

1. PROVED. For n=3, every d≥3 and 0<β<(d−2)/d, the kernel K is continuous, symmetric and orthogonally invariant, its σ-averaged kernel is $(\beta /3)\mathrm{tr}(A(x)A(y))$ and conditionally positive definite, and $I_K({\delta }_e)=((d-1)/d)(\beta -(d-2)/d)<0=I_K(\sigma )$. Thus σ is not a minimizer. These are the conclusions and proof dependencies of Result.family_refutes, consumed by Result.result; the displayed energies are the private Result.energy_sigma and Result.energy_dirac, and the averaged formula is Energy.avgKernel_family_eq.

2. PROVED. The uniform second moment is $\int x{x}^{\mathsf{T}}d\sigma =I/d$ (Moments.integral_coord_mul_coord), and $I_K(\mu )=\beta \mathrm{tr}(Q^2)-\mathrm{tr}(Q^3)$ (Energy.energy_family_eq).

3. PROVED in prose. For every n≥3, take

   $$K_{d,n,\beta }=\frac{\beta }{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}\sum _{i<j}L(x_i,x_j)-\frac{1}{\left(\genfrac{}{}{0ex}{}{n}{3}\right)}\sum _{i<j<k}T(x_i,x_j,x_k).$$

   Independence gives the same energy polynomial for each pair and triple. Averaging n−2 inputs kills every term containing an integrated feature, leaving $\beta L(x,y)/\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. Hence the same counterexample holds. This extension is mathematical reasoning beyond the three-input Lean family.

   There is also an exact spectral threshold. Let M=$\int x{x}^{\mathsf{T}}d\mu$, let λ₁,…,λ_d≥0 be its eigenvalues, so ${\sum }_i{\lambda }_i=1$, and put $S_k={\sum }_i({\lambda }_i-1/d{)}^k$ and c=(d−2)/d. Expanding and using that sum gives

   $$c{S}_2-S_3=\frac{1}{d}\sum _{i<j}({\lambda }_i-{\lambda }_j{)}^2(1-{\lambda }_i-{\lambda }_j)\ge 0.$$

   Each factor is nonnegative. Also $S_2={\sum }_i{\lambda }_i^2-1/d\le (d-1)/d$. Therefore $I_K=(\beta -c)S_2+(c{S}_2-S_3)$ is minimized for β<c by rank-one M, and for β>c by M=I/d. Equality ${\sum }_i{\lambda }_i^2=1$ means rank one; its measure is supported on an antipodal pair. Conversely every measure on an antipodal pair gives rank one. For β>c the minimizers are exactly isotropic measures. At β=c the energy is nonnegative and σ is a minimizer, giving the exact threshold for minimization by σ. Finally $|\mathrm{tr}(Q^3)|\le \Vert Q{\Vert }_{\mathrm{op}}\mathrm{tr}(Q^2)$ implies local minimality of σ for β>0: Q depends continuously on μ in the weak topology, and $\Vert Q{\Vert }_{\mathrm{op}}<\beta$ makes the energy nonnegative. The spectral certificate, minimizer descriptions and local statement are mathematical reasoning, not additional Lean theorems.

4. COMPUTED. At d=n=3 and β=1/10 the exact Dirac energy is −7/45 and the averaged kernel is $((x\cdot y{)}^2-1/3)/30$. These rational evaluations specialize the displayed formulas.

5. OPEN. Whether necessity, proved in the source, admits a converse under an additional convexity or uniqueness hypothesis beyond the source’s local converse remains unresolved here.

ASSUMED-UNVERIFIED

The all-arity extension, spectral threshold, descriptions of minimizers and weak local minimality argument are proved in prose and are not additional Lean conclusions. The literature scope is not exhaustive. The additional-hypothesis converse question remains open.