# Verification report

**Manuscript:** Sharp Positivity Tests for Polynomial Hermitian Kernels.  
**Author:** Alper Ferudun, Mercury Software GmbH.  
**Source identifier:** AIM-SEVERAL_COMPLEX_VARIABLES-0026.  
**Date:** 17 October 2026.

## Scope of acceptance

The accepted written theorem concerns globally polynomial Hermitian kernels: rank at most 2k makes k-point positivity sufficient for a squared norm; a univariate polynomial space of dimension r has exact stability index ceil(r/2); an indefinite polynomial kernel in P_k has at least 2k positive eigenvalues. A Hankel-moment family attains that signature. At coefficient rank at most four, the existence of any squared-norm power is equivalent to positive or negative semidefiniteness of the coefficient matrix, with least exponent one or two respectively. Rank five supplies the explicit sharp obstruction.

The unrestricted AIM classification remains unresolved. Finite sampled domains and arbitrary entire kernels are not included in the polynomial theorem. The zero polynomial is a trivial separate case. Exponents are positive integers.

## Load-bearing proof checks

The self-audit checks independence of the polynomial support, uniqueness of polarization, the nonzero Wronskian and confluent determinant, dominance and its constructible-image consequence, Euclidean closure, interpolation along a polynomial curve in several variables, the compact-cone sharpness argument, the positive-signature reduction, the singular Hankel inequality, the roots-of-unity witness, every coefficient of the squared family, the sign normalization, and both directions of the rank-four criterion. A nonsingular Jacobian alone is not used to infer global positivity: constructibility and Euclidean density are explicitly required.

## Supplementary exact computation

One standard-library Python implementation performs 22,703 positive checks per execution:

| Check family | Count |
| --- | ---: |
| Distinct-degree confluent determinants | 510 |
| Confluent product identities | 100 |
| Sparse and mixed polynomial Jacobians | 90 |
| Gaussian-integer moment inequalities | 1,200 |
| Rational Hankel determinants | 140 |
| Cyclotomic moment witnesses | 168 |
| Square coefficients against the closed formula | 20,295 |
| Cancellation congruences and all-principal-minor tests | 200 |

Negative controls reject the finite-domain extension, the use of leading principal minors alone, the k=1 squared-norm-square extension, and preservation of positivity after interpolation from a finite sample. In the last example the interpolated two-point determinant is -1071/2.

Normal and optimized executions return the same expected JSON. The source ZIP is extracted into a fresh directory, its LaTeX rebuild is compared page by page with the original text and content streams, and its replay driver reruns both modes. All six PDF pages were directly visually inspected. The machine-readable build record and manifest bind the exact package bytes.

The computations are supplementary, not a proof of universal quantifiers. This is originating-researcher self-audit, not independent human review, a second independent implementation, or formal certification. The article records AI assistance. No mathematical gap was identified within the stated scope; no absolute novelty certificate is claimed.