Motzkin convolution Hankel refutation

Abstract

The printed universal first equality chain of Conjecture 7 in arXiv:2502.21050v1 is false at r = 4, n = 0. This does not refute a possibly intended restriction r >= 8.

Wang and Zhang, Hankel determinants for convolution powers of Motzkin numbers, arXiv:2502.21050v1, Conjecture 7, prints no lower bound on r. The authors may have intended r >= 8, since Theorems 1–6 cover r = 2 through 7. That interpretation is unverified. This module refutes the printed universal first chain and makes no claim that the authors’ intended conjecture is false.

Theorem 1.1 (The printed universal first chain is false).

Lean statement: D5/S0/Certificates/MotzkinConvolutionHankelRefutation.result

Proof. Machine-checked in Lean as D5/S0/Certificates/MotzkinConvolutionHankelRefutation.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

Motzkin numbers are defined by the source’s Catalan-binomial sum. Cauchy convolution defines every natural power, and Matrix.det defines every Hankel size, including the empty determinant. At r = 4, n = 0, the conjectured common value would equate H0 = 1 with H4 = -1. The latter is a private kernel-checked numerical computation. The paper’s Theorem 4 is corroboration only and is not a proof premise.

References

• Truth anchor: D5/S0/Certificates/MotzkinConvolutionHankelRefutation.result