Vanishing Outside the Threshold Triples

Abstract

Sparse binary kernel vectors force the Hankel determinant at t = -2 to vanish outside the threshold triples.

Theorem 1.1 (Vanishing away from the triples).

Lean statement: D5/S3/Combinatorics/DigitHankel/BinaryDigitHankelStructure.zero_direction

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/BinaryDigitHankelStructure.zero_direction (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Bartosz Sobolewski, Maciej Ulas (2026). Hankel determinants of weighted binary sums of digits. DOI: 10.48550/arXiv.2607.09376. URL: https://arxiv.org/abs/2607.09376v1.

Commentary.

For every integer n at least two, if there is no nonnegative integer k for which n + 1 = n_k, n = n_k, or n = n_k + 1, then H(n,-2) = 0, where n_k = ceil(2^(k+2)/3). Splitting the binary digit sum into blocks gives sparse nonzero vectors in the kernel of the Hankel matrix at these sizes, forcing its determinant to vanish.

References

• Truth anchor: D5/S3/Combinatorics/DigitHankel/BinaryDigitHankelStructure.zero_direction
• Dependency: D5/S3/Combinatorics/DigitHankel/BinaryDigitHankelDefs