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Sparse Kernels at Cyclotomic Parameters

Abstract

Sparse Kernels at Cyclotomic Parameters

Theorem 1.1 (Sparse kernel interval criterion).

Lean statement: D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelVanishing.sparse_kernel_interval

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelVanishing.sparse_kernel_interval (✓ 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.

If two finitely supported coefficient sequences have the stated support, nonzero witnesses, zero total sum, and both digit-sum convolution identities, then the Hankel determinant at the indicated size and parameter t is zero.

Theorem 1.2 (Odd carry kernel).

Lean statement: D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelVanishing.odd_carry_kernel

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelVanishing.odd_carry_kernel (✓ 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 a field of characteristic zero, if f changes by one between every even and following odd index, then every positive odd size has a nonzero vector in the kernel of the corresponding carry-difference matrix.

Theorem 1.3 (Root base kernel).

Lean statement: D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelVanishing.root_base_kernel

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelVanishing.root_base_kernel (✓ 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 positive d and k and a parameter zeta with zeta to the d equal to one, there is a coefficient sequence beginning with one, vanishing on the final prescribed interval, summing to zero, and annihilating the digit-sum Hankel rows together with the odd carry rows.

Theorem 1.4 (Root second kernel).

Lean statement: D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelVanishing.root_second_kernel

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelVanishing.root_second_kernel (✓ 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 positive d and a parameter zeta with zeta to the d equal to one, there is a coefficient sequence beginning with one, vanishing on the final interval of length 2 to the d minus one, summing to zero, and annihilating every digit-sum Hankel row through size 2 to the 2d.

References