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Reflection and Block Conjugation

Abstract

Reflection and Block Conjugation

Theorem 1.1 (Determinant-one block conjugation).

Lean statement: D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelReflection.block_conjugation

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelReflection.block_conjugation (✓ 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 sequence over a commutative ring satisfying the two binary carry relations, with 2 to the k below n at most twice that power, there is a determinant-one matrix whose conjugation transforms the Hankel matrix into the stated binary block form with weights w and x minus 2w.

Theorem 1.2 (Weighted reflection recurrence).

Lean statement: D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelReflection.weighted_reflection

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankelReflection.weighted_reflection (✓ 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.

Under the carry relations, k at least one, and n in the reflection range, the Hankel determinant H at n and bordered determinant E at n reduce to the corresponding determinants at 2 to the k plus reflected index, with the explicit powers and signs in the recurrence.

References