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A negative answer to the final question of Problem 7.2

Abstract

Weak increase of a Gaussian quotient permits an extra negative coefficient.

Definition 1.1 (Weak increase from coefficient one).

Formalization. D5/S3/Combinatorics/GaussianQuotient/MirzavaziriYaqubiMonotonicityRefutation.increasingFromOne (✓ std3).

Citation. Madjid Mirzavaziri, Daniel Yaqubi (2026). Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler. DOI: 10.48550/arXiv.2610.19249. URL: https://arxiv.org/abs/2610.19249v1.

Commentary.

Problem 7.2, p. 16: “Characterize the quadruples for which is a polynomial whose only negative coefficients are at and at . In particular, is this always the case when is weakly increasing from to its middle coefficient?”

The source asks whether the first quotient is “weakly increasing from q¹ to its middle coefficient” (Problem 7.2, p. 16). The comparisons begin with the coefficients of degrees one and two and end at the natural-number quotient of natDegree U by two.

Definition 1.2 (The proposed coefficient signs).

Formalization. D5/S3/Combinatorics/GaussianQuotient/MirzavaziriYaqubiMonotonicityRefutation.twoExceptionSignPattern (✓ std3).

Citation. Madjid Mirzavaziri, Daniel Yaqubi (2026). Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler. DOI: 10.48550/arXiv.2610.19249. URL: https://arxiv.org/abs/2610.19249v1.

Commentary.

The source requests a polynomial “whose only negative coefficients are −1 at q¹ and at qᴰ⁻¹” (Problem 7.2, p. 16). Here D is natDegree P. Every other natural index, including the zero coefficients past the degree, is required to be nonnegative.

Definition 1.3 (The final universal sufficiency question).

Formalization. D5/S3/Combinatorics/GaussianQuotient/MirzavaziriYaqubiMonotonicityRefutation.claim (✓ std3).

Citation. Madjid Mirzavaziri, Daniel Yaqubi (2026). Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler. DOI: 10.48550/arXiv.2610.19249. URL: https://arxiv.org/abs/2610.19249v1.

Commentary.

Problem 7.2, p. 16: “Characterize the quadruples for which is a polynomial whose only negative coefficients are at and at . In particular, is this always the case when is weakly increasing from to its middle coefficient?”

N, k, a and b are natural numbers with k ≤ N and a,b ≥ 1; U and P are integer polynomials. The two multiplication equalities express exact polynomial quotients. gauss is the existing q-Pascal Gaussian polynomial, and qInteger is the finite geometric sum, mapped from natural to integer coefficients. The question concerns weak increase through the middle of U; the exceptional degree D−1 is calculated from P. Nat.div(m,2) is natural-number division, equal to floor(m/2). All index subtraction is natural subtraction, truncated at zero.

Theorem 1.4 (The universal assertion is false).

Proof. Machine-checked in Lean as D5/S3/Combinatorics/GaussianQuotient/MirzavaziriYaqubiMonotonicityRefutation.result (✓ std3). ∎

Resolves. Problems/mirzavaziri-yaqubi-2026-gaussian-quotient-monotonicity-problem-7-2 (refuted) by D5/S3/Combinatorics/GaussianQuotient/MirzavaziriYaqubiMonotonicityRefutation.result.

Source. Repository-derived.

Acknowledgement. Madjid Mirzavaziri, Daniel Yaqubi (2026). Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler. DOI: 10.48550/arXiv.2610.19249. URL: https://arxiv.org/abs/2610.19249v1.

Commentary.

At (N,k,a,b)=(30,3,7,10), exact quotients U and P have degrees 75 and 66. U increases weakly from coefficient one through coefficient 37, but the coefficient of degree 11 in P is −1. That index differs from both 1 and 65. The Gaussian hook and symmetry identities establish the first quotient, and explicit polynomial arithmetic establishes the second. This answers the final sufficiency question; the classification of all quadruples and an unbounded family are not asserted by this theorem.

References