bibkey: vishnyakova2026polynomially authors: Anna Vishnyakova year: 2026 title: Polynomially Deformed Normalized Pochhammer Sequences Having Generating Functions With Only Real Non-positive Zeros doi: 10.48550/arXiv.2608.03723 claim: Definition 1.4 specifies the normalized falling-Pochhammer linear operator; Example 6.2 exhibits the two quadratic inputs whose images are perfect squares; Example 6.4 defines its real-root parameter sets; Conjecture 6.5 proposes an upper bound on their even-degree extent. strata_touched:

• D5/S3/Zeros/PochhammerDeformation/QuadraticInterval license: citation-only triage: anchor

Polynomially Deformed Normalized Pochhammer Sequences

The attribution covers the operator and the parameter-set definitions. The quadratic closed form and its small-positive-parameter refutation are derived in the repository. The refutation concerns the strict upper bound at degree two, while the asserted interval shape at that degree holds. It makes no claim about higher degrees, their monotonicity, or their limiting extent.

Antecedent to distinguish: Example 6.2

Example 6.2 of the source already exhibits two inputs. Write s = sqrt(a(a+1)). The two branches carry different constant-term signs and must be quoted separately; compressing them into one ± formula flips the negative branch:

• P(x) = (x + (a + s)/2)^2 gives L_a(P)(x) = (s x + (a + s)/2)^2
• P(x) = (x + (a - s)/2)^2 gives L_a(P)(x) = (s x - (a - s)/2)^2

Each image is a perfect square whose double root lies in [-1,0] (for a > 0 one has s > a, so (a - s)/2 < 0 and the second root (a - s)/(2s) is negative, of modulus at most one). Because s = sqrt(a^2 + a), the two parameters (a ± s)/2 are exactly the two endpoints of the interval classified here. The source therefore already establishes that both endpoints belong to the degree-two parameter set, at the repeated-root boundary.

What this repository adds is disjoint from that antecedent:

1. Necessity / whole-interval classification — a proof that the parameter set equals the closed interval between those endpoints, not merely contains them. The source does state this equality, but only conjecturally: Conjecture 6.5 asserts M_{2k}(a) = [-c_{2k}(a), a + c_{2k}(a)] for every even degree. What it does not do is establish it at any even degree; it writes that for even n “the situation is much more complicated”. The increment here is therefore the proof of that interval-shape clause at k = 1, not the statement of it.
2. The closed form of the extent c_2(a) = (sqrt(a^2+a) - a)/2. Stated precisely: the source gives no closed form at any even degree, and its sentence The proof of this fact and the possible value of the limit ... remain open refers to the general even-degree conjecture and to the limit, not to a separately declared open problem at k = 1. The increment here is a specific-case classification, not the resolution of a question the source singled out.
3. The sharp threshold c_2(a) < 2a iff 1/24 < a, with equality at a = 1/24, refuting Conjecture 6.5’s strict clause 0 < c_{2k}(a) < 2a throughout 0 < a <= 1/24. The source neither proves nor conjectures any sharpness statement in this direction.

This distinction was raised by an independent review seat reading the source, not by the implementation; it is recorded here so the increment is not overstated.

Search Log

• 2026-09-06: Read the v1 HTML, including the mathematical alttext for Definition 1.4, Example 6.2, Example 6.4, and Conjecture 6.5. The last explicitly states the strict bound 0 < c_{2k}(a) < 2a.
• 2026-09-06 (review round): an independent review seat identified Example 6.2 as an antecedent supplying both interval endpoints. Re-read and confirmed against the source alttext; the antecedent and the increment are separated above.
• Searched pinned Mathlib, the repository’s D5 declarations, installed Lean packages, and GitHub repositories for Lean Pochhammer formalizations. The unrelated GaussianWhoWhere project studies finite Hermite-Pochhammer translation rigidity. No matching interval classification was found in the searched scope. This is not a claim of worldwide priority.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2608.03723
• Version inspected: https://arxiv.org/html/2608.03723v1
• Definition: https://arxiv.org/html/2608.03723v1#S1.count4
• Conjecture: https://arxiv.org/html/2608.03723v1#S6.count5