bibkey: byun2026unimodality authors: Seok Hyun Byun and Svetlana Poznanović year: 2026 title: Unimodality and log-concavity of generalized Glasby-Paseman sequences doi: null url: https://arxiv.org/abs/2604.14639v1 claim: Conjecture 1.1(d) specifies the exact asymptotic maximum of the powered-sum ratio (1.4) for every fixed positive real a and positive integer l; the paper proves the case l=2 and a=1. strata_touched:

• D5/S3/AnalyticClosure/BinomialMovingEndpoint
• D5/S3/AnalyticClosure/BinomialMaximumLocalization
• D5/S3/AnalyticClosure/BinomialPoweredRatioLocalization
• D5/S3/AnalyticClosure/BinomialPoweredRatioMaximum license: citation-only triage: anchor

Generalized Glasby–Paseman maximum

The source is the version-1 preprint dated 16 April 2026. Page 2, equation (1.4), defines, for positive integers m,l and a positive real a,

[ R_{m,l,a}(r)= \frac{\sum_{i=0}^{r}(\binom mi a^i)^l} {\sum_{i=0}^{r}(\binom ri a^i)^l},\qquad 0\le r\le m. ]

Conjecture 1.1(d) on the same page states that its maximum is asymptotic to

[ \frac{\sqrt l}{\sqrt{2\pi m}} \frac{\sqrt{1+2a}(1+a)a^{(l-2)/2}}{(1+a)^l-1} \left(\frac{1+2a}{1+a}\right)^{(m+1/2)l}. ]

Footnote 1 defines “asymptotically” as the ratio tending to one as m tends to infinity. The subtraction in (l-2)/2 is real subtraction, so l=1 gives the exponent -1/2. The maximum ranges over all integer indices including 0 and m; ties do not change its value.

The paragraph immediately after Conjecture 1.1 credits prior results for l=a=1 and for l=1 with positive integer a. Theorem 1.2 proves the l=2,a=1 case in this paper, with the asymptotic in equation (1.6). These cases are prior results, not separate new candidate resolutions.

The source also conjectures unimodality (a), eventual log-concavity (b), and an exact three-position unique-peak restriction for integer a (c). The repository maximum target concerns only (d), for the full real-a parameter range. Its slope localization does not establish (c).

Verified locator

• Versioned source: https://arxiv.org/abs/2604.14639v1
• Source body: https://arxiv.org/pdf/2604.14639v1, page 2, equation (1.4), Conjecture 1.1(d), footnote 1, and Theorem 1.2(c).
• Registration and bounded source audit: https://github.com/the-omega-institute/trureturing/issues/9357

The source audit supplied with #9357 reports that the arXiv abstract still listed v1 on 2026-09-21 at 21:46 UTC. The versioned abstract and pages 1–3 were also consulted for this note on 2026-09-21 UTC. That identifies the source and its stated conjecture; it does not establish worldwide priority or rule out an uninspected settlement. This note is not a typed resolution claim or evidence of canonical Freeze.