GPT 6 Astra Ultra attempt on Math-690

Problem 145: a boundary lemma from the moment asymptotic itself

1) OBJECTIVE AND SOURCE REVIEW

Attempt dated 2026-09-23. If $s_k$ are consecutive squarefree integers and $g_k=s_{k+1}-s_k$, the target is existence of finite limits $x^{-1}\sum_{s_k\le x}g_k^\alpha$ for every $\alpha\ge0$. I read gpt_pro_5.4/145.tex fully. Its fixed-pattern and uniform-integrability reduction is useful. I remove the extra short-interval theorem from its transfer between endpoint conventions.

2) WORK

The following elementary jump lemma applies to any positive increasing integer sequence $s_k$ with $s_{k+1}/s_k\to1$. Suppose, for a fixed $\gamma\ge0$, \[ F_\gamma(y):=\sum_{s_{k+1}\le y}g_k^\gamma=B_\gamma y+o(y). \] At $y=s_{k+1}$ the jump of $F_\gamma$ is exactly $g_k^\gamma$. Apply the asymptotic both at $s_{k+1}$ and at $s_{k+1}-1/2$. Subtracting gives \[ g_k^\gamma=B_\gamma/2+o(s_{k+1})=o(s_{k+1}). \] If $s_k\le x<s_{k+1}$, the left-endpoint and right-endpoint sums differ by precisely this one jump. As $s_{k+1}/x\to1$, their normalized difference is $o(1)$. Therefore the same asymptotic, with the same constant, holds with cutoff $s_k\le x$.

For squarefree integers, their positive asymptotic density $6/\pi^2$ gives $s_{k+1}/s_k\to1$: a gap from $s_k$ to at least $(1+\delta)s_k$ would contradict the difference between the counting asymptotics at these two arguments. Thus any established right-endpoint moment asymptotic transfers automatically. In particular Chan's checked theorem for $0\le\gamma<3.75$ transfers throughout that same interval, without importing a particular pointwise squarefree-gap exponent.

For higher moments the original obstruction persists: existence of all fixed-gap frequencies $p_m$ does not bound \[ \limsup_{x\to\infty}\frac1x\sum_{s_k\le x,\ g_k>Y}g_k^\alpha. \] The jump lemma controls only the final gap, not this whole tail.

3) ADVERSARIAL VERIFICATION

Subtracting two $o(y)$ errors at nearby endpoints remains $o(y)$; it is not asserted to be $o(1)$ before normalization. The half-integer endpoint ensures exactly one integer endpoint has entered. The ratio condition is stated explicitly and follows from squarefree density. No moment outside Chan's verified interval is inferred.

4) FINAL

UNRESOLVED. The boundary transfer now has a self-contained proof from existing asymptotics. It simplifies the supplied argument but does not enlarge the known moment interval or prove uniform integrability for arbitrary $\alpha$.

5) SOURCES CHECKED

Complete supplied file. T. H. Chan, On moments of gaps between consecutive squarefree numbers, https://arxiv.org/abs/2310.08448, checked 2026-09-23, for the exact exponent range and right-endpoint cutoff. The jump argument above is independent of any claimed latest short-gap estimate.

6) COMPLETION ESTIMATE

Subjective progress toward all moments, not a percentage of exponent values. COMPLETION: 20%