Geometric Moments and Rational Lambert Values: Proofs and Further Results

Will Cook

Long paper

Geometric Moments and Rational Lambert Values: Proofs and Further Results

Erdős #1049 By Will Cook September 2026 Authorship, AI use and citation

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Summary. The record proves the fixed-base Hankel asymptotic for Zudilin's normalised moments and the rational-base irrationality criterion log b/log a<0.4056830213840605…. The determinant proof separates geometric node configurations from positive weights; the irrationality proof instead specialises Zudilin's 2004 polynomial forms after cancellation. Coefficient moments, evaluated lattices and congruence conditions distinguish integer divisibility from nonzero remainder decay. A broader ratio-limit argument and a rational-value model remain unformalised. None of these arguments settles $F (3 / 2)$ .

This paper covers the sharp fixed-base Hankel asymptotic and rational-base irrationality proofs, coefficient moments and arithmetic obstructions, with ordinary ratio-limit and rational-value models distinguished from checked results.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open.

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In this paper

A denominator-exponent model for Padé-type forms

For a sum of Padé-type terms, a common denominator must clear each summand. The two exponent expressions below are bounded by $(3 n^{2} - n) / 2$ , as their differences factor over $Z$ . We double all exponents to avoid fractions and write ${\tilde{E}}_{n} = 3 n^{2} - n$ . These are algebraic comparisons only: no approximation family is specified, so integral coefficients and a nonzero remainder estimate would still be needed for an arithmetic application.

Proposition 9.1 (exponent model: summand bound and exact gap). Let ${\tilde{E}}_{n} = 3 n^{2} - n$ and put

\tilde{P} (n, k) = 2 (k (n - k) + n k) + k (k - 1),

\tilde{Q} (n, m) = 2 (n^{2} - n) + j^{2} + 2 j m + j - m^{2} + 3 m, j = n - m - 1 .

Then, for integers $n, k, m$ :

if $0 \leq k \leq n$ , then $\tilde{P} (n, k) \leq {\tilde{E}}_{n}$ , and the gap factors as ${\tilde{E}}_{n} - \tilde{P} (n, k) = (n - k) (3 n - k - 1)$ ;
${\tilde{E}}_{n} - \tilde{Q} (n, m) = 2 (n + m (m - 1))$ identically;
if $n \geq 0$ and $m \geq 1$ , then $\tilde{Q} (n, m) \leq {\tilde{E}}_{n}$ .

Part (1) is the summand exponent bound, part (2) the exact gap identity, and part (3) the maximal exponent bound. The gap in (2) is an identity in $Z [n, m]$ , so (3) follows from $m (m - 1) \geq 0$ and $n \geq 0$ . The range $m \geq 1$ is sufficient, not necessary for this polynomial inequality: for integer $m$ , one also has $m (m - 1) \geq 0$ when $m \leq 0$ . No claim about a summand of an unspecified approximation is needed.

Example 9.2. At $n = 2$ the proposed doubled exponent is ${\tilde{E}}_{2} = 10$ , and $\tilde{P} (2, k)$ takes the values $0, 6, 10$ at $k = 0, 1, 2$ . The three gaps are $10, 4, 0$ , matching the factorisation $(2 - k) (5 - k)$ of part (1); the summand at $k = 2$ is the one that saturates the proposed denominator. For part (2), at $m = 1$ we have $j = 0$ and $\tilde{Q} (2, 1) = 6$ , with gap $4 = 2 (2 + 1 \cdot 0)$ .

These are inequalities between polynomials in the exponents. They establish that the proposed exponent ${\tilde{E}}_{n}$ dominates the two displayed summand exponent expressions, and nothing further. Positivity of the remainder, its rate of decay, and the comparison of that rate against the denominator height are the analytic obligations, and none of them is treated here, so nothing in this section is an irrationality measure.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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