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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A finite spectral consequence.

Write $A_{N} = (α_{i + j})_{0 \leq i, j < N}$ and $B_{N} = (β_{i + j})_{0 \leq i, j < N}$ . We study the matrix pencil $Y A_{N} - B_{N}$ , where $Y$ is a scalar variable, through the roots of its determinant. Let $D = diag (1, p (p - 1)^{3}, \dots, [p (p - 1)^{3}]^{N - 1})$ . Then $A_{N} = p D (s_{i + j}) D$ . For $p > 1$ this is an invertible positive diagonal congruence. The remainder representation already proves $F (p) A_{N} - B_{N} = (v_{i + j}^{*}) > 0$ at every rank.

Proposition 3.9 (finite coefficient positivity and pencil roots). For every real $p > 1$ and $1 \leq N \leq 8$ , $A_{N}$ is positive definite and all roots of $det (Y A_{N} - B_{N})$ are real and strictly less than $F (p)$ . The roots at consecutive ranks $N, N + 1 \leq 8$ interlace non-strictly.

Proof. The leading principal minors of $(s_{i + j})_{i, j < N}$ are exactly $D_{k, 0} (p)$ for $1 \leq k \leq N$ ; their coefficient positivity proved above, together with Sylvester’s criterion, gives positive definiteness. The diagonal congruence gives $A_{N} > 0$ . The real symmetric matrix $A_{N}^{- 1 / 2} B_{N} A_{N}^{- 1 / 2}$ has these pencil roots as eigenvalues, and $F (p) I - A_{N}^{- 1 / 2} B_{N} A_{N}^{- 1 / 2} > 0$ puts them strictly below $F (p)$ . The minimum–maximum principle for $x^{T} B_{N} x / (x^{T} A_{N} x)$ on nested coordinate subspaces gives non-strict interlacing. These are subspaces for the original pencils; the separately conjugated symmetric matrices need not be principal submatrices of one another. ◻

Proposition 3.9 uses only the eight unshifted certificates $D_{k, 0}$ , not an all-rank coefficient measure. The shifted certificates $D_{k, 1}$ are used below for a truncated Stieltjes moment representation and the first fifteen continued-fraction coefficients, not for this application of Sylvester’s criterion. An infinite-support coefficient measure would extend the pencil argument to all ranks. At the certified ranks, the largest root is a nondecreasing lower bound for $F (p)$ ; convergence to $F (p)$ , simple roots, strict interlacing and denominator control are not established. At $p = 1$ the congruence degenerates and $F$ diverges, so the proposition excludes that endpoint.

The pencil is different from the Jacobi matrix of the coefficient moments. This is already visible at rank two. The defining formulas give

β_{0} = 0, β_{1} = p^{3} (p - 1)^{2} (p + 2) > 0 (p > 1) .

Consequently $det B_{2} = - β_{1}^{2} < 0$ . Since $A_{2} > 0$ , the product of the two real pencil roots is $det B_{2} / det A_{2} < 0$ : one root is negative and the other positive. In contrast, the finite Stieltjes construction below has only positive nodes. Positivity of the coefficient Hankel matrix therefore does not identify these two spectral constructions.

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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