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

PDFOpen the PDF, 80 pages TeXLaTeX source Source on GitHub Problem page

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.

Start here, by the editorial route

In this paper

Rescaling integer rows

For a single irrationality form, the essential conditions are integral coefficients and a nonzero remainder tending to zero. Dividing a nonzero integer row $(U, V)$ by $gcd (| U |, | V |)$ preserves integrality and divides its remainder by the same amount; the resulting row is primitive. A useful content division removes a factor already shared by the two entries. Multiplying a row first merely introduces a factor that also multiplies its remainder. For two rows, the same factors multiply their determinant. The identities below separate this rescaling from genuine cancellation. A height estimate is needed for an irrationality measure, not for the elementary single-form criterion. For a real target $S$ , put

L_{S} (U, V) = U S - V, Δ ((U_{n}, V_{n}), (U_{m}, V_{m})) = U_{n} V_{m} - U_{m} V_{n} .

Here $L_{S} (U, V)$ is the error of the integer linear form, not the rational approximation error $| S - V / U |$ when $U \neq 0$ . The determinant eliminates $S$ exactly:

Δ = U_{m} L_{S} (U_{n}, V_{n}) - U_{n} L_{S} (U_{m}, V_{m}) .

If this integer determinant does not vanish, then

1 \leq | Δ | \leq | U_{m} | | L_{S} (U_{n}, V_{n}) | + | U_{n} | | L_{S} (U_{m}, V_{m}) |,

and the two errors cannot both be smaller than $1 / (| U_{n} | + | U_{m} |)$ . The next theorem compares the divisor introduced by rescaling with the resulting change in the determinant’s absolute value.

Theorem 4.1 (rescaling rows and their determinant). Let $S$ be real, let $(U_{n}, V_{n})$ and $(U_{m}, V_{m})$ be pairs of integers, and let $c_{n}, c_{m}$ be integers. Then

L_{S} (c_{n} U_{n}, c_{n} V_{n}) = c_{n} L_{S} (U_{n}, V_{n}),

Δ (c_{n} (U_{n}, V_{n}), c_{m} (U_{m}, V_{m})) = c_{n} c_{m} Δ ((U_{n}, V_{n}), (U_{m}, V_{m})),

and consequently

| Δ (c_{n} (U_{n}, V_{n}), c_{m} (U_{m}, V_{m})) | = | c_{n} | | c_{m} | | Δ ((U_{n}, V_{n}), (U_{m}, V_{m})) | .

In particular $c_{n} c_{m}$ divides the scaled determinant. Multiplication by these scalars therefore introduces a divisor whose absolute value is exactly the factor multiplying the determinant’s absolute value.

Proof. Expand the linear form and use the bilinearity of the determinant. The original integer determinant is the quotient witnessing divisibility; it need not be primitive. ◻

Example 4.2. Take $(U_{n}, V_{n}) = (1, 2)$ and $(U_{m}, V_{m}) = (3, 5)$ , so that $Δ = 1 \cdot 5 - 3 \cdot 2 = - 1$ . Multiplying the first row by $c_{n} = 6$ and the second by $c_{m} = 10$ gives the rows $(6, 12)$ and $(30, 50)$ , whose determinant is $6 \cdot 50 - 30 \cdot 12 = - 60$ . That determinant is now divisible by $60$ , which looks like a local gain of $60$ ; and its absolute value has risen from $1$ to $60$ , which is a cost of exactly the same size.

Lean checks the error identity in error scaling, the determinant identity in content factorisation, the exact absolute-height identity in absolute determinant scaling, the divisor statement in content-product divisibility, and the elimination identity in exterior determinant identity.

The identities allow zero scalars, but cancelling the scalar factors requires them to be nonzero. Multiplication alone gives no improvement in the comparison between a divisor and the determinant’s absolute value. The theorem neither asserts that the determinant is nonzero nor estimates the original approximation error. Cancelling polynomial factors before evaluation, finding factors common to several minors, and adding different rows remain separate operations.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

Prefer the manuscript?

PDFOpen the PDF, 80 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:1f04580702ed8f8c. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.