Geometric Moments and Rational Lambert Values: Proofs and Further Results

Where the earlier criterion stops.

Both regions are cut out by the same quantity. The published cutoff $1/2-1/{\pi }^2$ of [6] is the reciprocal of ${\mu }_{\mathrm{BV}}=2{\pi }^2/({\pi }^2-2)$, the constant Van Assche later recovered as an integer-base irrationality-exponent bound for $F(p)$ [18], and the cutoff ${\theta }^{\ast }$ of Theorem 2.2 is the reciprocal of the smaller integer-base bound $\mu =C_1/C_0$ printed at [8]. For Zudilin’s family this reciprocal relation uses both the exact degree limit $d=C_1-C_0$ and the stronger evaluation bound $|U_n(p)|\le p^{W_n}\exp (O_p(n))$ proved in Corollary 2.4, together with decay exponent $\sigma =C_0$. A general exponential coefficient-height bound would contribute an additional term to the integer-base exponent estimate; it cannot simply be discarded. The discussion following Theorem 2.5 makes that distinction explicit. The rational-base proof clears the denominators of the already cancelled polynomials. It does not infer value irrationality merely by taking the reciprocal of a published exponent bound. The bases gained are exactly the strip $s^{\mu }<r\le s^{{\mu }_{\mathrm{BV}}}$, which is empty for $s=2$ and $s=3$ and is first occupied at $s=4$ by the single coprime numerator $31$. The integer comparisons and rational bracket in Section 2.5 establish membership in the region. The irrationality conclusion in Theorem 2.3 also uses the constructed forms and their proved analytic estimates. This application does not improve either inherited integer-base exponent bound.