FiniteSlitBankMoments
Abstract
A strictly signed continuous bank on a finite interval dominates a faster geometric error.
Theorem 1.1 (A fixed positive bank mass dominates the outer scale).
Lean statement: D5/S3/AnalyticClosure/Polylogarithm/FiniteSlitBankMoments.bank_dominance
Proof. Machine-checked in Lean as D5/S3/AnalyticClosure/Polylogarithm/FiniteSlitBankMoments.bank_dominance (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. Ce Xu and Jianqiang Zhao (2026). Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers. DOI: 10.48550/arXiv.2609.11072. URL: https://arxiv.org/html/2609.11072v1.
Commentary.
Let R>1 and let h be continuous and nonnegative on [1,R], with h(x)>0 for 1<x<R. For every real B, eventually in n, B/R^n is strictly less than (1/pi) times the integral from 1 to R of h(x)/x^(n+1). The threshold may depend on h, R and B; B need not be positive.
Choose a=(2+R)/3 and b=(1+2R)/3. The compact interval [a,b] lies inside (1,R), so h has a strictly positive minimum there. Integral monotonicity gives a constant c>0 with c/b^n below the full bank moment for every n. Since b<R, B*(b/R)^n tends to zero and eventually lies below c/pi. This fixed interior mass supplies the strict comparison even when h vanishes at the endpoint one.
This is the real integral estimate used after a finite slit contour has been deformed. The estimate itself neither assumes nor proves the Cauchy contour identity, its orientation, its connector limits, or an outer-circle bound. It therefore asserts no Taylor coefficient sign. No coefficient asymptotic equivalent or effective threshold is asserted.