CompositionBanks
Abstract
Every positive composition and positive reciprocal power has its source-faithful endpoint and strict conjugate banks on a sufficiently small closed half-collar.
Theorem 1.1 (Endpoint limits and strict conjugate banks).
Lean statement: D5/S3/AnalyticClosure/Polylogarithm/CompositionBanks.result
Proof. Machine-checked in Lean as D5/S3/AnalyticClosure/Polylogarithm/CompositionBanks.result (✓ 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.
For every positive head, positive-entry tail and positive power ell, the theorem uses the actual continued source branch and its depth-normalized reciprocal power A. It produces a real radius rho with 0<rho<1. On the intersection of the full principal slit domain with the closed rho-ball about one, the continued branch is nonzero. Along the full slit-domain filter at one, A tends to the reciprocal zeta power when the head is admissible, and to zero when the head is one.
The same radius supports jointly chosen upper and lower functions continuous on the closed upper and lower half-collars. They agree with A wherever the corresponding half-collar lies in the slit domain, take the common endpoint value at one, and are related by complex conjugation throughout the lower closed half-collar. For every real t with 0<t<=rho, the upper boundary value at 1+t has strictly negative imaginary part. Thus the strict sign is proved on the entire punctured boundary interval supplied by rho, not only eventually or at a selected point.
The proof combines source-weight induction, radial and arc integral control, leading-one polynomial transport, the ordinary-head step and local analytic representations of the actual branch. This is an intermediate analytic bridge with zero solved-problem credit. It does not assert a general power-log asymptotic. Its rho is local and composition-dependent; it does not supply the still-missing normalized extension on a fixed slit disk, the Taylor or formal-inverse coefficient identification, finite-contour transfer, or the all-j assembly required for Xu-Zhao Conjecture 1.3. The full conjecture remains open, with no resolution, novelty or priority claim.