W‑Scanner Exact Ledger: A Reusable Audit Engine for Exact Arithmetic, Finite Verification, and Reproducible Mathematical Computation

Abstract This paper introduces W‑Scanner Exact Ledger, a reusable audit engine designed to separate arithmetic semantics from logical coverage in finite mathematical verification. Exact rational or symbolic computations may be correct for every executed instance, yet they provide only finite coverage; they are not, by themselves, proofs of universally quantified theorems. The engine enforces this distinction by recording arithmetic modes, coverage modes, theorem‑input roles, verification rules, limitations, and blockers.

The promotion policy rejects floating‑point or diagnostic values as theorem inputs, distinguishes padded conclusions from certified conclusions, and prevents randomized exact tests from being promoted to universal theorems. The source package implements exact rational statistics, scalar Schur ledgers, polynomial and Bernstein ledgers, finite Abel summation, formal root‑of‑unity sine arithmetic, Gram‑kernel checks, SHA‑256 manifests, command‑line tools, tests, and archived project‑specific checkers. Several finite identities from analytic number theory are included only as case studies; no analytic asymptotic estimate or statement about the Riemann Hypothesis is certified.

Version 8 builds on the earlier W‑Scanner framework by adding a strict separation between arithmetic exactness and logical theoremhood, implemented in a minimal Python package. The framework is not a theorem prover and does not replace proof. Its narrower purpose is to make finite computational evidence replayable, scope‑aware, and difficult to overstate, thereby providing a reproducible bridge between exploratory computation and reviewable finite mathematics.

Keywords certified computation; exact arithmetic; symbolic verification; audit ledgers; reproducibility; interval arithmetic; finite certificates; validated numerics