Finite Witness Architecture for Computational Proof Programs: Separating Analytical Reduction, Rigorous Numerical Certification, and Residual Analytical Work

Computational mathematics increasingly relies on large numerical calculations, interval arithmetic, symbolic reduction, automated theorem proving, and machine-assisted verification. These tools can establish highly reliable finite results, but a recurrent methodological failure occurs when a rigorously certified computation is allowed to conceal an unproved analytical bridge between the finite object and the original mathematical claim. This paper names and formalizes a finite witness architecture for computational proof programs. The architecture separates four logically distinct layers: