A Finite-Dimensional Bounded-Perturbation Certificate

Saddle-point approximations replace an action by its quadratic part, but transferring a quantitative error statement between applications requires an explicit remainder estimate. Here (C,ρ) is a supplied certificate pair, not a canonical invariant. Let (Γ, B,λ) be a measure space, let measurable A_2,R:Γ→ R, let K_0>0, and suppose 0<FGauss=K_0∫_Γ e- A_2\,dλ<∞. If |R|≤ C/ρ^3≤1 almost everywhere, then F = FGauss · (1 + δ), |δ| ≤ 2C/ρ^3 where F=K_0∫_Γ e- A_2-R\,dλ is positive and finite. The proof is an elementary change to the normalized reference probability measure followed by |e^x-1|≤2|x| for |x|≤1. The exponent three is part of the remainder parameterization, not a derived universal physical exponent. The paper records a domain-validation agenda across twelve areas. None currently supplies the measurable normalized integral and full-domain remainder estimate needed to instantiate the theorem. Numerical calculations illustrate quartic, cubic, product, magnetization-Laplace, and Poisson/Stirling asymptotics, but do not validate those hypotheses. The companion checks 25 scalar consequences of supplied inequalities; it does not formalize integration, normalization, or the manuscript theorem. The established contribution is therefore the conventional finite-dimensional certificate and a falsifiable agenda for future domain work. Keywords: saddle-point approximation, Gaussian approximation, uniform remainder bound, bounded perturbation, reproducibility MSC 2020: 41A60, 81S40, 60H30, 11M06, 91A10 Maturity: Draft. Target venue: Zenodo preprint. Part of The Latent research program. Related papers in this program: Core The Latent, Fin Correlated Cos V2, Math Additive Correlative Duality, Phy Navier Stokes Regularity, Rough Volatility, Universal.