Fixed-Point-Corrected Numerical Schemes for Reverse-Time Diffusion Sampling: Stability and Error Decomposition

We study fixed-point-corrected numerical schemes for reverse-time diffusion sampling. Instead of treating the reverse sampler only as an explicit Euler–Maruyama discretization, we formulate each reverse-time step as a local implicit equation and approximate its solution by a finite number of fixed point corrections. After fixing the backward-time drift convention, we prove well-posedness of the local implicit step, contraction of the inner solver for sufficiently small step sizes, and a conditional global error decomposition separating terminal initialization, score approximation, assumed implicit time-discretization error, and fixed point truncation error. The estimate clarifies how additional inner corrections reduce numerical residuals at the cost of extra score evaluations. Multi-seed exact-score experiments, implicit-gap diagnostics, controlled score perturbations, a fitted Gaussian-mixture score example, and a neural-score Fashion-MNIST experiment with inpainting demonstrate the distinction between algebraic solver consistency and distributional sample quality. The paper is intended as a solver-level numerical analysis of reverse diffusion dynamics, not as a new large-scale generative architecture or a complete convergence theorem for diffusion models.