Asymptotic-preserving terminal completion

1. Integration point: modify the sampler's final denoising segment, not the neural network weights. Given a pretrained diffusion model with denoiser output D_theta(x,sigma), choose a switching scale a larger than the requested terminal scale epsilon. Integrate the existing Euler, Heun, DPM-Solver, or flow-matching solver only down to sigma=a, then perform one terminal completion call. For a VE/EDM-style parameterization, use the residual r=x_a-D_theta(x_a,a) and return x_epsilon=D_theta(x_a,a)+(epsilon/a)r. For a sampler whose model predicts score or noise rather than x0, first convert that prediction to the equivalent denoiser D_theta using the model's standard preconditioning formulas.

2. Pseudocode:

input x, sigma_max, epsilon, base_resolution h, solver order p
choose a = clip(C_a * h**(p/2), epsilon, sigma_max)
while sigma > a:
    sigma_next = base_grid_step(sigma, h)
    x = base_solver_step(x, sigma, sigma_next, model)
D = denoiser(model, x, a)
x = D + (epsilon/a) * (x-D)
return x

If epsilon=0, return D directly. The formula above is the normal-mode terminal map; the O(a^2) estimate is the geometric truncation error on smooth compact boundaryless manifolds.

3. Compute versus estimate: the terminal map and cutoff scaling are prescribed mathematically. Estimate the unknown constant C_a and the bounded transfer factor empirically on a calibration batch by sweeping a; estimate the actual convergence order from endpoint errors at h, h/2, and h/4 using identical random seeds. Do not let a fall below epsilon.

4. First experiment: use CIFAR-10 or ImageNet-64 with a small EDM checkpoint, comparing the original sampler against Euler or Heun plus this completion at equal FID and equal same-seed endpoint error. Sweep NFE and epsilon, recording FID, reconstruction or Wasserstein proxy, and function evaluations. The falsifiable target is that NFE becomes essentially independent of epsilon, while log-log endpoint error versus h retains the baseline slope p and the terminal residual scales approximately quadratically with a.