One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversion

One implicit DDIM inversion step is the cheapest probe of whether a pretrained diffusion model encodes local manifold geometry. It is the stationarity condition of an explicit potential, x-G(x)=nablaPsit(x), strongly convex at the Bayes limit with modulus exactly e-ht for the step's log-SNR gap ht - for every data law, schedule and point, with no manifold, reach or unimodality hypothesis. Three consequences must be kept apart. (i) The solution is unique at the Bayes limit; a second one requires the trained score to violate the posterior-covariance bound by 1/(1-e-ht), a hypothesis-free certificate of model error; the same bound makes contraction a schedule constant, ρgstar=1-e-ht<0.326 throughout the standard DDPM schedule. (ii) The solver can still fail: Picard iteration is unit-step gradient descent on Psit, unstable wherever λmax(nabla2Psit)>2, so oscillation certifies nothing; damping below 2/λmax cures it. (iii) The geometry lives in the convergence domain: on the scale-free depth w=rkappamax the oscillation shell sits at w=tfrac12, schedule-free, and the divergence shell at w=1/(1+ρgstar), with a measured finite-noise correction in |II|2. Exact scores reproduce both to within 0.54% on three classes; no trained score we probe shows a shell - a derived limitation, not a null result: the Fermi window conflicts with the model's own training support by 3.6-5.6×, and the trained Hessian-Lipschitz constant is 2-12% of the curvature the law reads, 0 on a ReLU net. Finally the unconditional ceiling σtλmax(sym,J)≤1, from Cov(x0mid xt)succeq0 alone, holds for the exact score to 3×10-7 but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by 1.26-4.66×.