Integrated score error can miss every Euler evaluation

In the finite-dimensional Gaussian credible-ball question reproduced below, choose $d=n=1$, $R=1$, $\sigma =1$, $T=1$, $\beta (t)=1$, and $\gamma =1\mathrm{/}4$, with arbitrary positive $p,s,a$. Take the numerical operator equal to the true operator, so ${\delta }_A=0$. For every integer $K\ge 1$ there is a jointly Borel finite score field with integrated score error exactly zero, whose Euler terminal law is $N(1,K^{-1})$ for every observation. With $M=K$ independent terminal draws, the resulting empirical ball has frequentist coverage tending to zero at truth $\theta =0$, although the stated uniform boundary-margin condition holds with exponent $\kappa =1$. The exact credible-ball baseline satisfies ${\delta }_{1,1\mathrm{/}4}^0<3\mathrm{/}4$. Consequently no bound in that question can have $\mathrm{\Phi }\to 0$ uniformly over its original score class as $({\delta }_A,{\epsilon }_s,K^{-1},M^{-1\mathrm{/}2})\to (0,0,0,0)$. The original formulation is retained verbatim under Original question.

Knapik, van der Vaart, and van Zanten, full paper

Szabó, van der Vaart, and van Zanten, full paper

Giné and Nickl, full textbook

Narnhofer et al., full paper

Fix integers $d,n\ge 1$, parameters $p,s,a,R,\sigma >0$, a credibility error $\gamma \in (0,1)$, and a continuous schedule $\beta :[0,T]\to (0,\mathrm{\infty })$ with $T>0$. Let ${\mathrm{\Theta }}_s(R)=\{\theta \in {\mathbb{R}}^d:{\sum }_{j=1}^d{j}^{2s}{\theta }_j^2\le R^2\},A_p=\mathrm{diag}(1^{-p},2^{-p},\dots ,d^{-p})\mathrm{.}$ Observe $Y_i=A_p\theta +{\epsilon }_i$, $1\le i\le n$, with independent ${\epsilon }_i\sim N(0,{\sigma }^2{I}_d)$. Use the proper Gaussian prior ${\mathrm{\Pi }}_0=N(0,\mathrm{diag}(j^{-1-2a}{)}_{j=1}^d)$, and let ${\mathrm{\Pi }}_n(\cdot \mid Y)$ be its exact posterior under the stated forward operator and noise law. Let $m_n(Y)$ be the posterior mean and let $r_{n,\gamma }(Y)$ be the unique radius for which the Euclidean ball $C_{n,\gamma }(Y)=\{u\in {\mathbb{R}}^d:\mathrm{\parallel }u-m_n(Y){\mathrm{\parallel }}_2\le r_{n,\gamma }(Y)\}$ has posterior probability $1-\gamma$. Define the exact frequentist coverage $c_{n,\gamma }(\theta )=P_{n,\theta }\{\theta \in C_{n,\gamma }(Y)\}$ and its baseline error ${\delta }_{n,\gamma }^0={\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}\mathrm{\mid }{c}_{n,\gamma }(\theta )-(1-\gamma )\mathrm{\mid }$.

For a deterministic numerical matrix ${\widehat{A}}_n\in {\mathbb{R}}^{d\times d}$ with ${\delta }_A\ge 0$ and $\mathrm{\parallel }{\widehat{A}}_n-A_p{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\delta }_A$, let ${\widehat{\mathrm{\Pi }}}_n(\cdot \mid Y)$ be the Gaussian posterior formed with ${\widehat{A}}_n$ and the same prior and noise variance. Write ${\widehat{p}}_{n,t}^Y$ for the density obtained by applying the forward variance-preserving diffusion $d{X}_t=-\frac{1}{2}\beta (t)X_{t}dt+\sqrt{\beta (t)}d{W}_t$ to $X_0\sim {\widehat{\mathrm{\Pi }}}_n(\cdot \mid Y)$, and set ${\widehat{s}}_{n,t}^Y=\mathrm{\nabla }\log {\widehat{p}}_{n,t}^Y$. A jointly Borel learned score field $s_{n,t}^Y$ is required to satisfy ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}{E}_{Y\sim P_{n,\theta }}[\frac{1}{T}{\int }_0^T{\int }_{{\mathbb{R}}^d}\mathrm{\parallel }{s}_{n,t}^Y(x)-{\widehat{s}}_{n,t}^Y(x){\mathrm{\parallel }}_2^2{\widehat{p}}_{n,t}^Y(x)dxdt]\le {\epsilon }_s^2\mathrm{.}$ For $h=T\mathrm{/}K$, define the reverse-time Euler chain by $Z_0\sim {\widehat{p}}_{n,T}^Y$ and, for $k=0,\dots ,K-1$, $Z_{k+1}=Z_k+h[\frac{1}{2}\beta (T-kh)Z_k+\beta (T-kh)s_{n,T-kh}^Y(Z_k)]+\sqrt{h\beta (T-kh)}{\zeta }_k,$ where the ${\zeta }_k$ are independent $N(0,I_d)$ vectors independent of $Z_0$ and the observations. Draw $M$ independent copies of this chain conditionally on $Y$ and denote their terminal values by $X^{(1)},\dots ,X^{(M)}$. Let $\bar{X}=M^{-1}{\sum }_i{X}^{(i)}$, let $\widehat{r}$ be the empirical $1-\gamma$ quantile of $\{\mathrm{\parallel }{X}^{(i)}-\bar{X}{\mathrm{\parallel }}_2{\}}_{i=1}^M$ (the smallest order statistic whose empirical cumulative mass is at least $1-\gamma$), and put ${\widehat{C}}_{n,K,M}(Y)=\{u:\mathrm{\parallel }u-\bar{X}{\mathrm{\parallel }}_2\le \widehat{r}\}$. Suppose the exact credible-ball boundary satisfies the uniform margin condition ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}{P}_{n,\theta }(\mid \mathrm{\parallel }\theta -m_n(Y){\mathrm{\parallel }}_2-r_{n,\gamma }(Y)\mid \le u)\le C_{\mathrm{a}\mathrm{c}}{u}^{\kappa }(0<u\le u_0)$ for fixed $C_{\mathrm{a}\mathrm{c}},u_0>0$ and $\kappa >0$. Determine whether a minimax-sharp nonasymptotic function $\mathrm{\Phi }$ exists uniformly over all such score fields and, if it does, find it as a function of $d,n,p,s,a,R,\sigma ,T,\gamma ,C_{\mathrm{a}\mathrm{c}},u_0,\kappa ,{\delta }_A,{\epsilon }_s,K^{-1}$, and $M^{-1\mathrm{/}2}$ such that ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}\mid P_{n,\theta ,\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{r}}\{\theta \in {\widehat{C}}_{n,K,M}(Y)\}-(1-\gamma )\mid \le {\delta }_{n,\gamma }^0+\mathrm{\Phi },$ with matching lower bounds that identify when $\mathrm{\Phi }\to 0$. The probability includes observation noise and sampler randomness. The target concerns the additional coverage error introduced by the approximate forward operator, learned score, finite reverse discretization, and finite posterior sample; it does not assume that the exact Bayesian credible ball is already calibrated on every truth class.

This resolves the requested uniformly vanishing-error guarantee negatively under the original score hypothesis. Time-integrated error does not control pointwise values at a deterministic solver grid. A constant worst-case coverage bound remains possible; no impossibility of all numerical functions is asserted. No Bayesian calibration claim is imported from a conformal theorem.