Information-Geometric Blind Spots of Conformal Test Martingales

Conformal test martingales (CTMs) are e-processes for sequentially testing exchangeability via p-values produced by a conformity function $A$. However, any distribution shift that preserves the push-forward $P\circ A^{-1}$ leaves the p-value distribution invariant, rendering it invisible to every CTM based on $A$. We formalise this blind spot through the \emph{$A$-equivalence class} $[P{]}_A$ and show that, for any target distribution $T$, there is a unique closest element $Q^{\ast }\in [P{]}_A$ in KL divergence—the \emph{$A$-cryptic I-projection}—with explicit density $q^{\ast }(z)=t(z)p_S(A(z))\mathrm{/}{t}_S(A(z))$ and a Pythagorean decomposition into detectable and cryptic components. When $A$ is the Neyman-Pearson likelihood ratio, the cryptic component vanishes, connecting our result to optimal e-value design. We demonstrate the theory on high-dimensional image data by empirically constructing this I-projection. By conditionally resampling EMNIST letters to identically match the conformity scores of MNIST digits, we demonstrate that a distribution can be shifted to a disjoint support while leaving the test martingale mathematically blinded.