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 A-1 leaves the p-value distribution invariant, rendering it invisible to every CTM based on A. We formalise this blind spot through the A-equivalence class [P] A and show that, for any target distribution T, there is a unique closest element Q in[P] A in KL divergence---the A-cryptic I-projection---with explicit density q(z) = t(z),p S(A(z))/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.