Testing the "Benign Running Statistics" Hypothesis: Class-Conditional Batch-Normalization Substitution Rejects the Distribution-Shift Account at $p \lt10^{-19}$

Batch-normalization (BN) running statistics are widely treated as a benign approximation to the true population mean/variance $\textemdash$ a numerical convenience whose perturbation should affect accuracy in the same way as any other distribution shift. We pose this as a falsifiable null hypothesis and test it. $H_0$ (distribution-shift account): substituting any non-global statistics at inference should hurt accuracy roughly symmetrically; in particular, substituting same-class statistics (closer to global) should hurt no more than substituting wrong-class statistics (further from global). We design a paired three-condition intervention $\textemdash$ global / same-class / wrong-class $\textemdash$ across four CNN architectures and three datasets, totalling 1,320 paired-trial comparisons over 22 trained models. The data reject $H_0$ sharply: same-class accuracy is $3.5$ vs. wrong-class $65.4$ on CIFAR-10/SmallResNet ($t(9) = -39.1$, $p < 10^{-10}$ paired; same-vs-global $t(9) = -374$, $p = 3.5 \times 10^{-20}$). We then run three pre-registered alternative-explanation tests: (i) a linear-probe test on the same post-substitution features (probe accuracy $\geq 99.7$ rules out "the encoder broke"); (ii) a GroupNorm architectural-control test (no running statistics; unaffected, rules out "generic running-stat perturbation"); (iii) a temperature-scaling test (ECE halves from $0.394$ to $0.151$ but accuracy unchanged, rules out "logit-scale issue"). A within-architecture layer-wise ablation tests the multiplicative-compounding hypothesis: single-layer drops sum to $29.4$ but the all-layer drop is $86.8$, a $2.95\times$ super-additive factor that replicates seed-to-seed ($n = 3$). A per-statistic disentangling experiment tests a sub-hypothesis of the algebraic argument below: substituting only $\mu$ leaves $76.7$ accuracy, only $\sigma^2$ leaves $27.2$, and the joint substitution reaches $2.8$ $\textemdash$ neither statistic alone reproduces the catastrophe, the two interact super-additively, and contrary to the one-layer linearization $\sigma^2$-substitution alone is more damaging than $\mu$-substitution alone. We close with a one-line algebraic identity that restates the surviving asymmetry as an exact zero in a one-layer linearization. The paper is intended as both an empirical contribution and a worked example of intervention-design hypothesis testing as ML methodology.