Same Output, Different Process: Ornstein's d-Distance as a Computational-Equivalence Metric for Model Diffing

When are two neural networks the same computation? Existing model-comparison tools answer at the level of static representation geometry (CKA, SVCCA, crosscoders, the Platonic hypothesis) or of the conjugacy of a deterministic linearized system (Dynamical Similarity Analysis, DSA). Both levels can be blind to the distinction between processes: two models can emit statistically identical symbols, occupy indistinguishable representation geometries, and share a linearized flow, while implementing different stochastic processes. We introduce the first application of Ornstein’s ̄d-distance — a nonparametric metric on stationary processes from ergodic theory, estimated by optimal transport between empirical n-block distributions under Hamming cost — to model comparison, on deliberately low-entropy symbolic readouts where the estimator is consistent. In a pre-registered study on GRUs trained to emit hidden- Markov processes, we constructed twins of each base model (noise-injected, distilled, pruned, recoded) that preserve its output by every static measure (unigram total variation ≤ 0.0076, next-symbol cross-entropy within 1.6%). A 50%-pruned, fine-tuned twin with indistinguishable outputs (CKA = 1.000 ± 0.000; DSA inside its own null range) is separated from its base by ̄d at ≥ 2.6× the same-process floor on every one of 5 seeds (sign-consistent, p = 1/32; mean 18.0 ± 12.1×). Independent training seeds — called identical by CKA (1.000 ± 0.000) and DSA (0.080±0.014) — differ by 16.4±6.1× floor; sweeping the prune fraction from 0.1 to 0.9 keeps DSA at its null throughout while ̄d separates at every fraction. Conversely ̄d correctly certifies GRU distillation as process-preserving at the output level (transformer distillation is not: students separate on every seed of two grammars, while DSA rates them as more different than models of entirely different tasks), and a different-computation control with identical entropy rate and unigram marginal is separated at 194.9±21.3× floor while the pre-registered DSA configuration cannot distinguish it from its own null (a configuration-robustness check shows alternative configurations separate that control, yet every tested configuration still ranks the noise twin above the genuinely different computation — none yields scores readable as a process distance). We report the pre-registered gate verdict (which failed as written), the disclosed analysis amendments that repair it, every same-process floor, and the block-length consistency wall n ≲ log2 N/hˆ that bounds all claims. Per Ornstein–Weiss, ̄d is not a general isomorphism detector; the claim is a process-level distance within a fixed readout — and at that level it sees changes every static and deterministic-conjugacy metric we tested misses.