Same Function, Different Mechanism: Implementation-Local Geometry Beyond Accuracy
Miroslav Lžičař
Abstract
Two models can make exactly the same predictions yet respond differently when their implementations are perturbed. A lookup table can change one case independently; a shared rule couples changes across cases. We compare these mechanisms through an implementation chart: a specified local family of logit perturbations around a fixed predictive distribution. Our main result counts the independent prediction changes allowed by exact finite-group sharing: the active dimension is the number of input-output orbits minus the number of input orbits. For softened modular addition, lookup and shared cyclic charts therefore have ranks $p^2(p-1)$ and $p-1$ despite identical predictions. The classical Fisher pullback connects these dimensions to local KL curvature; mechanism-preserving reparameterizations leave the rank unchanged. We also bound the spectral tail under approximate sharing and separate quadratic rank from higher-order learning coefficients. Trained modular-addition models verify that chart rank is measurable, while controlled compiled transformers distinguish compiler-supported intermediate charts from matched alternatives. Negative controls show where this discrimination fails. The framework evaluates specified mechanism hypotheses; it does not infer a unique mechanism from predictions or predict training dynamics.