The Missing Normalization: A Le Cam Factorization of Rate and Mechanism in Metastable SPDEs

Normalized transition-path laws describe how rare events occur, but need not identify how often they occur. We formalize this separation for the small-noise one-dimensional stochastic Allen--Cahn equation. We construct a two-parameter family in which a thermal coordinate $\eta$ shifts the barrier by $\eps\eta$, while a mechanism coordinate $\kappa$ perturbs the stable saddle Hessian; a Fredholm kinetic gauge cancels the resulting $\kappa$-dependent Eyring--Kramers prefactor. Along a parameter-independent admissible Galerkin refinement, the physical exit experiment converges in Le Cam distance to $\{\ExpLaw(e^{-\eta})\otimes\Normal(0,\Qsk^{-1})\}$. Consequently, timing and transition-state fields are orthogonal statistical coordinates: either marginal is strictly deficient for the joint parameter, the local Fisher information is $\diag(1,c''(\kappa))$, and explicit estimators from repeated independent cycles attain the local asymptotic minimax bound. We further show, along a reactive-admissible refinement, that complete normalized $A\to B$ reactive-path laws become Hellinger-indistinguishable across $\eta$, while their stationary reactive intensities retain the factor $e^{-(\eta_{1}-\eta_{0})}$; distinct $\kappa$ remain detectable. Thus faithful metastable generative learning requires the unnormalized reactive intensity measure---event rate times conditional path law---not the normalized path distribution alone.