Ordinal Locality Transition Tensor: A Primitive Locality Construct for Information and Transition Geometry

This paper develops a measure-theoretic and geometric construction of the Local Transition Tensor, an object that encodes local transition structure in probabilized metric spaces before the introduction of labels, classes, or scalar summaries. Starting from a primitive rankwise locality kernel and its aggregation over measurable partitions, we derive an S_K-equivariant tensor that characterizes neighborhood transitions between partition cells. We formulate admissibility and exclusion theorems establishing the uniqueness of this construction under locality, measurability, equivariance, and linearity assumptions. We then study tensorial contractions and characterize exactly when the transition tensor induces a strictly positive definite Riemannian metric: positivity holds precisely on the open locus where the complete weighted transition embedding has full row rank. The paper further shows that transition geometry and Fisher–Rao information geometry arise as distinct descendants of the same primitive locality datum. The former preserves metric-neighborhood structure, while the latter factors through a geometry-blind quotient and a categorical score lift. A pseudo-locality classification theorem identifies the categorical observables invariant under geometry forgetting as exactly the functions of the label marginal. Finally, a controlled labeled-partition experiment holds the primitive kernel T_j fixed while varying twelve partitions with analytically known Hausdorff boundary content. The local off-diagonal slope of T^Y identifies that content with median within-block Spearman correlation 0.9545 and R²(within) = 0.9417, whereas a geometry-blind F₂ control collapses to R²(within) = 0.0203. This separates an intrinsic geometric complexity of the labeled transition tensor from geometry-blind or purely computational proxies.