Higher-Order Return and Persistent Silence: Fisher–Rao Geometry as a Weighted Hidden-Flow Quotient

Abstract

We study finite-dimensional observational return beyond the quadratic residue level. Starting from a hidden-defect filtration relative to a diagonal observational algebra, we introduce higher cyclic return operators and distinguish temporary quadratic silence from persistent algebraic silence. Quadratic silence is characterized as a restricted orthogonality condition, while persistent silence under finite algebraic completion is shown to correspond to an exact commutant obstruction. We further formulate the observable residue sector as a quotient of hidden flow by silent circulation. Equipping hidden edge flows with a weighted energy yields a natural quotient metric determined by the weighted graph Laplacian. For complete support with weights w_ij = p_i p_j, this minimum-energy quotient metric coincides exactly with the Fisher–Rao metric on the probability simplex. The results separate rank saturation from metric saturation: hidden connectivity determines the dimension of visible geometry, while the support and weighting structure determine its metric form. The paper remains entirely finite-dimensional and does not derive the Fisher weights dynamically; rather, it identifies the precise hidden-flow quotient structure under which Fisher–Rao geometry emerges.