Sharp Gaussian information-response frontiers: finite p-adic realization and noncommuting harmonic quenches

Abstract

How strongly can a positive precision update change a Gaussian response at a given information cost? Standard log-determinant and inverse-update identities do not by themselves determine the admissible costs when the worst modal contraction, total contraction, and active-rank cap are fixed. On a finite-dimensional physical space, for each directed Gaussian relative entropy separately, we derive its complete sharp attainable interval under these three constraints, including all equality and degenerate cases. A fixed probe also retains eigenvector orientation, but its normalized response is bounded by the largest contraction. Consequently, vanishing relative entropy forces every normalized individual response change to vanish, while absolute or extensive responses can still diverge if scale or rank is uncontrolled. We construct an exact finite hierarchical testbed for the same update problem on quotients of Zpd. A selected Taibleson-type hierarchy has closed radial graph and Green kernels; resistance recovers its cylinder tree, and baseline transfer resistance vanishes across separated cylinders. Point-shortcut response then reduces to endpoint valuations and an m×m matrix without allocating vertex-indexed objects. For the corresponding finite harmonic stiffness family with a common positive-definite mass matrix, we require no commutativity. An exact position-measurement bound holds at all temperatures, while a Bernoulli expansion through eighth order resolves the lowest-order inequivalent cyclic trace words within the expansion. For each fixed nontrivial quench, the joint-position data-processing gap is (βℏ)2Tr⁡[(ΔS)2S0−1]/24+O((βℏ)4), whereas the full-state semiclassical correction Dq−Dcl relative to the classical limit begins at order (βℏ)4; asymptotic relative-entropy tightness in ratio follows as βℏ→0.