Mixed Hessians, Polynomial Communication, and the Discovery of Tensor Trees

This preprint studies the discovery of exact tree tensor network representations of bounded-degree polynomial phase tensors over finite fields, without a pre-supplied variable tree. The manuscript develops a stationary-completion construction relating mixed-Hessian rank to interactive polynomial evaluation and tensor unfolding rank. It then combines these comparisons with a connectivity-function approximation and an exact fixed-tree compiler to obtain a proposed topology-discovery and compilation procedure. A Fourier-transform application to fully observed functional relations is also presented. The revised manuscript expands the algorithmic and complexity arguments, including specialization, threshold search, equivalence testing, and counterexample extraction. The accompanying package contains source files and exact small-instance validation scripts. This is a research preprint. The mathematical arguments have not been independently peer reviewed, and the comparison with prior work is not exhaustive. AI tools assisted with mathematical exploration, drafting, programming, and revision.