Probability Graph Dynamics with Phi-Harmonic Structure

This work introduces Probability Graph Dynamics, a mathematical framework for selection-driven evolution on infinite weighted graphs. A probability graph is defined as a complete edge field equipped with a normalized weight measure, a harmonic frequency assignment constrained by powers of the golden ratio, and a threshold mechanism that induces sparse manifest subgraphs. The framework formalizes selection operators that redistribute probability mass while preserving normalization, enabling temporal dynamics without modification of the underlying probability field.Key results include proofs of manifest sparsity under thresholding, conservation of probability under admissible selection operators, invariance of the probability field across temporal updates, and accessibility of latent edges through operator-driven reweighting. A phi-harmonic shell decomposition induces a hierarchical organization of edges and motivates conjectures regarding shell weight decay and finite manifestation bandwidth.The construction is presented as a self-contained contribution to graph theory and probability dynamics, extending classical weighted graph models through infinite edge sets, thresholded subgraphs, and operator-based evolution. Interpretive, cognitive, physical, or phenomenological applications are intentionally excluded and deferred to companion works.