Fundamental Dynamic Structures I: Pre-Geometric Probability Oscillators and the Two- Flow Network Dynamics

We give a mathematical formalization of the Fundamental Dynamic Structures (FDS) framework, in which physical structure emerges from a pre-geometric network of probability oscillators. The primitive object is a decorated graph X = (G, ρ, θ): nodes carry an occupation ρ_i ≥ 0 and a cyclic phase θ_i ∈ S¹; links carry couplings A_ij; no space, time, or metric is presupposed. Physical states are defined as equivalence classes of decorated graphs under decorated-graph isomorphism and global phase shift, [X] = X/(Iso(G) × U(1)) — eliminating hidden coordinates and hidden absolute phase by construction. On this state space we define a single energy functional E[ρ, θ, A] and prove/organize the **two-flow structure**: the symplectic flow of E, in canonically conjugate variables (ρ_i, θ_i), is a network Schrödinger equation (the closed-system, reversible, λ-ordered dynamics); the metric (gradient) flow of the associated free energy, with an Onsager mobility respecting occupation conservation, is a Kuramoto-type synchronization dynamics (the reduced dynamics of open subsystems). The symplectic flow is generated by E and the dissipative flow by the free energy F = E − T·S; at fixed occupations the phase sectors of the two flows are generated by the same phase-dependent part of E, whose bond current is exactly the Kuramoto sine coupling (an exact algebraic identity). We state the coarse-graining assumptions under which the continuum limit is Madelung quantum hydrodynamics with effective mass m = ħ²/(2Ka²), and define **candidate emergence maps** for time (ordering parameter λ; physical time as the phase of a reference-clock subsystem, Page–Wootters) and for geometry (stable relational distinguishability; premetric scaffolds calibrated by a structured reference sector) — emergence routes, not completed derivations. The paper's claims are tabulated by status — definition, theorem, numerically verified, assumption, open — with the dynamic-graph sector E_A[A], dimension selection, spin/statistics, and gauge fields explicitly open.