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 ρᵢ ≥ 0 and a cyclic phase θᵢ ∈ S¹; links carry couplings Aᵢj; 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 Eρ, θ, A and prove/organize the two-flow structure: the symplectic flow of E, in canonically conjugate variables (ρᵢ, θᵢ), 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). Both flows are generated by the same functional; their identity in (ρ, θ) variables has been verified numerically to machine precision in the umbrella work. 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 EAA, dimension selection, spin/statistics, and gauge fields explicitly open.
Evgeny Sametskiy (Wed,) studied this question.