Fundamental Dynamic Structures: Quantum Hydrodynamics, Matter Selection, and Emergent Relativity from a Synchronizing Network of Probability Oscillators

We develop a framework — Fundamental Dynamic Structures (FDS) — in which physicalphenomena emerge from a network of abstract nodes, each carrying two variables: anoccupation ρ (probability) and a cyclic phase θ. Requiring (i) conservation of totaloccupation, (ii) pairwise coupling along network links, and (iii) reversibility, we show thatthe minimal node dynamics is a discrete Schrödinger equation on the graph, and that theclassical Kuramoto synchronization model and quantum dynamics are, respectively, thegradient and symplectic flows of one and the same network energy functional (verifiednumerically to machine precision, 10⁻¹⁵). The continuum limit yields Madelung quantumhydrodynamics with an effective mass m = ħ²/ (2Ka²). A logarithmic nonlinearity emergeswith coefficient b = kB·T from Boltzmann counting of quanta over nodes (Monte Carloverified) ; the exact per-cell potential is T·ψ₀ (n+1): the logarithmic coupling switches off atlow mode occupancy, leaving only a weak contact-like residual (apparent amplitude ~kB·T·n̄), reconciling the framework with single-neutron interferometry bounds onlogarithmic extensions of quantum mechanics. The logarithmic vacuum phase has anequation of state P = bρ, giving a density-independent sound speed: we verify numericallythat a region of doubled density is acoustically invisible (speed change < 2%), whereas acubic medium refracts as √ρ. Lorentz-violating corrections are quadratic only; existinggamma-ray bounds translate into an upper bound on the network spacing a < 1. 5×10⁻²⁷ m. Topological excitations (quantized vortices, Γ = 2πħm verified to 0. 4%) model matter: Bogoliubov–de Gennes analysis shows that vortices cannot survive in isolation (“no medium— no soliton”) and that within the medium unit charge is stable at all tested modes whilehigher charges are dynamically (m = 2) or energetically (m = 3) disfavored: the numericssuggest a selection pressure toward unit topological charge — a candidate account ofcharge quantization; full proof and 3D stability remain open. Phase-mediated vortexinteractions reproduce the two-dimensional Coulomb law (dipole speed v·d = 1. 010; pairenergy slope → 2π), yielding a phase-interaction prototype with Coulomb-like behavior, while the dissipative (synchronization) flow renders soliton attraction universal and massindependent (equivalence-principle test: accelerations equal to 0. 4% for a 4: 1 mass ratio), yielding a synchronization-attraction prototype with equivalence-principle-like behavior, suggestively of Painlevé–Gullstrand (inflow) form. We state falsifiable predictions, mostimmediately an effective logarithmic nonlinearity b ≈ kB·T in Bose–Einstein condensatesat temperature T (≈ 4. 3×10⁻¹² eV at 50 nK — of the order of typical BEC chemicalpotentials), whose distinguishing signature is its logarithmic density dependence.