Fundamental Dynamic Structures (FDS) is a program in which the primitive substrate is a finite weighted graph decorated by occupation ρᵢ ≥ 0 and cyclic phase θᵢ ∈ S¹, with no background metric or manifold. The complex field ψᵢ = √ρᵢ·e^iθᵢ is a bookkeeping representation. The closed symplectic flow of a graph energy E yields a network Schrödinger equation; at fixed occupations, the phase-gradient flow of the same bond sector yields an occupation-weighted Kuramoto sine coupling. The full reduced metric flow −M∇F, F = E − TS, is Postulate P7, and its microscopic derivation remains open. Under lattice-like coarse-graining the closed sector approaches Madelung hydrodynamics with effective kinetic mass mₑff = ħ²/ (2Ka²). A companion thermal module derives, under Maxwell–Boltzmann/Gibbs counting, an occupancy-resolved repulsive logarithmic term at the equilibrium free-energy level: its high-occupancy coefficient is kB·T, while the dilute limit is contact-like; Bose counting instead predicts high-occupancy saturation. Promotion of this term to conservative dynamics and selection of the focusing branch remain open. In two-dimensional logarithmic media, finite-volume Bogoliubov–de Gennes calculations detect no negative-energy core structure for unit winding, a core–phonon Krein-resonance structure for double winding, and four localized negative-Krein modes for triple winding; the infinite-domain and three-dimensional status remain open. The phase channel approaches point-vortex/Coulomb-gas asymptotics. A separate focusing-gausson prototype exhibits similar conservative accelerations for three initially in-phase probes spanning an 8: 1 inertial-mass range, while mass independence after synchronization remains conjectural. The dissipative companion paper studies projected-energy descent and wrapped-winding relaxation in a phenomenological model that is not yet the P7 Onsager flow. The repulsive LogSE closure has P = bρ and hence density-independent homogeneous bulk sound speed; its regular-lattice dispersion has a leading quadratic correction. Identification of this acoustic sector with universal relativistic propagation, derivation of Lorentzian geometry, dynamic graph gravity, gauge fields, spin/statistics, and stable three-dimensional matter are open program goals.
Evgeny Sametskiy (Sun,) studied this question.