Lifted Logarithmic Wasserstein Dynamics I: Exact Counting Mobilities and Microscopic Origins of the FDS Two-Flow Structure

Abstract

We identify conditional microscopic routes to the reduced dissipative sector postulated in Fundamental Dynamic Structures (FDS). The first result is a role-separation theorem. A bilinear complex-field bond fixes the conservative amplitude factor \(\sqrt{\rho_i\rho_j}\), whereas an edge-local gradient flow of a strictly convex local counting functional \(f\) reproduces an exact linear occupation flux only with the divided-difference mobility \[\Theta_f(a,b)=\frac{b-a}{f'(b)-f'(a)},\qquad\Theta_f(a,a)=\frac1{f''(a)}.\] For factorial counting, \(f_\Gamma(\rho)=\ln\Gamma(\rho+1)-\rho\ln G\), this gives the finite-occupation digamma mobility \(\Theta_\Gamma\). It remains finite at the empty-cell boundary and converges, after extensive scaling, to the logarithmic mean of discrete Wasserstein geometry. At the path-large-deviation scale, the same reversible transfer process has an exact cosh dissipation potential with symmetric activity proportional to \(\sqrt{ab}\); the quadratic Onsager law is only its small-affinity limit. Finite-count mobility, path-space activity, and extensive quadratic geometry therefore form a hierarchy rather than competing definitions of one universal mean. We then construct and audit reversible microscopic parents. Independent one-quantum transfers have a factorial stationary law and an exactly closed first-moment equation. Finite phase clocks converge, under the required diffusive rate scaling, to a noisy occupation-weighted Kuramoto flow. A joint occupation-phase jump process has one Gibbs law and one cosh generalized-gradient parent; a globally phase-equivariant labelled cover strongly lumps to the physical relative-phase quotient. At zero occupation, normalized-Haar detailed balance yields a conditional von Mises phase-birth law centred on the local synchronization field. In zero gauge, on an oriented simple cycle with one empty vertex and non-antipodal occupied neighbours, this local kernel induces an explicit two-arc law for the post-birth winding: conditional on the surviving lifted path, the new sector differs from the retained integer by at most one. Closed oscillator and graph baths provide a second route: exact elimination gives memory and fluctuation-dissipation, while short-memory and overdamped limits yield phase mobility \(\gamma=1/\eta\). A uniform-chain bath is a quantitative counterexample to automatic Markovianity. Finally, reversible many-carrier multibaker dynamics produces coarse diffusion, factorial counting, \(\Theta_\Gamma\), and the cosh parent; one smooth Haar-random complex field instead gives Dirichlet statistics. Thus number-resolved composition, local equilibrium, transfer activity, and bath spectral organization remain genuine inputs. The result is a layered, falsifiable microscopic parent of P7, not a unique derivation from the bare FDS state alone.