A Semi-Analytical Framework for Multiscale Phase Transitions in Media: End-to-End Synthesis of Non-Local KP-II Solitons and Crypto-Topological Macro-Attractors of Jamming

This paper introduces a novel semi-analytical framework for modeling multiscale phase transitions in continuous media, completely eliminating empirical "clutches" and tune-up parameters of turbulent viscosity. Based on the concept of information-metric dualism, spatial coordinates and macroscopic stresses are derived as an emergent property of the information density of the vacuum/vortex network within Alain Connes' spectral triads. A closed meta-system of 30 partial differential equations has been developed and numerically implemented on a 3D staggered Marchuk–Harlow grid, describing the "flow-to-jam" transition under extreme packing conditions (\phi \rightarrow 0.74). The scientific novelty of the study is determined by four discovered critical points of the phase trajectory:1. Dynamic Kepler Blocking Threshold: the random close packing limit is redefined as a continuous function of the second invariant of the strain rate tensor, modeling shear dilatant slip of layers.2. Differential Integration of the KP-II Residual: the asymptotic infrared Milgrom profile a_0/R, extracted from the Special Theory of Dark Matter (STDM), is integrated into the momentum balance via the exact non-linear residual of the Kadomtsev–Petviashvili hierarchy acting as the network's elastic pressure.3. Self-Oscillatory Bifurcation Lumley Turn: a tensor switch of equation types is formalized when the anisotropy invariant F \rightarrow 0, transferring the longitudinal flow into the mode of transverse vortex auto-resonance near the Prandtl walls.4. Spectral Quantum Wigner Sieve: the discrete framework of force chains is closed via the Gaussian Unitary Ensemble (GUE) of random matrices and the Sato-Grassmannian, linking the Shannon entropy generation rate with the porous geometry of Mandelbrot micro-voids and an autonomous Entropy Brake. Validation of the computational core against the experimental Jülich Data (Germany) dataset confirmed the asymptotic convergence of auto-blocking profiles with a precision accuracy of 98.4% in extreme regimes where standard commercial computational fluid dynamics (CFD) packages (ANSYS, OpenFOAM) demonstrate gradient catastrophe and a total computational deadlock.