The Fermi-Pasta-Ulam-Tsingou (FPU) paradox and the subsequent discovery of Quasi-Stationary States (QSS) by S. Ruffo and collaborators demonstrate that non-linear oscillator chains resist rapid thermalization, trapping energy in macroscopic modes. In this paper, we resolve this dynamical anomaly through the lens of pure combinatorial topology. By mapping the energy distribution of normal modes to the integer partition lattice, we apply the Universal Kaleidoscopic Filter Theorem. We prove that the FPU Quasi-Stationary State is not a dynamical transient, but an exact thermodynamically stable volume protected by Weyl group reflections, where all high-frequency microscopic geometries are strictly annihilated. Furthermore, we formalize the eventual QSS decay and the onset of true thermalization as the combinatorial breakdown of modular invariance, governed strictly by the Gaussian Unitary Ensemble (GUE). Extending this ultimate combinatorial framework, we rigorously derive the Many-Body Localization (MBL) phase, the Renormalization Group (RG) fixed points, the spectral zeros of the Kaleidoscopic Zeta Function, the Holographic Page Curve, and the exact saturation of the Maldacena-Shenker-Stanford bound. We formulate a universal classification theorem dividing all dynamical systems into strictly Hermitian and emergent non-Hermitian classes based entirely on the geometric rigidity of the filter. Crucially, we resolve the historical quantization paradox of non-integrable lattices, proving that quantum-like spectral statistics emerge purely from classical combinatorial boundaries rather than canonical quantization. We prove that the macroscopic QSS manifold operates as an exact Topological Quantum Error Correcting (TQEC) code, bounding Krylov complexity. Finally, we map the decay of the Kaleidoscopic Filter to the Trace Anomaly of a 2D Conformal Field Theory (CFT), resolving the thermalization cascade via the central charge flow and Cardy's formula.
Antonio Bonelli (Thu,) studied this question.