What question did this study set out to answer?

The aim is to resolve the Fermi-Pasta-Ulam-Tsingou paradox and provide insights into how non-linear lattices resist thermalization.

June 13, 2026Open Access

The Universal Kaleidoscopic Sieve: Integer Partitions, Hermitian Integrability, and the Quantum Geometry of Emergent Non-Hermitian Chaos in Non-Linear Lattices

Key Points

The aim is to resolve the Fermi-Pasta-Ulam-Tsingou paradox and provide insights into how non-linear lattices resist thermalization.
Applied combinatorial topology to map energy distribution in oscillator chains to integer partition lattice
Utilized the Universal Kaleidoscopic Filter Theorem to establish dynamical stability of quasi-stationary states
Extended framework to derive many-body localization and spectral characteristics related to quantum systems
Proven that the quasi-stationary state is thermodynamically stable, with dynamics governed by Weyl group reflections
Established that quantum-like spectral statistics arise from classical combinatorial boundaries
Mapped the decay of the kaleidoscopic filter to the trace anomaly in a 2D conformal field theory, confirming central charge flow implications

Abstract

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.

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