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This book reinterprets the Fermi–Pasta–Ulam–Tsingou (FPUT) paradox through the Trojan framework of protected nonlinear coherence. The classic puzzle—that a weakly nonlinear oscillator chain, started in a low-frequency mode, fails to thermalize and instead returns near its initial state—is reframed as a problem of transport rather than mere nonlinearity. Coupling exists; what is surprising is that it does not immediately become global modal transport.

The central claim is local, structural, and finite-time: near low-mode recurrent states and q-breather periodic orbits, FPUT dynamics exhibit a coherent modal backbone, near-integrable normal-form organization, suppressed action drift, and exit through resonance. Nonlinear coupling is present but geometrically contained. Recurrence is therefore protected nonlinear coherence; eventual thermalization is resonant thickening of that coherence.

Two pillars of evidence support the argument. Optical experiments by Pierangeli et al. demonstrate more than three FPUT-type recurrences in nonlinear spatial waves, controlled by amplitude and phase and governed by exact nonlinear Schrödinger recurrence laws; recurrence vanishes as integrability is lost. In the discrete lattice, q-breathers—localized periodic orbits continued from linear normal modes—supply the internal Trojan skeleton. Floquet stability, exponential modal localization, and resonance-driven bifurcations into composite periodic orbits show how protection persists and how it fails. Resonances of the form ω1=kωm \omega_1 = k\omega_m ω1=kωm collapse normal-form divisors, open transport channels, and enlarge the coherent packet without instant equipartition.

The book develops precise diagnostics: coherent modal blocks and failure regions, the FPUT thinness ratio measuring leakage against a failure margin, finite-time confinement estimates, spectral entropy as a transport observable, recurrence fidelity, and a transport-graph formulation of connectivity. A conditional admission criterion states when a recurrent or metastable regime satisfies Trojan architecture. Falsification criteria are given so the interpretation remains testable.

A final chapter extends the same transport logic to intramolecular vibrational energy redistribution. The thinness ratio distinguishes formal anharmonic coupling from chemically effective IVR, identifies a selectivity window, and shows why Fermi resonances may remain thin or become thick depending on network connectivity and chemical timescale.

The contribution is synthetic rather than a new global solution of FPUT. Existing results on solitons, Toda dynamics, q-breathers, NLSE recurrence, and metastability are organized under one architecture: stability is the containment of transport; thermalization begins when coupling acquires reach. FPUT thereby becomes a model case of protected nonlinear coherence rather than an isolated historical anomaly.

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