Universal Logarithmic Quantization Law of Localization Length in Anderson Systems via Spectral Chaos Theory

Understanding the spatial confinement of quantum waves in disordered media—the cornerstone of Anderson localization—remains a fundamental challenge at the intersection of condensed matter physics and quantum chaos. Here, we present the first exact, analytically solvable chaotic framework directly formulated via Takenaka-Malmquist bases, establishing that spatial localization length in disordered potential landscapes undergoes discrete topological quantization. By mapping phase-locking resonances within chaotic potentials onto a prime hierarchy K, we derive a closed-form, parameter-free universal localization law, L(K) = L0 for K = 2,3,5,7,...,251, lnK wherein the macroscopic spatial extent of localized quantum states scales inversely with the chaotic ca- pacity ln K, anchored solely by the fundamental reference scale L0 at K = 2. Rigorously benchmarking this formula against observational data across nine distinct disorder regimes from the landmark one- dimensional Bose–Einstein condensate (BEC) experiment by Billy et al. [Nature 453, 891 (2008)] [1], the theory quantitatively reproduces the experimental localization lengths without any free fitting param- eters, achieving an average relative error of just 2.3%. Crucially, this scaling relation holds seamlessly across two orders of magnitude in K up to K = 251, accurately predicting the minimal localization length (0.236 mm) with sub-percent precision (<0.1%). Remarkably, this logarithmic hierarchy mir- rors the universal algebraic basis governing black hole horizon ringdown dissipation [2], completing an axiomatic triad connecting gauge-theoretic mass gaps (∆m = m0 ln K ) [3], gravitational dissipation (νth ∝ 1/ ln K), and quantum transport (L(K) ∝ 1/ ln K). By providing parameter-free empirical validation of discrete spectral chaos [4], our results establish a conceptual framework for topologically digitalized wave engineering, overcome the critical power and miniaturization bottlenecks of optical lat- tice clocks to enable satellite-borne “Ultra-GPS”, and achieve the cross-domain unification of quantum transport and spacetime geometry.