Quantization of Anderson Localization Length: A Spectral Chaos Proposal and Its Modeling

We propose a quantization of Anderson localization length in which physically selected integer sectors carry the hierarchy Lloc(K) = L0/ ln K. The proposal is motivated by the exact relative transfer spectral radius K−s of a selected mean-zero angular Sobolev sector in Spectral Chaos. Identifying its logarithmic attenuation scale with spatial wave attenuation per fixed effective block predicts the hierarchy with L0 = a/s. This spatial identification and the physical selection of sectors are hypotheses. A comparison with nine benchmark point estimates from the localization measurements of Billy et al. [2], using assigned prime labels and one-point calibration, gives a mean absolute relative deviation of 2.26%, or 2.55% excluding the calibration point. We then model the proposed hierarchy explicitly. For a cotangent- map Schrödinger family, we derive the exact spatial transfer exponent at arbitrary real energy and positive coupling. At fixed coupling it is independent of the integer map degree K. The prescription Γ(K) = 2J sinh[(a/L0) ln K] gives the proposed length exactly as a band-center Lyapunov scale. A separate Aubry–André realization has established exponentially localized eigenfunctions under standard arithmetic and phase conditions. The models establish realizability; independent identification of K, effective coupling, and sector selection is needed to test localization-length quantization in a physical medium.