Topological Genesis of Quantum Chaos

Abstract

Topological Genesis of Quantum Chaos: Multifractality and the Poisson–GOE–Poisson Crossover in Anderson Lattices We present numerical evidence that macroscopic quantum chaos emerges as a continuous phase transition driven solely by the topological degradation of the underlying spatial substrate, dictating the thermodynamic limits of fault-tolerant AI hardware. ◆ Key Features Lattice topology: 2D toroidal mesh representing highly coupled interconnects (N = L \times L sites, L = 20–50). Anderson Disorder: Tight-binding Hamiltonian evaluating the localization crossover (W as the primary control parameter). Edge Percolation Channel: Independent topological degradation (p=0.10) ensemble rigorously confirming the GOE transition (\langle r \rangle = 0.5309 \pm 0.0117). Hardware-in-the-loop validation: Forward-pass simulation of a 256-neuron Quaternary Neural Network (W_{ij} \in \{-2, -1, 1, 2\}) mapped directly onto the frustrated spatial mesh. On-device execution: Exact diagonalization on mobile ARM NEON hardware (Snapdragon) without spectral unfolding, achieving a ~25x speedup over conventional HPC workflows. ◆ Main Results Poisson–GOE–Poisson Crossover: The pristine toroidal mesh exhibits symmetry-induced degeneracy (Poisson-like). Moderate disorder (W \approx 1–4) collapses the system into strict Wigner–Dyson GOE statistics (\langle r \rangle \approx 0.531). Strong disorder (W \gtrsim 8) restores Poisson statistics via Anderson localization. Multifractal Criticality: The critical regime (W \approx 4–6) reveals strongly multifractal eigenstates, monotonic decay of generalized dimensions D_q, and an f(\alpha) spectrum peaking near \alpha \approx 2.2–2.3. Algorithmic Resilience: At the exact GOE transition threshold (10% physical link loss), the Quaternary Neural Network maintains a 93.20\% inference accuracy. The structural damage translates to a mean Hamming distance of only 17.40 \pm 1.65 bits (out of 256), proving that the network anchors information in stable energy minima despite physical chaos. ◆ Implications These results demonstrate that Wigner–Dyson level repulsion is a thermodynamic consequence of spatial frustration. The critical multifractal threshold (W \approx 4–6) defines a quantitative failure boundary for distributed tensor-processing architectures. Furthermore, empirical inference tests prove that ultra-low precision quantization (quaternary states) acts as a structural buffer, absorbing chaotic fluctuations in degraded routing topologies for next-generation edge AI accelerators.