Hardware-Induced Irreversibility and Entropy Generation in the $\alpha$-Gauss-Logistic Map: A Physical Refutation of Algorithmic Determinism

We demonstrate that digitally implemented chaotic maps are not physically deterministic. Al-though the α–Gauss–Logistic map exhibits mathematically chaotic behavior with a positive Lya-punov exponent, finite-precision digital hardware introduces irreversible quantization that fun-damentally alters the dynamics. We show that IEEE-754 rounding errors, when reinjected intothe map, act as a physical noise source that is exponentially amplified by chaos, converting thesystem into a genuine entropy generator. Fourier spectral diagnostics and NIST Statistical TestSuite results confirm that purely algorithmic chaos retains hidden correlations (p-values ≈ 0.003),while the proposed AGLE (Adaptive Gaussian–Logistic Entropy) architecture achieves statisticalindistinguishability from physical randomness (p-values ≈ 0.83). This establishes a hardware-levelrefutation of algorithmic determinism for chaotic systems.