Memory Erasure in Non-Invertible Dynamics: A Diagnostic Framework with Benchmark Validation

This work develops a diagnostic framework for distinguishing hysteresis accumulation from finite-memory mixing in non-invertible dynamical systems. Using a strongly non-invertible arithmetic map as a benchmark system, we systematically test whether asymmetry, bias, or path dependence can accumulate over time. A comprehensive battery of kill tests—including shuffle invariance, windowed distribution drift, spectral gap analysis, autocorrelation decay, mutual information decay, and history-conditioned statistics—shows that the benchmark system exhibits deterministic structure with rapid information loss, not hysteresis. While short-range history dependence is detectable, all memory effects decay exponentially and do not accumulate with iteration time. Asymmetries remain stable rather than growing, demonstrating that asymmetry and hysteresis are fundamentally distinct phenomena. These results clarify that non-invertibility alone does not produce hysteresis. Instead, strong projection typically destroys correlations and promotes mixing, yielding a finite-memory regime. Hysteresis requires additional mechanisms—such as coherent oscillations, slow relaxation, or external driving—that are absent in the benchmark but common in physical, geological, and engineered systems. The benchmark system is therefore positioned not as a model of physical reality, but as a reference extreme on a broader projection–memory spectrum. This framing provides practical tools for analyzing memory, asymmetry, and stability in physical systems, complex dynamics, and AI architectures, while avoiding inappropriate transfer of arithmetic-specific results.