TOPOLOGICAL ALIASING AND HOLOGRAPHIC READOUT: A NON-ARCHIMEDEAN FRAMEWORK FOR EMERGENT QUANTUM STOCHASTICITY

The foundational assumption of modern physics—that spacetime is a continuous manifold modeled by the real number line—remains an unverified extrapolation from the macroscopic scale to the Planckian regime. This paper proposes the "Topological Aliasing" hypothesis, which posits that quantum randomness is not an inherent property of nature but a statistical artifact arising from the projection of deterministic, non-Archimedean ($p$-adic) dynamics onto an Archimedean (real-valued) measurement metric. By reframing the measurement problem as a topological mismatch between the latent state space and the observer's readout method, we provide a number-theoretic origin for wave-function collapse and stochasticity. We further integrate this framework with modern AdS/CFT holography, identifying the canonical Monna map as a boundary projection of a bulk $p$-adic tensor network. Our methodology utilizes a computational simulation of $p$-adic isometries acting on the ring of $p$-adic integers $\mathbb{Z}_p$. These deterministic sequences are projected onto the real interval $[0,1]$ via the Monna map, serving as a proxy for the Archimedean readout of a non-Archimedean reality. High-resolution analysis ($N=10,000$) of the Adelic summation of multiple prime-scale structures reveals that the emergent noise converges to a quasi-Gaussian stable distribution with platykurtic (sub-Gaussian) residuals ($p = 0.0002$), suggesting that quantum noise retains a structural "memory" of its discrete origins in the form of flatness relative to the normal distribution. We identify a robust "Correlation Flip" signature in linear systems, while non-linear $p$-adic dynamics exhibit a universal anti-correlated "repulsive" stochasticity. These findings address critical gaps in $p$-adic mathematical physics and offer a falsifiable framework for an Adelic physics where the "unreasonable effectiveness" of mathematics is a consequence of metric alignment.