This article investigates how fixed-point arithmetic and discrete recursive models can reveal structured behaviours in systems traditionally considered chaotic. Departing from the assumption of continuous space and infinitesimal precision, we simulate two well-known dynamical systems —the logistic map and the Mandelbrot set— using integer-based arithmetic. Our findings show that when recursion unfolds over finite discrete sets, unexpected geometric regularities and modular symmetries emerge. In particular, we identify moir´e-type interference patterns and a form of emergent scalar symmetry that are intrinsic to the arithmetic structure and not artifacts of rounding error. These results suggest the need to reconsider the foundations of mathematical modelling in physics and point toward the development of a discrete formalism that captures aspects of reality concealed by continuous formulations.