Prime Numbers as Topological Absences

This work begins with a simple question: how can discontinuity ( N) arise from continuity (R) ? The guiding idea is that a continuous space can undergo a progressive dimensional descent. During this descent, it does not move abruptly from the continuous to the discrete. It passes through intermediate regimes, partly continuous and partly fragmented. At the end of this process, a discrete framework appears. This framework can be interpreted as the support on which the integers become visible. The paper then introduces composition. Once a discrete support exists, multiplication acts as a filling operation: it occupies the positions that can be reached by combining two smaller elements. The positions filled by composition correspond to composite numbers. But some positions remain unfilled. These vacancies are not secondary accidents. They are the central objects of the interpretation. They occur exactly at the addresses traditionally called prime numbers. The main thesis is therefore that primes can be understood not first as ordinary positive numbers, but as absences of number: structured vacancies left open by the compositional covering. A prime is not merely a value in a list. It is the address of a topological absence, defined by the boundaries of what composition succeeds in filling. The work belongs to a geometric and topological reading of arithmetic. It proposes that composite numbers are presences produced by composition, while prime numbers are the residual absences that composition leaves behind.