Compact Topology and the Past Hypothesis: Configurational Capacity in the Early Universe

We present a topological perspective on the so-called Past Hypothesis, understood as a proposed explanation for the low-entropy character of the early universe, by analyzing the accessible phase space of quantum fields in a closed Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) universe. On spatial slices with \(S^3\) topology, field configurations must satisfy global boundary conditions, leading to a discrete mode spectrum and an infrared (IR) cutoff set by the curvature radius, \(R(t)\). By comparing the resulting compact spectrum with the volume-matched flat-space continuum at fixed physical ultraviolet (UV) cutoff, we show that compact spatial topology possesses a lower configurational entropy capacity for accessible momentum configurations than its non-compact counterpart. This restriction is strongest at early times, when \(R(t)\) is small, and is progressively relaxed as cosmic expansion densifies the allowed spectrum. Our result identifies compact spatial topology as a geometric regulator of IR phase space and suggests a kinematical mechanism by which closed topology may contribute to the low entropy assumed in the Past Hypothesis.