Odd Distinct Covering Systems and Discrete Novelty A Conditional Impossibility via Filtration Semantics

This paper presents a conditional impossibility result for odd distinct covering systems (ODCS), isolating the problem into explicit arithmetic and semantic components. Unconditionally, we show that finite prime support forces finite harmonic mass. Conditionally, we introduce two independent hypotheses: Prime–Incidence Hypothesis (PIH) — a quantitative constraint on weighted prime incidence controlling arithmetic supply. Memory–Time Constraint (MTC) — a medium law for refinement under a filtration, excluding completion by asymptotically vanishing informational updates. A finite-alphabet quantization argument converts non-measurability into positive conditional entropy on survivor mass. Under a standard entropy-budget (dependency-control) regime, this yields a contradiction to completion. The argument cleanly separates: arithmetic content (PIH), medium semantics (MTC), and dependency control (entropy interface), reducing the Erdős ODCS problem to a single, falsifiable boundary condition. All assumptions are explicit and isolated; no probabilistic or heuristic arguments are used.