Discrete Novelty Under Arithmetic Refinement: A Critical Examination of the Memory–Time Constraint

This paper examines the Memory–Time Constraint (MTC), an axiomatic principle asserting that completion under a filtration-governed refinement process requires infinitely many stages of genuinely new, finite-alphabet information on positive survivor mass. The work is a companion and stress-test to a conditional impossibility result for odd distinct covering systems. It does not assume MTC to be true. Instead, it analyzes whether MTC is an unavoidable feature of arithmetic refinement or an artificial obstruction. We test MTC against concrete arithmetic mechanisms, including: prime sieve processes, highly dependent congruence refinements, prime-power and composite constructions, and explicitly constructed non-arithmetic “phantom protocols.” We show that: MTC holds automatically for all known arithmetic covering mechanisms, MTC fails only for explicitly non-arithmetic refinement protocols, any genuine arithmetic violation of MTC would require fundamentally new structure. The paper cleanly separates: medium laws (filtration semantics), arithmetic supply conditions (e.g. prime-incidence budgets), and dependency control (Shearer/LLL-type regimes), thereby localizing the remaining obstruction in the theory of odd distinct covering systems to a single, falsifiable frontier. This work does not claim to resolve Erdős’ covering system problem. It precisely identifies what would be required to do so.