When Order Requires Measure: The Logical Necessity of Quantitative Structure

If irreversibility differs by degree, quantitative measure becomes unavoidable. This paper identifies the precise structural pressure under which magnitude is forced: if two admissible irreversible transitions produce non-equivalent restriction effects on future reachability, then an ordinal partial order is insufficient to represent the difference. To make “degree of restriction” explicit without physical assumptions, the paper defines the future cone of a state as the set of reachable descendants under reflexive transitive closure and introduces a minimal degree functional in the finite case via future-cone cardinality. A canonical monotone valuation is then constructed that increases with constraint…