Scalar Reward Collapse: A General Theory of Eigenstructure Evaporation in Closed-Loop Systems

Version 1.0.1: corrects citation/reference metadata We establish a general dynamical result for systems governed by scalar reward optimization in closed loops. Given a finite state space X, a scalar reward function r: X → ℝ, and a multiplicative reweighting operator T that updates probability distributions according to reward, we prove three fundamental theorems: (1) Eigenstructure Evaporation (Lemma 2.1): all non-reward-maximal modes decay exponentially under repeated application of T; (2) Fixed-Point Convergence (Theorem 2.5): closed-loop dynamics p_{t+1} = T(p_t) converge to a fixed point concentrated on reward-maximal states with reduced entropy; (3) Irreversibility (Corollary 2.7): once collapse occurs, the same operator T cannot restore suppressed modes without exogenous forcing or structural modification. We provide a Δt interpretation showing that scalar optimization forces multi-timescale systems into single-mode temporal collapse. These results supply the mathematical foundation for observed phenomena including platform content homogenization, AI model mode collapse, market herding dynamics, and institutional monoculture formation.