The Coherence Lagrangian: σ-Minimization as the Variational Principle Underlying Mathematics, Physics, and Consciousness

Proposes that the dynamics of every system maintaining identity over time is governed by a single variational principle: minimize accumulated distortion S = ∫σ dt subject to the coherence constraint K ≥ K_crit. Derives Shannon’s optimal ½ input distribution for the binary symmetric channel directly from σ-minimization, showing that ½ is not a property of the channel but of σ-minimization itself. Predicts the Yang–Mills mass gap as the minimum σ-transition Landauer cost between confinement and asymptotic freedom — a falsifiable prediction. Recovers the least-action principle (Feynman), Shannon channel capacity, the σ-Instability Millennium classification, the Selection Bottleneck theory of consciousness, and Jaynes’s maximum entropy as special cases of one variational structure. Connects to Friston’s free-energy principle as a biological special case.