The Law of Existence: Proven Uniqueness of J‑Cost Minimization Across Mathematics, Physics, Biology, and Consciousness

Abstract

The Coercive Projection Method (CPM) is presented as a universal, parameter‑free architecture by which possibilities collapse to actuality: structured modes minimize a unique convex cost J, exponential weights select, and aggregation commits. A minimal formalization—structured set S (J‑minimizers), a defect functional Def (code surplus = squared distance to S), and a quadratic energy E (total description length)—yields three core theorems with explicit constants: (A) projection–defect via finite nets and a rank‑one/Hermitian bound (C_proj = 2); (B) coercivity factorization E(x) − E* ≥ c · Def(x) with c = (C_eng · C_proj · K_net)^{-1}; and (C) aggregation, upgrading local positivity to global membership x ∈ S. The framework unifies Darwin/MDL (selection as code‑length descent) with mathematical existence, quantum measurement, information flow, and constants—stating falsifiable, cross‑domain predictions.