The Recognition Validity Theorem A Machine-Verified Forcing Chain for the Structural Foundations of Physics

We present the RS Validity Theorem, a machine-verified mathematical chain deriving the structural foundations of physics from a single functional equation. Starting from the requirement that any cost of deviation from unity be symmetric and multiplicatively consistent, we prove that the functional equation (the Recognition Composition Law) is the unique polynomial form, and that its solution J(x) = 1/2 (x + x⁻¹) − 1 is the unique cost function. A formal forcing chain (T0–T8) then derives existence, discreteness, the double-entry ledger, the golden ratio φ = (1 + √5)/2, the 8-tick cycle, and D = 3 spatial dimensions strictly from this cost function. This framework yields zero adjustable parameters at the foundational level. We present the derivation of the inverse fine-structure constant α⁻¹ and demonstrate its containment of the CODATA 2022 value. We clearly distinguish between mathematical calibrations (internal consistency checks) and genuine physical predictions, and document the open tensions in lepton mass ratios requiring sub-leading corrections. The entire structural argument—7,895 compilation units—compiles in the Lean 4 proof assistant with zero sorry and zero axioms beyond those of Lean's type theory and Mathlib.