The Recognition Forcing Theorem: From Distinguishability to Physical Constants

We present a self-contained derivation of the core Recognition Science forcing chain in ordinary mathematical prose. The starting point is not a physical postulate, a smooth manifold, a Hilbert space, or a list of constants, but the existence of a non-trivial distinction. From this floor we state the minimal conditions required for consistent comparison, introduce recognition work as a constrained cost of distinction, derive the reciprocal composition law, and prove that the unique calibrated continuous cost on positive ratios is J(x) = ½(x + x⁻¹) − 1 = cosh(log x) − 1. The resulting cost geometry forces the golden ratio as the unique self-similar scale, an eight-tick recognition cycle, and spatial dimension three through the compatibility of linking, tick closure, and the gap structure. We then give the narrow physical interpretation used in this paper: time is the recognition iteration, the causal speed is one adjacency per tick, the action quantum is the coherence quantum times a tick, and the gravity scale is the curvature response of the same cost geometry. Finally we present a small set of zero-parameter stress tests: the fine-structure constant, the native-unit forms of ℏ and G, the mass ladder, the baryon-to-photon ratio rung and band, the high-temperature relativistic degree count g⋆ = 427/4, and the dark-energy fraction Ω_Λ = 11/16 − α/π. The paper separates theorem, definition, convention, and empirical test at each step. It does not claim to derive every later application of Recognition Science; it claims that the core spine and its first constant checks are forced once the stated comparison conditions are accepted. Previously published mathematical-foundation papers are used as support for the ratio geometry and cost-rigidity background, but the logical chain needed for this paper is reproduced here.