Explanation of Composition_Normalization_implies_symmetry

(1) Plain English

The theorem asserts that any function F : ℝ → ℝ obeying the Composition axiom (F(xy) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y) for x, y > 0) together with the Normalization axiom (F(1) = 0) must satisfy F(x) = F(1/x) for every x > 0. In other words, the cost functional is invariant under reciprocal transformation.

(2) Why it matters in Recognition Science

Symmetry under reciprocals is a structural property of the Recognition Composition Law. It is invoked inside uniqueness_specification to establish that any sufficiently regular F satisfying the three cost axioms equals the canonical J. This symmetry therefore participates in the derivation of the unique cost functional J(x) = (x + x⁻¹)/2 − 1, which in turn forces the golden ratio and the RS-native constants.

(3) How to read the formal statement

theorem Composition_Normalization_implies_symmetry
    (F : ℝ → ℝ) [Composition F] [Normalization F] :
    ∀ {x : ℝ}, 0 < x → F x = F x⁻¹

The parameters are a real-valued function F equipped with type-class instances for the Composition and Normalization classes. The conclusion is a universal quantification over positive reals asserting equality of F at x and at its multiplicative inverse.

(4) Visible dependencies or certificates in the supplied source

The proof applies Composition.dAlembert at (1, x) and rewrites using Normalization.unit_zero. The same module contains the parallel statement J_symmetric for the concrete J and the theorem uniqueness_specification that consumes the symmetry result. No external axioms are required; the proof is a direct algebraic manipulation.

(5) What this declaration does not prove

It does not establish that J is the unique solution (additional hypotheses of continuity, strict convexity, and ODE regularity are needed and supplied only inside uniqueness_specification). It does not invoke the Calibration axiom, does not derive the existence of φ, and does not connect the cost axioms to the forcing chain or physical constants.