Explanation of Jcost_reciprocal_symmetric

(1) What the declaration says in plain English

The theorem states that the Recognition Science cost function Jcost is reciprocally symmetric: for every positive real x, Jcost(x) = Jcost(1/x). This is the key algebraic property of the cost function J.

(2) Why it matters in Recognition Science

This symmetry is the starting point for showing that the Mellin transform of a reciprocally symmetric function inherits a reflection symmetry s ↔ 1-s. In RS it underpins the derivation of functional equations for partition functions Z_RS and provides the structural bridge to L-functions and the Langlands program without introducing fitted parameters.

(3) How to read the formal statement

theorem Jcost_reciprocal_symmetric : ReciprocalSymmetric Jcost := by
  intro x hx
  exact Jcost_symm hx

• ReciprocalSymmetric f is the predicate ∀ x : ℝ, 0 < x → f x = f x⁻¹.
• The proof is a one-line application of the imported lemma Jcost_symm under the positivity hypothesis hx.

(4) Visible dependencies or certificates in the supplied source

The theorem is defined directly after the predicate ReciprocalSymmetric. It is one of the four conjuncts in the master certificate mellin_pullback_certificate. Related results in the same module include Jcost_log_even (log-coordinate evenness) and mellin_pullback_pointwise (pointwise integrand symmetry). The module imports Jcost_symm from IndisputableMonolith.Cost and has zero sorry.

(5) What this declaration does not prove

It does not define or prove convergence of any Mellin integral, does not construct the completed zeta function ξ(s), and does not establish the full functional equation ξ(s) = ξ(1-s). Those steps require complex-analytic machinery (theta-function identities, Poisson summation) absent from this module.