Explanation of Jcost_mellin_reflection

(1) In plain English: The declaration asserts that whenever a JCostMellinBridge is supplied (a record packaging an admissible Mellin kernel for the J-cost function), the associated transform M is invariant under the reflection map s ↦ 1-s. In other words, the Mellin transform inherits the reciprocal symmetry of J-cost at the level of the transform parameter.

(2) Why it matters in Recognition Science: Recognition Science starts from the unique reciprocal-symmetric cost J(x) = (x + x⁻¹)/2 - 1. This theorem lifts that symmetry into the Mellin-transform domain, supplying the transform-level reflection law that later phases use to reach functional equations for partition functions and L-functions. It is the explicit bridge between the cost foundation and the analytic side of the RS zeta program.

(3) How to read the formal statement:

 theorem Jcost_mellin_reflection (bridge : JCostMellinBridge) :
     ∀ s : ℝ, bridge.pkg.M s = bridge.pkg.M (mellinReflect s) :=
   mellin_reciprocal_reflection bridge.pkg

• JCostMellinBridge is a structure containing a MellinAdmissibleKernel Cost.Jcost.
• mellinReflect s is defined as 1 - s.
• The proof is a direct application of the general reflection theorem mellin_reciprocal_reflection, which itself extracts the substitution field of the admissible-kernel structure.

(4) Visible dependencies or certificates in the supplied source: The theorem is proved by mellin_reciprocal_reflection, which relies on the substitution axiom of MellinAdmissibleKernel. The bridge is populated via Jcost_mellin_reciprocal (which asserts ReciprocalSymmetric Cost.Jcost) together with the kernel-inversion lemmas mellinKernel_inversion and mellinIntegrand_after_inversion. The whole package is summarized in the certificate structure MellinPhase3Cert.

(5) What this declaration does not prove: It does not construct or verify an explicit integral definition of M; the transform is treated as an abstract field inside MellinAdmissibleKernel. It does not instantiate the kernel with a theta function, establish convergence, or derive the completed zeta functional equation. Analytic continuation, complex parameters, and the full Langlands correspondence remain outside its scope.