jcost_xiMap_functional_symmetry
plain-language theorem explainer
J-cost remains invariant when the defect map sends sigma to 1 minus sigma. Researchers reformulating the Riemann hypothesis inside Recognition Science cite this to equate the xi functional equation with reciprocal symmetry of the cost functional. The proof is a one-line term rewrite that applies the reflection property of the defect map followed by the inversion symmetry of J-cost on positive reals.
Claim. Let $x = e^{2(σ - 1/2)}$ denote the defect-coordinate map. Then for every real $σ$, $J(x(1-σ)) = J(x(σ))$, where $J$ is the J-cost functional.
background
The module XiJBridge identifies the completed Riemann xi functional equation ξ(s) = ξ(1-s) with the algebraic symmetry J(x) = J(1/x) under the change of variable x = e^{2(Re(s) - 1/2)}. This map sends the critical line Re(s) = 1/2 to the unique minimum x = 1 of J and converts functional reflection into inversion of the positive real coordinate. The upstream lemma Jcost_symm states that Jcost x = Jcost x^{-1} whenever x > 0, which supplies the required algebraic identity once the map is shown to interchange σ with 1-σ by sending x to its reciprocal.
proof idea
The proof is a one-line term-mode wrapper. It first rewrites the left-hand side via the reflection lemma that converts xiMap(1-σ) into the reciprocal of xiMap σ, then applies Jcost_symm to the positive quantity xiMap σ.
why it matters
The result supplies the direct algebraic link between the xi functional equation and J-cost symmetry, thereby supporting the module claim that RH is equivalent to all zeros having zero J-cost. It instantiates the reciprocal symmetry that follows from the Recognition Composition Law and from T5 J-uniqueness in the forcing chain. The theorem closes one step in the bridge that converts the analytic statement ξ(s) = ξ(1-s) into the cost statement J(x) = J(1/x).
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