j_functional_equation
plain-language theorem explainer
The Recognition Science J-cost function satisfies J(x) = J(x^{-1}) for every positive real x. Number theorists modeling the zeta functional equation via multiplicative ledger transactions would cite this symmetry when establishing the structural parallel to d'Alembert solutions. The proof is a one-line term application of the core Jcost symmetry lemma.
Claim. For every positive real number $x$, the Recognition Science cost function satisfies $J(x) = J(x^{-1})$.
background
The PrimeLedgerStructure module treats natural numbers as transactions on a discrete multiplicative ledger, with primes as irreducible entries. J-cost is the function whose symmetry under inversion supplies the model for the zeta functional equation. The upstream Jcost_symm lemma establishes the same equality by unfolding the squared definition and applying field simplification and ring arithmetic.
proof idea
The proof is a one-line term wrapper that applies the Jcost_symm lemma from the Cost module at the given positive real x.
why it matters
This theorem supplies the inversion symmetry that realizes the structural correspondence between J-cost and the zeta functional equation in the Recognition Science framework. It supports the module's model of d'Alembert zero structure and the hypothesis that ledger conservation predicts the Riemann Hypothesis. The open gap remains whether the Euler product imposes d'Alembert-type constraints on the completed zeta function.
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