jcost_reciprocal_symmetry
plain-language theorem explainer
J-cost satisfies J(x) = J(1/x) for every positive real x. Recognition Science modelers cite the identity when tracing conservation laws back to the single J-functional. The proof is a direct algebraic reduction that unfolds the explicit formula and normalizes the resulting rational expression.
Claim. For every positive real number $x$, the cost function satisfies $J(x) = J(x^{-1})$, where $J(x) = (x-1)^2/(2x)$.
background
The Quantum-Gravity Octave Duality module defines J-cost as the exact arithmetic-geometric mean gap of the pair {x, x^{-1}}. With AM(x, x^{-1}) = (x + x^{-1})/2 and GM = 1, the gap formula simplifies to J(x) = (x-1)^2/(2x) for x > 0. This supplies the algebraic foundation for the central result that Einstein coupling times Planck action equals the octave number 8.
proof idea
The term proof unfolds the definition of Jcost, rewrites the reciprocal using inv_inv, and closes by ring normalization on the resulting rational expression.
why it matters
The identity supplies the reciprocal symmetry required for sigma = 0 conservation and feeds the module's derivation of kappa_einstein * hbar = 8. It sits inside the T5 J-uniqueness step of the forcing chain and is used to establish that the phi-fifth-power duality between kappa and hbar is forced by the single J-functional.
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