Jcost_one
plain-language theorem explainer
J(1) = 0 follows by direct substitution into the recognition cost J(x) = (x + x^{-1})/2 - 1. Workers on the φ-ladder cutoff for Navier-Stokes regularity cite this normalization as axiom A1 to anchor equilibrium at unit scale. The proof reduces immediately via simplification on the explicit formula.
Claim. $J(1) = 0$, where $J(x) = (x + x^{-1})/2 - 1$ for $x > 0$.
background
The recognition cost is defined by Jcost x := (x + x^{-1})/2 - 1. This module develops the algebraic and combinatorial tools showing that the φ-ladder supplies an ultraviolet cutoff that terminates the Navier-Stokes energy cascade on the discrete lattice of Recognition Science. Jcost satisfies J(1) = 0 by normalization axiom A1. Upstream results include the reciprocal automorphism in CostAlgebra and the reciprocal event in LedgerForcing, which encode the symmetry J(x) = J(x^{-1}). The local setting is the formalization of Navier-Stokes regularity via φ-ladder cutoff, with main results including nonnegativity of Jcost and monotonicity of cascade depth.
proof idea
This is a one-line wrapper that applies simp to the definition of Jcost, which directly yields zero at argument one.
why it matters
This normalization anchors the equilibrium characterization in Jcost_eq_zero_iff and enables the zero-cost result for the self-similar fixed point in simplifiedCascadeCost_phi. It fills the base case for the φ-ladder cutoff argument in the Navier-Stokes regularity paper. Within the Recognition Science framework it corresponds to the T5 J-uniqueness step where J(x) = cosh(log x) - 1 evaluates to zero at x = 1.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.