J
Researchers deriving the recognition wavelength from bit-cost and curvature-cost equality in the Planck-scale matching module cite this definition of the canonical cost functional J(x) = (x + x^{-1})/2 - 1. It supplies the algebraic form used to obtain λ_rec ≈ 0.564 ℓ_P by equating J(φ) to the curvature term 2λ². The definition is introduced directly as the standard J-cost expression from the forcing chain.
claim$J(x) := (x + x^{-1})/2 - 1$ for real $x > 0$.
background
The J-cost functional originates in the unified forcing chain at T5, where J-uniqueness fixes the form J(x) = (x + x^{-1})/2 - 1, equivalently cosh(log x) - 1. This module places the definition inside the conjecture C8 derivation: bit cost J_bit = J(φ), curvature cost J_curv(λ) = 2λ² distributed over the eight faces of the Q3 hypercube, and the extremum condition that determines the recognition wavelength before restoring SI dimensions via the factor 1/π.
proof idea
The definition is a direct algebraic expression with no lemmas or tactics applied. It implements the standard J-cost form exactly as required by the forcing chain and the module's derivation steps.
why it matters in Recognition Science
This definition anchors the bit-cost side of the Planck-scale matching argument, enabling the equality J_bit = J_curv(λ_rec) that yields λ_rec = ℓ_P / √π. It connects directly to the T5 J-uniqueness landmark in the forcing chain and supports the subsequent steps to the eight-tick octave and D = 3. The module references the Discrete Informational Framework Paper, Conjecture C8.
scope and limits
- Does not prove uniqueness of this functional.
- Does not evaluate J at φ or any other point.
- Does not derive the cosh(log x) - 1 form.
- Does not address the curvature cost or the full extremum condition.
formal statement (Lean)
45noncomputable def J (x : ℝ) : ℝ := (x + x⁻¹) / 2 - 1
proof body
Definition body.
46
47/-- J equals the standard Jcost. -/