J_bit_explicit
The bit cost at the self-similar scale equals (φ + φ^{-1})/2 - 1. Researchers deriving the recognition wavelength from ledger curvature in the Planck-scale matching chain cite this identity to fix the left-hand side of the extremum condition J_bit = J_curv(λ). The proof is a one-line reflexivity that holds by the definition of J_bit_val together with the closed form of the J functional.
claimJ(φ) = ½(φ + φ^{-1}) - 1
background
In the Planck-Scale Matching module the bit cost is introduced as J_bit_val := J(φ), where J(x) = ½(x + x^{-1}) - 1 is the unique cost functional. The module derives λ_rec ≈ 0.564 ℓ_P by equating this bit cost to the curvature cost 2λ² over the eight faces of the Q3 hypercube and restoring SI dimensions via the factor 1/π. The upstream definition states that J_bit_val supplies the fundamental cost of a single ledger bit transition evaluated at the golden ratio.
proof idea
The proof is a one-line reflexivity. It equates J_bit_val directly to the algebraic expression (phi + phi^{-1})/2 - 1 by unfolding the definition J_bit_val = J phi and the explicit form of J.
why it matters in Recognition Science
This identity supplies the concrete left-hand side for the first step of the Conjecture C8 derivation chain. It feeds the extremum condition that determines λ_rec and rests on J-uniqueness together with the forcing of φ as the self-similar fixed point. The module doc-comment explicitly lists the step as “Bit Cost (J_bit): From the unique cost functional J(x) = ½(x + x^{-1}) - 1, evaluated at the self-similar scale φ”.
scope and limits
- Does not compute numerical values or approximations.
- Does not extend the equality to other scales or rung values.
- Does not derive or use the curvature cost J_curv.
- Does not restore physical units or introduce the 1/π factor.
formal statement (Lean)
80theorem J_bit_explicit : J_bit_val = (phi + phi⁻¹) / 2 - 1 := rfl
proof body
Term-mode proof.
81
82/-- Using φ + 1/φ = φ + (φ - 1) = 2φ - 1 (from φ² = φ + 1 ⟹ 1/φ = φ - 1).
83 Therefore J_bit = (2φ - 1)/2 - 1 = φ - 3/2.
84
85 **Note**: This is exact. 1/φ = φ - 1 (from φ² = φ + 1).
86 So φ + 1/φ = 2φ - 1.
87 J_bit = (2φ - 1)/2 - 1 = φ - 3/2 ≈ 1.618 - 1.5 = 0.118. -/