J_bit_pos
The theorem proves that the fundamental bit cost, defined as the natural logarithm of the golden ratio, is strictly positive. Researchers deriving physical constants or building astrophysical models in Recognition Science cite this result to secure positivity in energy and curvature steps. The proof is a direct one-line application of the standard lemma that the real logarithm is positive for arguments greater than one.
claimLet $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$ be the recognition cost. Let $J_0 = 1$ and let $J(x) = (x + x^{-1})/2 - 1$
background
Recognition Science starts from the Composition Law and extracts the unique cost function $J(x) = (x + x^{-1})/2 - 1$, also written cosh(ln x) - 1. The golden ratio φ arises as the self-similar fixed point of this cost. In the ConstantDerivations module the fundamental bit cost is introduced by the definition $J_bit := ln φ$. The module then derives the remaining constants as algebraic ratios in φ: c = 1, ħ = φ^{-5}, G = φ^5/π, and α^{-1} inside the observed band. The positivity statement is the first concrete numerical property required before any energy or length scale can be formed.
proof idea
The proof is a term-mode wrapper that applies the Mathlib lemma establishing positivity of the real logarithm for every argument strictly greater than one, instantiated at the already-established fact that φ > 1.
why it matters in Recognition Science
This result sits at the base of the constant-derivation chain that runs from the Composition Law through the eight-tick octave to the Planck-scale matching certificate. It is invoked by the extremum_condition and planck_scale_matching_cert theorems in PlanckScaleMatching and by the phi_forcing_principle in PhiForcing. The positivity supplies the sign needed to place the coherence quantum E_coh = φ^{-5} on the positive real line and to guarantee that subsequent curvature and length quantities remain positive.
scope and limits
- Does not compute a numerical approximation for the bit cost.
- Does not derive positivity from the full Composition Law.
- Applies only to the principal real logarithm.
- Does not address extensions to complex arguments or other cost functions.
formal statement (Lean)
72theorem J_bit_pos : J_bit > 0 := Real.log_pos φ_gt_one
proof body
Term-mode proof.
73
74/-- The coherence quantum: E_coh = φ^(-5).
75 This is the minimum energy for coherent recognition. -/