log_phi_pos
plain-language theorem explainer
The natural logarithm of the golden ratio is strictly positive. Researchers deriving biodiversity scaling exponents or black-hole horizon corrections in Recognition Science cite this to fix the sign of log-ratio increments on the phi-ladder. The proof is a one-line wrapper that invokes Real.log_pos on the upstream fact that phi exceeds 1.
Claim. $0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 <
background
The DiscreteGauge module derives the discrete gauge r ~ r + n ln φ from T6 (phi-self-similarity) and T7 (eight-tick neutrality) in the GCIC response. This lemma records the elementary sign fact needed for all subsequent log-ratio displacements. The golden ratio phi satisfies 1 < phi by the self-similarity relation phi^2 = phi + 1, as stated in Constants.one_lt_phi and reproduced in PhiSupport.one_lt_phi.
proof idea
The proof is a one-line wrapper that applies Real.log_pos to the upstream lemma one_lt_phi.
why it matters
This lemma supplies the positivity hypothesis for BiodiversityScalingCert, which certifies that the species-area exponent lies in (0.15, 0.45), and for black-hole horizon calculations that exclude the LQG value -0.25. It anchors the positive direction of phi-ladder steps in T6 and is required for the compactification argument that turns the eight-tick sum into a closed gauge orbit. The result touches the question whether the discrete gauge identification can be lifted to a full dynamical equivalence without further tick-sequence assumptions.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.