log_phi_lt_sevenTenths
plain-language theorem explainer
This theorem proves that the natural logarithm of the golden ratio φ is strictly less than 0.70. Ecologists deriving species-area scaling from J-cost self-similarity in Recognition Science cite it to place the Arrhenius exponent z = log φ / (1 + log φ) inside the empirical band. The short tactic proof chains log φ < log 2 from the imported φ < 2 fact with the decimal bound log 2 < 0.6932 and closes by linarith.
Claim. The natural logarithm of the golden ratio satisfies $log φ < 0.70$.
background
In the Biodiversity Scaling from J-Cost module a regional ecosystem is a recognition graph on its species inventory. σ-conservation forces φ-self-similar bond density so that species count obeys S(A) ∝ A^z with structural exponent z = log φ / (1 + log φ). The module proves 0 < z < 1/2 and the tighter numerical band z ∈ (0.15, 0.45) that sits inside the empirical Arrhenius range (0.10, 0.50).
proof idea
The proof first applies Real.log_lt_log to the positive φ and the imported phi_lt_two lemma to obtain log φ < log 2. It then invokes Real.log_two_lt_d9 to bound log 2 below 0.6932 and finishes with linarith.
why it matters
The bound is collected into BiodiversityScalingCert, which discharges species_area_exponent_in_band and thereby certifies that the structural exponent lies inside the observed Arrhenius interval. It supplies the numerical closure for the RS reading of ecosystems as recognition graphs whose J-cost scaling reproduces the classic species-area law. The parent step is the derivation of z from φ-self-similarity in the forcing chain.
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