IndisputableMonolith.Ecology.BiodiversityScalingFromJCost
This module derives the Arrhenius species-area exponent z = log φ / (1 + log φ) from J-cost conservation on a φ-self-similar inventory. Ecologists modeling biodiversity scaling laws would cite it for its RS-native derivation of the classic z ≈ 0.25 value. The module consists of supporting lemmas establishing positivity, bounds, and a certification theorem for the exponent.
claimThe species-area exponent is $z = {log φ} / (1 + log φ)$, where φ is the self-similar fixed point satisfying J(φ) = 1 under σ-conservation on a φ-self-similar inventory.
background
Recognition Science encodes recognition cost via the J-cost function J(x) = (x + x^{-1})/2 - 1, which satisfies the Recognition Composition Law. The module applies this under σ-conservation to a φ-self-similar inventory, where φ is the fixed point forced in the T5-T6 steps of the UnifiedForcingChain. The imported Constants module supplies the RS-native time quantum τ₀ = 1 tick as the base unit for all scaling.
proof idea
This is a definition module, no proofs. It introduces the species_area_exponent definition together with auxiliary facts (log_phi_pos, one_plus_log_phi_pos, species_area_exponent_pos, species_area_exponent_in_band) and the BiodiversityScalingCert wrapper.
why it matters in Recognition Science
The module supplies the biodiversity scaling exponent that connects J-cost to observable species-area relations in the Recognition framework. It realizes the Arrhenius form z = log φ / (1 + log φ) as a direct consequence of φ-self-similarity and σ-conservation, feeding downstream ecological models that rely on the phi-ladder and eight-tick octave structure.
scope and limits
- Does not derive the exponent without assuming σ-conservation.
- Does not treat non-self-similar or non-φ inventories.
- Does not include empirical data fitting or statistical validation.
- Does not extend to higher-dimensional or time-dependent biodiversity models.
depends on (1)
declarations in this module (13)
-
def
species_area_exponent -
theorem
log_phi_pos -
theorem
one_plus_log_phi_pos -
theorem
log_phi_lt_one -
theorem
two_log_phi_gt -
theorem
log_phi_gt_threeTenths -
theorem
log_phi_lt_sevenTenths -
theorem
species_area_exponent_pos -
theorem
species_area_exponent_lt_half -
theorem
species_area_exponent_in_band -
structure
BiodiversityScalingCert -
def
biodiversityScalingCert -
theorem
biodiversity_scaling_one_statement