biodiversity_scaling_one_statement
plain-language theorem explainer
Recognition Science derives the Arrhenius species-area exponent z = log φ / (1 + log φ) from σ-conservation on a φ-self-similar species inventory. Ecologists modeling biodiversity scaling would cite this result for its structural prediction that z lies strictly between 0.15 and 0.45 while remaining below 1/2. The theorem supplies a non-trivial constraint: observed exponents outside this band would falsify the underlying conservation law. The proof is a direct term that conjoins the four upstream bounds on z.
Claim. Let $z = {log φ}/(1 + log φ)$. Then $0 < z < 1/2$ and $0.15 < z < 0.45$.
background
In the Recognition Science framework a regional ecosystem is a recognition graph on its species inventory. σ-conservation on this graph forces φ-self-similar bond density, so that species count S(A) scales as A^z with the structural exponent z = log φ / (1 + log φ). The module proves this identity and verifies that z sits inside the empirical Arrhenius band (0.15, 0.45), which brackets the canonical range (0.20, 0.40).
proof idea
The proof is a term-mode construction that returns the four-tuple (species_area_exponent_pos, species_area_exponent_lt_half, species_area_exponent_in_band.1, species_area_exponent_in_band.2). No tactics are executed; the conjunction is assembled directly from the already-established positivity, sub-half, and band-membership results for the exponent.
why it matters
This declaration packages the key bounds on the Arrhenius exponent into one statement, completing the structural part of the biodiversity scaling theorem. It supplies the concrete numerical constraint z ∈ (0.15, 0.45) that can be tested against GBIF or BioTIME data. In the broader framework it shows how the self-similar fixed point φ propagates into ecological scaling, yielding a falsifiable prediction. The module doc-comment records the remaining open question of per-taxon dispersion across phyla.
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