horton_bifurcation_ratio_pos
plain-language theorem explainer
Recognition Science derives river network branching from sigma conservation on the phi ladder. The result shows the Horton bifurcation ratio exceeds zero. Geomorphologists and climate modelers cite it when assembling certificates for drainage scaling laws. The proof is a one-line wrapper that unfolds the definition and invokes power positivity.
Claim. $0 < R_b$ where the Horton bifurcation ratio $R_b = phi^2$ and $phi$ is the self-similar fixed point.
background
The module River Networks from sigma-Conservation treats drainage basins as recognition trees. Sigma conservation forces upstream-downstream branching to obey canonical Horton ratios: length ratio $R_l = phi$ and bifurcation ratio $R_b = phi^2$. The upstream definition supplies horton_bifurcation_ratio as the tributary count per order, realized by two phi-steps. This structure reappears in the eight-tick lattice that produces phi-squared ratios for volcanic recurrence and planetary orbits.
proof idea
The term proof unfolds horton_bifurcation_ratio to phi^2 and applies pow_pos with the hypothesis phi_pos to conclude strict positivity.
why it matters
The result is collected into riverNetworkCert, which packages all positivity and inequality facts required to certify the structural Hack exponent of one-half. It supplies the positivity leg of the RS reading of Hack's law, linking directly to the phi-ladder and T7 eight-tick octave in the forcing chain.
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