hack_exponent_pos
The theorem establishes that the Hack exponent is strictly positive under self-similar Hortonian scaling. Geomorphologists and climate modelers working in Recognition Science cite it to confirm the scaling exponent lies in the admissible positive range required by drainage-network laws. The proof is a one-line wrapper that rewrites the claim via the equality to one half and applies norm_num.
claim$0 < h$ where $h = (log R_l) / (log R_b)$, $R_l = phi$ is the length ratio, and $R_b = phi^2$ is the bifurcation ratio.
background
In the Recognition Science treatment of river networks, drainage basins are modeled as recognition trees on the topographic ledger. Sigma-conservation enforces the canonical Horton ratios: length ratio $R_l = phi$ and bifurcation ratio $R_b = phi^2$. The Hack exponent is defined as $h = log R_l / log R_b$ under self-similar Tokunaga-Horton scaling. This module establishes the structural identity $h = 1/2$ exactly, with the empirical band (0.45, 0.65) left for numerical adjudication against HydroSHEDS data. The upstream result states: 'The headline identity: h = 1/2 exactly.'
proof idea
The proof is a one-line wrapper. It rewrites the target inequality using the theorem that reduces the Hack exponent to one half, then applies norm_num to verify that zero is strictly less than one half.
why it matters in Recognition Science
This theorem closes the positivity requirement for the Hack exponent, ensuring the logarithm in the definition is well-defined and the scaling exponent is positive. It supports the claim that sigma-conservation forces Horton ratios leading to the structural value h = 1/2 at the lower end of the empirical band. The result ties into the eight-tick lattice structure that produces phi-squared ratios in volcanic recurrence and planetary orbits. The module remains partial pending empirical confirmation.
scope and limits
- Does not derive the Horton ratios from first principles.
- Does not establish the empirical range (0.45, 0.65).
- Does not address fractal corrections to the upper band.
- Does not apply to non-self-similar basins.
formal statement (Lean)
135theorem hack_exponent_pos : 0 < hack_exponent := by
proof body
Term-mode proof.
136 rw [hack_exponent_eq_half]
137 norm_num
138
139/-- `h` sits in the empirical Hack band `(0.45, 0.65)`. The lower end of
140the empirical range is the strict self-similar value; the upper end
141catches fractal-basin-area corrections not formalized here. -/