leopoldMaddockExponents
leopoldMaddockExponents supplies the empirical central-tendency triple (0.26, 0.40, 0.34) as a concrete instance of the hydraulic exponents structure. Fluvial geomorphologists cite it when comparing observed at-a-station width-depth-velocity scaling in alluvial channels to the discharge closure constraint. The definition assigns the three fractions directly and discharges the positivity and summation obligations by norm_num.
claimThe empirical central-tendency hydraulic geometry exponents are the record with width exponent $b = 0.26$, depth exponent $f = 0.40$, velocity exponent $m = 0.34$, satisfying $0 < b$, $0 < f$, $0 < m$ and $b + f + m = 1$.
background
The hydraulic exponents structure encodes the at-a-station scaling exponents for channel width (b), depth (f) and velocity (m). These three real numbers must each be positive and sum exactly to one; the sum-to-one condition follows directly from the discharge identity Q = w · d · v by taking logarithms and differentiating with respect to a reference discharge.
proof idea
The definition constructs the record by assigning b := 26/100, f := 40/100, m := 34/100. The three positivity fields and the closure field are each discharged by a single norm_num tactic.
why it matters in Recognition Science
This definition supplies the empirical witness that, together with the equipartition triple, populates the master hydraulic geometry certificate. It shows that the algebraic closure forced by σ-conservation on the discharge ledger accommodates the central tendency reported in Leopold & Maddock 1953 USGS PP 252 without additional parameters.
scope and limits
- Does not derive the numerical ratios from Recognition Science primitives.
- Does not guarantee that every single-thread reach obeys exactly these values.
- Does not extend to multi-thread or braided channels.
- Does not incorporate measurement uncertainty or regional variation.
formal statement (Lean)
125noncomputable def leopoldMaddockExponents : HydraulicExponents where
126 b := 26 / 100
proof body
Definition body.
127 f := 40 / 100
128 m := 34 / 100
129 width_pos := by norm_num
130 depth_pos := by norm_num
131 velocity_pos := by norm_num
132 closure := by norm_num
133
134/-! ## §3. Master certificate -/
135