PredictionRange
PredictionRange enumerates the five standard meteorological forecast horizons that map onto the canonical J(phi) band in Recognition Science. Meteorologists and RS modelers cite the enumeration when converting the Lorenz two-week predictability limit into a discrete configDim D = 5. The declaration is a plain inductive type whose deriving clause supplies the Fintype instance used by downstream cardinality and certification results.
claimLet $P$ be the finite set whose elements are the five weather prediction intervals: nowcast (0-2 h), short-range (2-24 h), medium-range (1-10 d), extended (10-30 d), and seasonal (>30 d). Then $|P|=5$ and $P$ carries a decidable equality and Fintype structure.
background
The module treats weather predictability through the ratio $r$ = (actual state) / (predicted state), which obeys J-cost dynamics whose cutoff lies inside the canonical J(phi) band. Five discrete ranges realize configDim D = 5, with the skill score predicted to decay as phi^(-t/T_c) where T_c approximates the 14-day Lyapunov doubling time. The module imports CanonicalJBand to supply the skill-threshold certificate that will later reference this enumeration.
proof idea
The declaration is an inductive definition that introduces exactly five constructors and derives DecidableEq, Repr, BEq, and Fintype in a single deriving clause; no explicit proof body is required.
why it matters in Recognition Science
PredictionRange supplies the five_ranges field inside WeatherPredictabilityCert and the target of the cardinality theorem predictionRangeCount. It directly encodes the RS claim that meteorological predictability is governed by the same J-cost and phi-ladder structure that fixes D = 3 in the spatial case and the eight-tick octave, thereby linking the Lorenz butterfly cutoff to the Recognition Composition Law.
scope and limits
- Does not derive the numerical value of the J-cost band edges from the functional equation.
- Does not model specific atmospheric differential equations or initial-condition sensitivity.
- Does not prove the phi^(-t/T_c) decay law; that remains a downstream claim.
- Does not address ensemble forecasting or data assimilation techniques.
formal statement (Lean)
23inductive PredictionRange where
24 | nowcast | shortRange | mediumRange | extended | seasonal
25 deriving DecidableEq, Repr, BEq, Fintype
26