WeatherPredictabilityCert
plain-language theorem explainer
The weather predictability certificate structure requires exactly five discrete prediction time ranges together with satisfaction of the canonical J-cost band conditions. Atmospheric modelers working in Recognition Science would cite it when mapping Lorenz unpredictability to J-dynamics decay. It is introduced as a bare structure definition that packages two fields with no lemmas or computational steps.
Claim. A weather predictability certificate consists of the assertion that the inductive type of five weather prediction intervals has cardinality five, together with an instance of the canonical J-cost certificate whose clauses include $J(1)=0$, reciprocity $J(x)=J(1/x)$, positivity at the golden ratio, the numerical band $0.11 < J(phi) < 0.13$, and positivity at $1/phi^2$.
background
The meteorology module models weather predictability via J-cost dynamics, where the ratio of actual to predicted states obeys the J functional equation and the cutoff occurs at the canonical J(phi) band. PredictionRange enumerates the five canonical intervals nowcast (0-2 h), short-range (2-24 h), medium (1-10 d), extended (10-30 d), and seasonal (30 d+), which the module identifies with configDim D = 5. The upstream CanonicalCert supplies the six-clause J-certificate reused across domains: matched zero at unity, reciprocity under inversion, positivity at phi, the explicit band 0.11 to 0.13, and recovery positivity at the inverse square.
proof idea
The declaration is a structure definition that directly declares the two fields cardinality condition on the five-range inductive type and reference to CanonicalCert. No lemmas are applied and no tactics are used; it functions as a record type with no proof obligations.
why it matters
This structure supplies the type instantiated by the concrete certificate weatherPredictabilityCert, which populates the five-range count via predictionRangeCount and the skill threshold via the canonical J-band instance. It encodes the Recognition Science claim that weather skill decays inside the five-range partition according to J-cost, linking directly to J-uniqueness and the phi fixed point that set the predictability cutoff at roughly 14 days. It touches the open calibration of Lyapunov doubling time to the phi-ladder rung.
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