ClimatePredictabilityCert
plain-language theorem explainer
The ClimatePredictabilityCert structure assembles the algebraic symmetries, nonnegativity, and numerical band of the forecast J-cost function together with mutual exclusion of horizon states. Climate dynamicists and recognition theorists cite it to certify that deterministic skill persists exactly while the uncertainty ratio stays below the golden-section quantum. The structure is populated directly from five supporting lemmas on the cost function and threshold definition.
Claim. A certificate for the climate predictability horizon consists of the statements that the J-cost on the uncertainty ratio vanishes at unity, satisfies $J(r)=J(r^{-1})$ for all $r>0$, remains nonnegative for positive ratios, lies in the open interval $(0.11,0.13)$ when evaluated at the golden ratio, and that the strict sub-threshold and super-threshold regimes on the ratio are mutually exclusive.
background
The module defines forecastCost r as the J-cost of the uncertainty growth ratio r. PredictabilityThreshold is defined as Cost.Jcost phi, the canonical golden-section quantum. IsWithinHorizon r holds precisely when forecastCost r lies strictly below this threshold, while IsPastHorizon r holds when the cost meets or exceeds it. The upstream theorem cost_nonneg from ObserverForcing establishes that J-cost is nonnegative for any recognition event.
proof idea
The structure is a record definition that collects five independent properties. It directly references the lemmas forecastCost_zero_at_unit, forecastCost_reciprocal_symm, forecastCost_nonneg, predictability_threshold_band, and horizon_states_exclusive to populate each field. No further reduction occurs inside the structure itself.
why it matters
This certificate supplies the structural claim that forecast skill is lost precisely when the J-cost on the initial-condition ratio reaches J(phi) inside the interval (0.11,0.13). It is instantiated by the downstream definition climatePredictabilityCert. The construction realizes the module prediction that the same recognition quantum gates climate forecast skill as it does plaque vulnerability, combustion ignition, and magnetic reconnection.
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