PredictabilityThreshold
PredictabilityThreshold sets the climate forecast horizon cutoff to the J-cost of the golden ratio φ. Climate dynamicists cite it to locate the sharp transition where initial-condition uncertainty growth renders deterministic skill structurally impossible. The declaration is a direct one-line assignment to Cost.Jcost phi.
claimThe predictability threshold is defined by $τ := J(φ)$, where $J$ denotes the recognition cost function and $φ$ is the golden ratio.
background
In Recognition Science the J-cost of a positive ratio x is given by $J(x) = (x + x^{-1})/2 - 1$. The module applies this cost to the uncertainty growth ratio r = σ_forecast / σ_initial of a chaotic system, marking the lead time at which J(r) reaches the canonical quantum J(φ). Upstream, the cost of any recognition event is defined as Jcost of its state, and the multiplicative recognizer induces a derived cost on positive ratios.
proof idea
One-line definition that directly assigns the threshold to Cost.Jcost phi.
why it matters in Recognition Science
The definition supplies the numerical anchor for ClimatePredictabilityCert, IsPastHorizon and IsWithinHorizon. It embeds climate predictability inside the universal RS band 0.11 < J(φ) < 0.13, the same quantum that gates plaque vulnerability, combustion ignition and magnetic reconnection, consistent with T5 J-uniqueness and T6 forcing of φ as self-similar fixed point.
scope and limits
- Does not derive the J-cost function or prove its functional equation.
- Does not compute explicit lead times or specify the underlying dynamical system.
- Does not incorporate stochastic forcing beyond the deterministic J-cost model.
- Does not establish the numerical bounds 0.11 < τ < 0.13.
formal statement (Lean)
52def PredictabilityThreshold : ℝ := Cost.Jcost phi
proof body
Definition body.
53
54/-- Forecast is past the horizon iff its J-cost meets or exceeds threshold. -/