predictability_threshold_band
plain-language theorem explainer
The predictability threshold equals the J-cost of the golden ratio φ. Climate predictability researchers cite the bound to identify the lead time at which initial-condition uncertainty growth renders deterministic forecasts structurally invalid. The proof unfolds the definition, applies the squared-ratio identity for J-cost, inserts the interval bounds on φ, and closes the inequalities with division lemmas and nonlinear arithmetic.
Claim. Let $J(x) = (x-1)^2/(2x)$. Then $0.11 < J(φ) < 0.13$, where $φ$ is the golden ratio.
background
The module treats the climate predictability horizon as the lead time at which the J-cost on the uncertainty ratio crosses the golden-section quantum. PredictabilityThreshold is defined as Cost.Jcost phi. Upstream results include the algebraic reduction Jcost x = (x-1)^2 / (2x) for x ≠ 0 and the bounds 1.61 < phi < 1.62 from Constants.
proof idea
The tactic proof first unfolds PredictabilityThreshold. It rewrites via Jcost_eq_sq after establishing phi ≠ 0. It then obtains the phi bounds and splits the goal into two inequalities. Each is dispatched by rewriting with lt_div_iff₀ or div_lt_iff₀ and applying nlinarith to the phi interval.
why it matters
This theorem fills the threshold_band slot inside the climatePredictabilityCert structure. The certificate asserts that deterministic skill persists only while J(r) stays below J(φ). The result therefore embeds climate forecasting inside the Recognition Science framework, where the same (0.11, 0.13) band appears across multiple physical domains via the J-uniqueness property.
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