horizonAtRung_succ_ratio
plain-language theorem explainer
The relation states that forecast horizons on the phi-ladder satisfy a multiplicative recurrence with factor phi inverse. Operational climate modelers comparing resolution-dependent skill decay at centers like ECMWF and GFS would cite this to derive adjacent rung ratios. The proof proceeds by unfolding the horizon definition, applying the zpow addition rule justified by phi non-zero, and simplifying with ring tactics.
Claim. For every natural number $k$, the forecast horizon at rung $k+1$ equals the forecast horizon at rung $k$ multiplied by $phi^{-1}$.
background
In the climate module, forecast skill horizons are placed on the phi-ladder. The function horizonAtRung at rung k is defined as referenceHorizon times phi raised to the power of negative k. This setup encodes the recognition science prediction that adjacent operational centers differ by exactly one phi factor in their useful skill duration, as illustrated by ECMWF at 10 days and GFS at approximately 6.2 days. The proof relies on the lemma that phi is nonzero, which appears in multiple modules including Constants and PhiLadderLattice. The module states that the recognition-Lyapunov-time prediction is that adjacent forecast-skill horizons across operational centers ratio by exactly φ per integer rung of model resolution.
proof idea
The tactic proof first unfolds the definition of horizonAtRung. It then obtains phi non-zero from Constants.phi_ne_zero and uses it to rewrite the exponent as a sum via zpow_add₀. An integer cast equality is established with push_cast and ring, after which the main equality is closed by rewriting and applying ring.
why it matters
This theorem supplies the one-step recurrence used by horizon_adjacent_ratio to obtain the exact phi inverse ratio and by horizonAtRung_strictly_decreasing to prove monotonicity. Together they populate the operationalForecastSkillCert. It directly implements the phi-ratio structural prediction for forecast horizons in the Recognition Science climate application, linking to the phi fixed point and the eight-tick octave structure.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.