horizon_adjacent_ratio
plain-language theorem explainer
The ratio of forecast horizons at consecutive rungs on the phi-ladder equals the reciprocal of the golden ratio. Operational meteorologists would cite this to relate skill horizons between centers such as ECMWF at rung 0 and GFS at rung -1. The proof rewrites the ratio via the successor relation and clears the denominator with positivity.
Claim. For every natural number $k$, the ratio of the forecast horizon at rung $k+1$ to the forecast horizon at rung $k$ equals $phi^{-1}$.
background
The module defines the forecast horizon at rung $k$ as referenceHorizon times $phi$ raised to the power of negative $k$. This builds the phi-ladder that assigns shorter horizons to lower-resolution models. The positivity theorem establishes that every such horizon is a positive real. The successor ratio theorem states that advancing one rung multiplies the horizon by $phi^{-1}$. The local setting concerns operational weather forecasting skill decay, with ECMWF at rung 0 and GFS at rung -1.
proof idea
The proof applies the successor ratio theorem to rewrite the numerator in terms of the denominator. It then invokes the positivity theorem to justify field simplification that isolates the reciprocal of phi.
why it matters
This theorem supplies the adjacent ratio equality required by the operational forecast skill certificate. It closes the phi-ladder structure for skill horizons and supports the prediction that horizons scale by phi per rung of model resolution. The result aligns with the self-similar fixed point phi in the recognition framework.
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