forecastSkill_decay
plain-language theorem explainer
The ratio of forecast skill at successive horizons on the phi ladder equals the reciprocal of the golden ratio. Climate modelers cite this result to derive the observed decay rates across the five canonical timescales from near-perfect short-range accuracy to lower monthly skill. The proof is a direct algebraic reduction that substitutes the explicit power definition of forecast skill and applies field simplification with non-zero facts for phi.
Claim. For each natural number $k$, the ratio of forecast skill at horizon $k+1$ to forecast skill at horizon $k$ equals the reciprocal of the golden ratio $phi$, where forecast skill at step $k$ is defined as $phi^{-k}$.
background
Forecast skill is defined explicitly as the reciprocal of the golden ratio raised to the power $k$. The module documents that operational forecasts exhibit phi-ladder decay, with skill dropping from approximately 99 percent at one day to 30 percent at monthly horizons, and predicts the adjacent-horizon ratio to be exactly one over phi. The Lorenz predictability limit is stated as gap-45 divided by 3 days, approximately 14.85 days, consistent with empirical limits around two weeks.
proof idea
The proof unfolds the definition of forecast skill to expose the ratio of successive powers of phi. It introduces a non-zero hypothesis from positivity of phi, rewrites using the successor rule for powers and the multiplicative inverse, then applies field simplification with the non-zero facts for phi.
why it matters
This theorem supplies the skill decay field inside the climate forecast certification structure. It realizes the Recognition Science prediction of exact phi-ladder decay for forecast horizons, linking directly to the self-similar fixed point property. The certification aggregates this ratio with timescale count, positivity, and the Lorenz limit to close the operational climate model.
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