cumulative_ratio_factors
plain-language theorem explainer
The theorem establishes that the cumulative recurrence ratio across any number k of VEI steps equals the single-step ratio raised to the k power. Modelers of volcanic interval statistics on the phi-ladder cite this identity when aggregating multi-class predictions from the Smithsonian GVP catalog. The proof is a one-line term reduction that unfolds the two defining expressions and applies the power-multiplication identity.
Claim. For every natural number $k$, the cumulative recurrence ratio across $k$ VEI steps equals the adjacent-VEI step ratio raised to the power $k$, that is, $phi^{2k} = (phi^2)^k$.
background
In the volcanic eruption recurrence ladder each VEI step corresponds to one octave on the recognition lattice J-cost spectrum. The single-step ratio is defined as $phi^2$ and the cumulative ratio across $k$ steps is defined as $phi^{2k}$. These definitions rest on the eight-tick octave (T7) together with the gap-45 frustration that forces the $phi$-squared scaling for long-period geophysical events.
proof idea
The proof unfolds the definitions of cumulative ratio and vei step ratio, then rewrites using the leftward power-multiplication rule to obtain the equality.
why it matters
This identity supplies the algebraic factorisation required by the master certificate EruptionRecurrenceCert and by the one-statement summary theorem that packages the full recurrence prediction. It closes the structural claim for Track E6, confirming that cumulative ratios remain inside the canonical band for arbitrary numbers of steps. The derivation inherits directly from the T7 eight-tick octave and the J-uniqueness condition of the forcing chain.
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