cumulative_ratio_one_step
plain-language theorem explainer
cumulative_ratio_one_step establishes that the cumulative recurrence ratio over one VEI step equals the adjacent-VEI step ratio phi squared. Volcanic recurrence modelers working in the Recognition Science program cite it to close the base case of the phi-ladder. The term proof unfolds the definitions of cumulative_ratio and vei_step_ratio then applies reflexivity.
Claim. The cumulative recurrence ratio across one Volcanic Explosivity Index step equals the adjacent-VEI step ratio: $phi^{2} = phi^{2}$, where the cumulative ratio for $k$ steps is defined by $phi^{2k}$ and the single-step ratio by $phi^{2}$.
background
The module treats volcanic eruptions as clustering on a phi-rational recurrence ladder. Each VEI step corresponds to one octave on the recognition lattice J-cost spectrum, yielding the ratio $T_{VEI(n+1)}/T_{VEI(n)} = phi^{2}$ from the eight-tick structure plus gap-45 frustration on long-period events. The cumulative ratio across $k$ steps is defined as $phi^{2k}$. The single-step ratio is defined as $phi^{2}$.
proof idea
The proof is a one-line wrapper that unfolds the definitions of cumulative_ratio and vei_step_ratio then closes by reflexivity.
why it matters
It is invoked inside the construction of eruptionRecurrenceCert and inside the proof of eruption_recurrence_one_statement, which packages the full claim that adjacent-VEI ratios lie in (2.59, 2.63) and that the cumulative ratio equals $(phi^{2})^{k}$. The result anchors the base case of the phi-ladder for geophysical events, consistent with the framework's eight-tick octave and the self-similar fixed point phi.
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