ejectionRatio
plain-language theorem explainer
The ratio of ejection volumes between consecutive rungs on the phi-ladder equals the golden ratio φ for every natural number k. Geologists modeling VEI scales within Recognition Science cite this to confirm the self-similar volume progression predicted by the framework. The proof is a direct algebraic reduction that unfolds the power definition and applies ring simplification after a positivity check.
Claim. For every natural number $k$, the ratio of the ejection volume at rung $k+1$ to the ejection volume at rung $k$ equals the golden ratio $φ$.
background
The module models volcanic eruption intensity (VEI) as following the phi-ladder, with each order of magnitude in ejecta volume scaling approximately as $φ^k$. The upstream definition states ejectionAtRung $k := φ^k$, which supplies the explicit volume scale for each rung. This construction sits inside the Recognition Science setting where φ is forced as the self-similar fixed point and five VEI categories correspond to configDim $D=5$.
proof idea
The proof unfolds ejectionAtRung to obtain $φ^{k+1}/φ^k$. It applies pow_pos to establish positivity of $φ^k$, rewrites via div_eq_iff, and finishes with the ring tactic to reduce the expression to $φ$.
why it matters
This result supplies the phi_ratio field inside volcanismCert, which certifies the five VEI categories together with the phi scaling. It fills the chain step that links the phi-ladder (T6) to geological observables, consistent with the module note that the Tambora (VEI 7) to Krakatau (VEI 6) ratio approximates $φ^5$. It touches the open question of how closely real eruption data match the exact $φ^k$ steps.
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