recoveryTime_pos
plain-language theorem explainer
The recovery time after a cascade of depth k is strictly positive for every natural number k. Ecologists working with finite recognition graphs cite this to guarantee positive durations in post-extinction recovery models. The proof is a one-line wrapper that unfolds the definition and invokes the strict positivity of powers of phi.
Claim. For every natural number $k$, $0 < phi^k$, where $phi$ is the golden-ratio fixed point of the recognition forcing chain.
background
In this ecology module an ecosystem is a finite recognition graph whose species carry rungs on the phi-ladder; extinction occurs when a rung drops below the life-ignition threshold Z_life = phi^19. The function recoveryTime supplies the recovery timescale after a cascade of depth k, expressed in phi-ladder units of the natural recovery scale tau_0. The upstream definition states recoveryTime (k : ℕ) : ℝ := phi ^ k.
proof idea
The proof is a one-line wrapper. It unfolds recoveryTime to obtain phi ^ k and applies the lemma pow_pos phi_pos k.
why it matters
This positivity result is required by the ExtinctionCascadeCert structure that certifies cascade closure on finite graphs. It is invoked in the deep-cascade lower bound phi^16 < recoveryTime 17, which calibrates mammal recovery after K-Pg extinction to 10^4-10^5 years under canonical tau_0. The lemma therefore anchors the recovery-time component of the ecology track and aligns with the phi-ladder fixed point (T6) and eight-tick octave of the Recognition Science forcing chain.
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