Z_life_pos
plain-language theorem explainer
The life-ignition rung equals phi to the nineteenth power and is strictly positive. Ecologists formalizing recognition cascades on finite species graphs cite this result to anchor the bankruptcy threshold above zero before invoking monotonicity. The proof is a one-line wrapper that unfolds the definition and applies the positivity of powers with positive base.
Claim. $0 < Z_ {life}$ where $Z_{life} := phi^{19}$ and $phi$ is the self-similar fixed point of the recognition composition law.
background
In the Recognition Science ecology module an ecosystem is a finite recognition graph whose vertices carry rungs drawn from the phi-ladder. The life-ignition threshold is defined by the upstream declaration Z_life := phi ^ 19, taken from the abiogenesis first-crossing result. Ledger bankruptcy occurs when removal of support edges drives a species rung below this threshold, and the module encodes the resulting cascade as a monotone fixed-point iteration on the live set.
proof idea
The proof is a one-line wrapper. It unfolds the definition of Z_life to obtain phi ^ 19 and then applies the lemma pow_pos phi_pos _ to conclude strict positivity.
why it matters
Z_life_pos supplies the first field of the ExtinctionCascadeCert structure that packages the invariants for the one-statement cascade theorem. It closes the structural half of the Q2 track by placing the threshold inside the positive reals, consistent with the phi-ladder mass formula and the eight-tick octave. The result touches the open calibration of biological recovery time against phi^k for cascade depth k.
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