Z_life
plain-language theorem explainer
Z_life sets the life-ignition threshold to φ^19 on the Recognition Science phi-ladder. Extinction-cascade analyses cite it as the bankruptcy cutoff below which a species rung triggers removal from the live set. The definition is a direct constant assignment with no lemmas or computation steps.
Claim. Define the life-ignition threshold by $Z_ {life} = φ^{19}$, where $φ$ is the golden-ratio fixed point of the forcing chain.
background
The module treats an ecosystem as a finite recognition graph in which each species carries a rung value Z(v) augmented by support from neighboring edges. Bankruptcy occurs precisely when total rung drops below the ignition threshold, after which the species is removed and its supports vanish. Z_life supplies that threshold as φ^19, matching the abiogenesis crossing referenced in the module header. The constant is imported from Constants and appears in the IsBankrupt predicate and the monotone cascade closure.
proof idea
Direct definition that assigns the constant as the nineteenth power of phi. No upstream lemmas are applied; the body is a single equality using the built-in real exponentiation.
why it matters
Z_life anchors the bankruptcy test inside ExtinctionCascadeCert and the one-statement cascade theorem. It supplies the explicit cutoff for the monotone fixed-point iteration on finite graphs and links the ecology track to the phi-ladder scaling (T6) and eight-tick octave (T7) of the unified forcing chain. The value also appears in the recovery-time lower bound φ^16 < recoveryTime 17 for deep cascades.
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