IsBankrupt
plain-language theorem explainer
Defines bankruptcy for species j in ecosystem E under live set L as the condition that total rung falls below the ignition threshold Z_life = phi^19. Ecologists and recognition theorists modeling finite-graph extinction cascades cite this predicate to mark when bond removal triggers collapse. The definition is a direct inequality comparison of totalRung against Z_life together with a decidability instance.
Claim. Species $j$ is bankrupt in ecosystem $E$ with live set $L$ precisely when its total rung satisfies totalRung$(E,L,j) < Z_mathrm{life}$, where $Z_mathrm{life} = phi^{19}$.
background
The module formalizes extinction cascades on finite recognition graphs. An Ecosystem consists of $n$ species together with positive baseline rungs and nonnegative support values between every pair; only species inside the live set $L$ contribute their support terms. totalRung of species $j$ under $L$ is therefore the baseline of $j$ plus the sum of supports received from members of $L$. Z_life is the constant $phi^{19}$, the life-ignition threshold referenced from AbiogenesisFirstCrossing.
proof idea
The definition is the single inequality totalRung E L j < Z_life. The accompanying instance declaration unfolds the predicate and delegates to the existing decidability instance for real-number comparison.
why it matters
IsBankrupt supplies the atomic test inside cascadeStep and therefore inside the monotone fixed-point iteration that defines the full cascade closure. It appears directly in ExtinctionCascadeCert and in the one-statement theorem that asserts monotonicity, termination in at most $n$ steps, and recovery-time scaling. The predicate therefore realizes the ledger-bankruptcy step of the Recognition Science cascade model on a finite species graph.
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