module module moderate

`IndisputableMonolith.Ecology.ExtinctionCascadeFromLedgerBankruptcy`

The module defines the framework for extinction cascades triggered by ledger bankruptcy in ecological systems under Recognition Science. It sets the life ignition rung at phi to the nineteenth power and supplies iteration operators to track bankruptcy propagation through ecosystems. Theoretical ecologists applying the framework would cite these constructions for threshold analysis in biological networks. The module consists of definitions and basic monotonicity lemmas with no complex proofs.

claimThe module defines the life-ignition rung $Z_{\rm life}=\phi^{19}$ (cf. abiogenesis first crossing), the bankruptcy predicate on ecosystems, and the cascade step and iteration operators that track propagation of bankruptcy.

background

Recognition Science derives structures from the phi functional equation and the phi-ladder. This module imports the Constants module, whose doc-comment states the fundamental RS time quantum (RS-native) $\tau_0=1$ tick. It introduces the life ignition rung at $\phi^{19}$ as the point where life emerges on the phi ladder, together with ecosystem total rung sums, the bankruptcy predicate, and iterative cascade steps that reduce viable rungs.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies foundational definitions for the ecology section of Recognition Science and directly encodes the life ignition rung $\phi^{19}$ referenced from AbiogenesisFirstCrossing. It enables modeling of how bankruptcy at critical rungs triggers extinction cascades, extending the core forcing chain into ecological applications. No immediate downstream theorems are listed yet.

scope and limits

Does not include continuous differential equations for population dynamics.
Does not validate against observed extinction events in nature.
Does not model recovery or adaptation mechanisms after bankruptcy.
Does not specify the precise mapping from ledger entries to biological rungs.

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (24)

def Z_life L71
theorem Z_life_pos L73
theorem Z_life_gt_one L77
structure Ecosystem L85
def totalRung L97
theorem totalRung_pos L100
def IsBankrupt L112
theorem isBankrupt_antimono L121
def cascadeStep L137
theorem cascadeStep_subset L141
theorem cascadeStep_card_le L147
def cascadeIterate L154
theorem cascadeIterate_zero L159
theorem cascadeIterate_succ L162
theorem cascadeIterate_subset L165
theorem cascadeIterate_subset_initial L170
theorem cascadeIterate_card_monotone L178
def recoveryTime L187
theorem recoveryTime_pos L189
theorem recoveryTime_strict_mono L193
theorem deep_cascade_recovery_lower L204
structure ExtinctionCascadeCert L214
def extinctionCascadeCert L241
theorem extinction_cascade_one_statement L262