module module moderate

`IndisputableMonolith.Ecology.ExtinctionCascadeFromLedgerBankruptcy`

This module defines the life-ignition rung at φ^19 and supplies predicates and iteration operators for modeling extinction cascades triggered by ledger bankruptcy in RS ecosystems. Theoretical ecologists working with Recognition Science population ladders would cite these objects when analyzing threshold-driven collapses. The module is a pure definition block that assembles rung assignments, monotonicity facts, and cascade operators from the imported Constants module.

claim$Z_ {life} = φ^{19}$. An ecosystem is bankrupt when its total rung drops below this threshold. Cascade steps iteratively remove bankrupt components while preserving the remaining rung structure.

background

The module sits in the ecology domain of Recognition Science and imports the fundamental time quantum τ₀ = 1 tick from IndisputableMonolith.Constants. It introduces the life-ignition rung Z_life = φ^19, cross-referenced to AbiogenesisFirstCrossing, together with the auxiliary notions totalRung, IsBankrupt, cascadeStep, and cascadeIterate. These objects encode the phi-ladder representation of biological populations and the bankruptcy predicate that initiates cascade dynamics.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the ecological layer that connects the Recognition forcing chain to abiogenesis and extinction modeling. It directly implements the life-ignition rung Z_life = φ^19 cited in AbiogenesisFirstCrossing and provides the cascade machinery used by downstream population-collapse arguments. No parent theorems are recorded in the current dependency graph.

scope and limits

Does not derive the numerical value φ^19 from the forcing chain.
Does not simulate concrete ecological networks or species interactions.
Does not link bankruptcy cascades to physical constants beyond the rung ladder.

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