cascadeIterate
The k-fold iteration of the single-step cascade operator on a finite ecosystem. Modelers of recognition-driven extinctions cite it to close the bankruptcy propagation from an initial seed set. It is introduced by recursion on the step count, with the base returning the seed and each successor applying the removal of newly bankrupt species.
claimLet $E$ be a finite ecosystem on $n$ species equipped with baseline rungs and non-negative bond supports. For an initial live set $L$, the $k$-step iterated cascade is the function $I$ satisfying $I(0)=L$ and $I(k+1)$ equals the set of species in $I(k)$ that remain non-bankrupt after removal of supports from extinct neighbors.
background
An ecosystem is the structure whose fields are a baseline rung vector (all positive) and a support matrix (all entries non-negative) on a finite species index set. The single-step operator removes every species whose total rung, computed from the current live set, falls below the life threshold. The module formalizes extinction cascades as the monotone closure of this removal process on the finite graph, starting from a seed extinction and propagating via lost recognition bonds.
proof idea
The declaration is a recursive definition by pattern matching on the natural-number argument. The zero case returns the input live set verbatim. The successor case applies the single-step removal operator to the result of the previous iterate.
why it matters in Recognition Science
This supplies the cascade closure operator required by the subset, cardinality-monotonicity, and zero/succ lemmas, which in turn feed the ExtinctionCascadeCert structure and the one-statement extinction theorem. It realizes the fixed-point iteration of the module's structural cascade theorem, linking ledger bankruptcy dynamics to the phi-ladder rung thresholds.
scope and limits
- Does not compute numerical recovery times or rung values.
- Does not assume specific numerical supports beyond non-negativity.
- Does not extend to infinite species sets or continuous time.
- Does not prove convergence rate or fixed-point uniqueness.
- Does not incorporate external forcing or mutation terms.
formal statement (Lean)
154def cascadeIterate {n : ℕ} (E : Ecosystem n) (L : Finset (Fin n)) :
155 ℕ → Finset (Fin n)
156 | 0 => L
157 | k + 1 => cascadeStep E (cascadeIterate E L k)
158