cascadeIterate_zero
The base case of the cascade iteration states that zero applications of the extinction operator to an initial seed set L in a finite ecosystem E returns L unchanged. Modelers of recognition-bonded extinction chains on finite species graphs cite this when initializing the monotone closure computation for ledger-bankruptcy cascades. The proof reduces directly to the recursive definition of the iterate via reflexivity.
claimLet $E$ be an ecosystem on $n$ species with baseline rungs and non-negative bond supports, and let $L$ be a finite set of species indices. Then the zero-fold iteration of the cascade operator satisfies $cascadeIterate(E, L, 0) = L$.
background
An ecosystem is a structure on $n$ species whose baseline rung map and support matrix satisfy positivity and non-negativity, respectively. The cascade iteration is defined recursively: at step zero it returns the seed set $L$, and at step $k+1$ it applies the single-step cascade operator to the result of step $k$. This supplies the base case of that recursion, ensuring the iteration begins from the initial extinct species as described in the module's account of ledger bankruptcy driving rung drops below the life threshold.
proof idea
This is a one-line wrapper that applies reflexivity to the base clause of the recursive definition of cascadeIterate.
why it matters in Recognition Science
This base case anchors the structural cascade theorem on finite recognition graphs. It supports the proof that the cascade closure is monotone and contains the seed set, as described in the module's account of ledger bankruptcy driving rung drops below the life threshold. No downstream uses are recorded yet.
scope and limits
- Does not establish monotonicity of the cascade operator.
- Does not bound the number of iterations needed for closure.
- Does not relate the cascade to specific rung values or the phi ladder.
formal statement (Lean)
159theorem cascadeIterate_zero {n : ℕ} (E : Ecosystem n) (L : Finset (Fin n)) :
160 cascadeIterate E L 0 = L := rfl
proof body
161