Z_of_window_zero
plain-language theorem explainer
The lemma shows that the integer count of ones in any empty window is identically zero. Workers on finite patterns and cylinder sets in Recognition Science streams cite this base case when building periodic extensions or inducting on window length. The proof is a one-line simplification that unfolds the summation definition over the empty Fin 0 type, which is zero by convention.
Claim. Let $w$ be any function from the empty set Fin 0 to Bool. Define the window sum $Z(w) :=$ sum over $i$ in Fin 0 of the indicator that returns 1 when $w(i)$ is true and 0 otherwise. Then $Z(w) = 0$.
background
The Streams module treats finite windows and their periodic extensions. Pattern n is the type of length-n windows, realized as maps from Fin n to Bool. Z_of_window sums the indicators of true entries to produce the integer count of ones in the window. The module doc states the setting as periodic extension and finite sums. Upstream results supply the Pattern abbrev from Measurement and the summation definition itself; the remaining upstream structures (nucleosynthesis tiers, ledger factorization, spectral emergence) supply the broader Recognition Science context but are not invoked here.
proof idea
The proof is a one-line wrapper that applies simp to the definition of Z_of_window. The sum over the empty type Fin 0 reduces directly to zero by the standard convention for empty summation in Mathlib.
why it matters
This base case closes the n=0 instance for the window functional, supporting downstream constructions of cylinder sets and periodic extensions (Cylinder_zero, extendPeriodic8_zero) within the same module. It supplies the zero-length anchor needed for finite-sum arguments in Recognition Science streams without touching the forcing chain, J-cost, or phi-ladder. No open questions are addressed; the result is fully discharged.
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