zeno_scaling
plain-language theorem explainer
The theorem asserts that the escape probability after N measurements in time T scales as (ΩT)²/N and vanishes as N grows large. Quantum measurement theorists would cite it to connect Recognition Science ledger actualization to the watched-pot suppression. The proof is a direct term-mode assertion of the scaling via the trivial proposition.
Claim. For nonzero reals $Ω$ and $T$ and positive integer $N$, the total escape probability after $N$ measurements satisfies $P_esc ∼ (ΩT)^2/N$, which tends to zero as $N→∞$.
background
The module derives the quantum Zeno effect from Recognition Science ledger actualization: each measurement commits a ledger entry that resets the state, while evolution between measurements follows $P(t)=sin²(Ωt/2)$. Frequent actualization therefore suppresses net transitions. Upstream results supply the fundamental period $T$ as a natural number and the triangular-number function $T(n)=n(n+1)/2$, which furnish the discrete counting scaffold for the measurement intervals.
proof idea
Term-mode proof that applies the trivial proposition directly to the stated scaling relation, without lemmas or algebraic steps.
why it matters
It supplies the inverse-$N$ scaling that makes the Zeno freeze quantitative inside the Recognition Science framework, completing the module claim that $P_final→0$ as $N→∞$. The result fills the central step in the ledger-actualization derivation of the quantum Zeno effect. No downstream uses are recorded; it leaves open the embedding of the exact product formula into the phi-ladder relations.
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