The Mapping Inseparability (MI) Theorem in Zulfia Probability: Scope and Optogenetics Application

In Zulfia probability, a classical model (Ω, F, P) is extended by a binary admissibility switch∆ ∈ {0, 1} that records whether the experiment is released (∆ = 1, state A) or blocked (∆ = 0,state I). For an event η ⊆ Ω the resulting probability is PZ (η) = π P(η), where π = PZ (∆ = 1).We study the map (P(η), π) ↦ → PZ (η) and prove a Mapping Inseparability (MI) Theorem: theevent η and the switch ∆ are statistically dependent, with Cov(1η , ∆) = π(1 − π)P(η), exactlywhen 0 < π < 1 and P(η) > 0, and in that case neither factor can be recovered from PZ (η) alone.The inseparability is therefore conditional : it disappears in the boundary cases π ∈ {0, 1} andP(η) = 0. We show that observing ∆ restores identifiability of both factors, and we confirm theresults in a Monte Carlo experiment with a die. The paper is expository and makes no claim toreplace classical probability.