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

In Zulfia probability, a classical model (Ω, F, P) is extended by a binary admissibilityswitch ∆ ∈ {0, 1} that records whether the experiment is released (∆ = 1, state A) orblocked (∆ = 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 MappingInseparability (MI) Theorem: the event η and the switch ∆ are statistically dependent,with Cov(1η, ∆) = π(1 − π)P(η), exactly when 0 < π < 1 and P(η) > 0, and in thatcase neither factor can be recovered from PZ (η) alone. The inseparability is thereforeconditional : it disappears in the boundary cases π ∈ {0, 1} and P(η) = 0. We showthat observing ∆ restores identifiability of both factors, and we confirm the results ina Monte Carlo experiment with a die. The paper is expository and makes no claim toreplace classical probability.