Zulfia Hypothesis Testing: Decomposition Framework with Zulfia Mapping Operator

AbstractClassical hypothesis testing has, for over 100 years, relied on two foundational but un-verified assumptions: (i) distributional assumptions imposed a priori on the null hypothesisH0, and (ii) the existence of exact null states where the deviation ∆ = 0 holds with pos-itive probability. Empirical data, however, consistently shows that no observation returnsexactly to a model-predicted null state; all measurements deviate from model predictions.This paper introduces Zulfia Hypothesis Testing (ZHT), a decomposition framework thateliminates these imposed assumptions and replaces them with a structurally generated in-ference procedure.We define the Zulfia Mapping Operator MZ : H × R+ → Z with a forward-onlyproperty, enabling the fundamental decomposition:Z(t) = MZ [η(t0)] + ∆(t)where η ∈ H is the hypothesis-supporting component and ∆(t) ∈ D is the deviation com-ponent. Theorem 2.2 establishes that ∆(t)̸ = 0 ∀ t > t0 under the non-invertibility of MZ .This result formally eliminates the 100-year assumption of exact null states and providesstructural inference via generated decomposition rather than imposed distributions.