Conjugate-Half Mixing Produces a Differential Interference Correction

T101 applies the conjugate-half mixing framework of T100 to a two-output interference experiment and derives the resulting structure of observable probability corrections. While T100 established that weak mixing between the observable sector and its conjugate counterpart produces first-order corrections constrained by total-probability conservation, T101 specializes those corrections to interference measurements and determines the form that any physically admissible deviation must take. The starting point is the standard two-output Born readout \[P_+⁽⁰⁾=cos^2θ,_-⁽⁰⁾=sin^2θ,\] where \(θ\) parameterizes the interferometer configuration and \(φ\) denotes the relative phase. From T100, observable probabilities in the presence of weak conjugate-half mixing satisfy \[P_±ᵒᵇˢ=P_±⁽⁰⁾+η Δ_±+O(η^2),\] together with the conservation law \[Δ_++Δ_-=0.\] The theorem shows that, in a two-output experiment, conservation immediately forces an antisymmetric correction structure: \[Δ_+=-Δ_-=f(θ,φ),\] yielding \[P_+ᵒᵇˢ=cos^2θ+η f(θ,φ)+O(η^2),\] \[P_-ᵒᵇˢ=sin^2θ-η f(θ,φ)+O(η^2).\] Thus, conjugate-half mixing cannot appear as an unconstrained additive deviation. Any increase in one output channel must be accompanied by an equal decrease in the other. The theorem next derives positivity constraints on the correction function. Since probabilities must remain between zero and one, the correction cannot remain finite at the interference endpoints \[θ=0,θ=π/2,\] where one channel has unit probability and the other has zero probability. Consequently, \[f(0,φ)=0,\!(π/2,φ)=0,\] or must be suppressed sufficiently rapidly near those points. A minimal admissible envelope satisfying these constraints is \[α(θ)=sin(2θ),\] which vanishes at both endpoints while remaining positive throughout the interior of the interference interval. Incorporating the leading phase dependence yields the simplest structured correction \[f(θ,φ)=c\,sin(2θ)cosφ,\] where \(c\) is a coupling parameter describing the strength of the projected conjugate-half contribution. Substitution gives the minimal interference-correction model \[P_+ᵒᵇˢ=cos^2θ+η c\,sin(2θ)cosφ+O(η^2),\] \[P_-ᵒᵇˢ=sin^2θ-η c\,sin(2θ)cosφ+O(η^2).\] The theorem therefore predicts that any observable signature of conjugate-half mixing must appear as antisymmetric redistribution between interference outputs rather than as excess total probability. The characteristic experimental signal is \[δ P_+=-δ P_-,\] with the correction vanishing near the interference endpoints and reaching maximal magnitude in the interior of the interference pattern. T101 serves as the interference-level realization of the conjugate-half mixing framework. T100 established the general conservation law governing first-order Born corrections. T101 translates that abstract constraint into a concrete interference prediction, showing that weak conjugate-half mixing manifests as differential redistribution of probability across interference channels while preserving total probability to first order.