A proposal for an experimental test of phase coupling in the bicomplex extension of quantum mechanics

The bicomplex algebra B = C ⊗R C has recently been identified as a minimal commutative and associative extension of the complex scalar algebra underlying quantum mechanics. This structure contains a proper ideal J ∼= C that admits a complex-like exponential representation and leads to an additive phase relation between the complex and ideal sectors. The present work explores the consequences of this algebraic phase coupling under the hypothesis that it may become physically effective in suitable experimental conditions. In particular, we consider an optical setup involving a nonlinear medium, where mixed terms between complex and ideal components may arise. Within this framework, a specific, testable prediction is derived: a lateral displacement of the source in a double-slit experiment could lead to an amplified fringe shift, whose magnitude depends on the geometric parameters of the setup. The predicted scaling differs significantly from the standard geometric expectation of conventional wave optics. The analysis is explicitly conditional on the assumed physical realization of the bicomplex phase coupling. The proposed experiment therefore provides a direct way to test whether the additional algebraic structure has observable consequences or remains purely formal.