Double-Slit Phase is Just a Sign: Interference and Decoherence from Block-Alternating Products

We propose a mathematical model for the double-slit experiment based on the block-alternating infinite products A(K,d) and B(K,d). The two products play strictly separated roles: B(K,d) encodes the background Minkowski spacetime (the stage), invariant under observation, while A(K,d) encodes the propagating matter wave (the actor), whose coherence parameter d acts as a phase switch. The coherence parameter d=1 corresponds to alternating phase (AC structure, full interference), while d=1/2 and d=infinity are algebraically identical and correspond to constant phase (DC structure, no interference). Observation---defined as physical interaction with a detector that makes path information available---transitions the observed path from d=1 to d=1/2. Unobserved paths remain at d=1 and continue to interfere. At the screen (K=1/2, the null boundary), A(1/2,d)=1 for all d, equalising the amplitudes. The interference pattern is then determined by d alone, yielding |psi|^2 = 4cos^2(pi*Delta/2) (unobserved) or |psi|^2 = 2 (observed). Combined with the envelope W[B] proportional to sinc^2(beta) arising from the Minkowski structure of B, the standard formula I proportional to sinc^2(beta)*cos^2(pi*Delta/2) is recovered as a mathematical consequence. The measurement problem is resolved algebraically: decoherence is not a collapse of the wave envelope but a transition of the phase parameter d within the A structure. The spacetime stage B is unaffected by the act of measurement.