Finite-Resolution Phase-Flow Coherence and the Structural Origin of the Weak Mixing Angle

This paper presents a finite-resolution formulation of Phase-Flow Coherence (PFC) that yields a deterministic and algorithmically closed definition of the weak mixing angle. In contrast to the fine-structure constant, which exhibits finite-resolution locking, the weak mixing angle is shown to belong to a distinct structural class: an information-optimal projection between partially coherent channels. As a consequence, finite locking is generically absent and scale dependence (running) emerges necessarily. All elements of the construction—discrete update channels, phase determination, coherence masking, coarse-graining, and Fisher-information minimization—are specified explicitly, without fitted parameters, renormalization assumptions, or external physical inputs. The work demonstrates that the qualitative difference between locked and running dimensionless parameters follows directly from finite-resolution structure, providing a unified geometric interpretation.