What Newton Did for Gravity, Stuart-Landau Does for Tau: A Phenomenological Law for Alzheimer's Pathology

When Newton watched an apple fall, he asked not why but how—and discovered a law that predicts planetary motion without explaining gravity's cause. We propose an analogous approach for Alzheimer's disease. The Stuart-Landau equation, the canonical normal form for symmetry-breaking phase transitions, provides a phenomenological law for tau aggregation dynamics: the order parameter |A| (pathological tau concentration) grows or decays depending on whether the control parameter μ (net aggregation drive) is positive or negative. This framework, rooted in Shimizu's 1972 instability theory and the Kuramoto tradition, achieves quantitative consistency with E2814 anti-tau antibody clinical data (μ ≈ −0.23 month−1 under treatment) and generates testable predictions for ongoing combination trials. Like Newton's law, which enabled spacecraft navigation centuries before Einstein explained gravity, Stuart-Landau dynamics offer a path from empirical biomarker data to rational drug design—regardless of whether we fully understand the underlying molecular mechanisms.