A Rigorous Proof of the Sharp Quantitative Stability for the L1-Poincaré-Wirtinger Inequality on the Circle

We present a rigorous and self-contained derivation of the sharp L1-Poincaré Wirtinger inequality on the unit circle T, utilizing the median to measure oscillation. We establish that the sharp constant is 1/4, correcting the value proposed in the original conjecture, and characterize the extremizers as the manifold E of two-level step functions on complementary arcs of length 1/2. We then prove the sharp linear quantitative stability of this inequality: the L1-distance to the manifold of extremizers is bounded by exactly 1/4 of the deficit. The proof relies on fundamental tools, including the layer cake representation and the coarea formula, which are rigorously derived. The stability argument crucially dependsm on a selection principle, established rigorously via the Bath-Tub Principle, which ensures the commutativity of integration and maximization in this context.