Quantitative Stability of the L1-Poincaré-Wirtinger Inequality: Information Geometry, Hamiltonian Duality, and the Geometric Uncertainty Principle

[This is a creative exploration of a concept with AI -it should not be taken as correct mathematics.]

Correction and provenance notice (5 August 2026). This published record is retained with its original files and authorship. The common-half-arc selection equality used in the underlying L1 stability proof line is false: an exact rational example has levelwise optimum 4/5 and best common-half-arc cost 9/10. The sharp coefficient 1/4 has a different coarea and nested-core proof in the maintained all-versions record, DOI 10.5281/zenodo.17010427. The wider information-geometric, Hamiltonian, and uncertainty-principle bridges are not audited by this notice and require independent proofs. This record is obsoleted and corrected by 10.5281/zenodo.17010427.

The quantitative stability of the L1-Poincaré-Wirtinger inequality on the unit circle is proved, demonstrating that a function's L1-distance to extremizers is directly related to the square root of its deficit. The methodology employs information geometry, interpreting the deficit as a functional on a Riemannian manifold of probability densities. A duality is established between geometric stability (governed by a position-based Hamiltonian and gradient flow) and dynamical stability (governed by a momentum-based Hamiltonian and unitary flow), whchi is then unified through Noncommutative Geometry and implies a Geometric Uncertainty Principle. This shows a fundamental trade-off between maximal geometric and quantum stability.