Sharp Root Anti-Concentration via Projective Incidence and Ordered Root Laws

This paper answers the one-dimensional local root anti-concentration questions posed by Balcan, Pegden, and Sharma in the context of online optimization of piecewise-Lipschitz functions. For a homogeneous feature curve and coefficients whose density relative to the uniform law on a symmetric convex body K is bounded by A, we show that the worst-case interval-hitting constant equals A times a section-averaged projective incidence speed. For cube-supported coefficients, this speed is equivalent, up to universal constants, to the projective Lipschitz constant. This yields a sharp, dimension-free characterization and removes the previous \sqrt N loss. For monic degree-d polynomials under arbitrary coefficient laws, we prove that the interval-hitting constant is finite if and only if the ordered real-root laws have bounded densities, with a factor-d comparison that is sharp. Conditional and joint coefficient-space area formulas, together with a two-chart certificate, make this criterion verifiable for dependent and singular coefficient laws. We also give two graph-learning applications that complete the transition-to-regret chain. A cost-sensitive Gaussian-RBF harmonic classifier uses the projective incidence theorem and achieves expected regret \widetilde O((An^2D e^{BD}/\ell+1)\sqrt T). A common-offset polynomial-kernel model uses rigid translation of the ordered roots and achieves \widetilde O((qn^2κ+1)\sqrt T) regret, even when the induced coefficient law is singular in the ambient coefficient space.