Online algorithm configuration needs a local guarantee that a random transition boundary does not cross a short parameter interval. Existing polynomial bounds assume a bounded joint coefficient density and are quadratic in the degree. Recent work asks for natural necessary and sufficient coefficient conditions, polynomial guarantees under independent bounded marginals, and a normalization for Pfaffian boundaries. We address these questions through coefficient flux. For an affine boundary $a(t)+\langle X,u(t)\rangle=0$ with $\|u(t)\|_2=1$, we identify two Radon quantities, a slice density and a tangential first moment. Their maximum is exactly the best root-hitting constant uniform over all normalized affine boundaries. If the coefficient law is dominated by an isotropic log-concave law, both quantities are universally bounded. This gives a dimension-free bound controlled only by the total variation of $a$ and the spherical length of $u$. For monic degree-$d$ polynomials, the resulting joint-density bound is linear rather than quadratic in $d$. For independent coefficients with marginal densities at most $\kappa$ and support radius $R$, a Rogozin–Ball argument gives $O(\kappa R d^{3/2})$, and we construct product laws attaining this rate. Consequently, online losses with random monic transitions have sharp dispersion coefficient $\Theta(\kappa R d^{3/2})$ and $\widetilde O(\sqrt T)$ regret. Marginal density without independence can force a deterministic root. Finally, the normalized feature curve $F/\|F\|_2$ supplies a sufficient Pfaffian normalization and yields general dispersion and regret bounds. For exponential-kernel graph learning, the previously hidden conditioning is exactly controlled by the exponent range.
Keywords: anti-concentration, data-driven algorithm configuration, online learning, log-concave distributions, dispersion, piecewise structure