Rational Probes of Spectral Geometry in Hermitian Matrix Models

Abstract

The planar loop equations of a Hermitian one-matrix model leave finitely many low moments undetermined. Hankel positivity constrains these moments but does not reveal how a fixed multicut family is embedded in moment space. We introduce a finite-plane diagnostic using the Cauchy kernels $f_n(x)=(z_n-x)^{-1}$ which we use as spectral probes. Their Gram matrix is the positive-semidefinite Pick matrix $P_{mn}$, whose positivity defines a finite-node bootstrap analogous to the Hankel bootstrap. For a fixed regular multicut topology, the nonbranching double zeros of the spectral discriminant, which we call \emph{dressed saddles}, impose transverse constraints on the corresponding moment-space locus, while its remaining directions are filling fractions. Writing $z=E+iη$, $E$ selects a spectral region and $η>0$ sets a continuous resolution scale. We use $P(z,z)$ as a finite-plane spectral response and interpret $\nabla_{\mathbf m}P(z,z)$ as a moment-space susceptibility. Near a regular real dressed saddle, this susceptibility is enhanced as $η^{-2}$ and aligns with a conormal to the filling-fraction manifold, whereas the leading enhancement cancels along tangent deformations. Responses near several independent saddles can therefore reconstruct the conormal space and, through their common kernel, its tangent space. Combining Pick positivity with reality and analyticity of the resolvent, we obtain alternative analytic derivations of known planar results in quartic and sextic models. The asymmetric quartic makes the one-dimensional filling-fraction geometry explicit, while sextic models exhibit independent conormal directions. These results clarify how local consistency conditions constrain the low moments before global period matching and the equilibrium variational inequality select the equilibrium measure.