Newton Support Functions and Metric Completion of Singular Conformal Metrics at Corners

We study singular conformal metrics \(g=Fg_0\) near codimension-two corners, where \(F\) is a finite positive sum of monomial singularities. The competing orders are encoded by the homogeneous Newton support function \(\mathcal H(p,q)=\max_j(pa_j+qb_j)\). We prove the sharp projective accessibility criterion \(\mathcal H(p,q)<2\min\{p,q\}\), and deduce that the corner lies at finite metric distance exactly when \(\max_j(a_j+b_j)<2\). At every accessible corner, all normal approaches over a fixed corner point determine one canonical completion point. The induced boundary metric is locally bi-Lipschitz to a snowflake of the smooth corner metric, yielding an explicit Hausdorff-dimension formula. For rational exponent data, the Newton decomposition is also realized by compatible rooted and monoidal modifications. The metric conclusions themselves require only positivity, bounded coefficients, and uniform ellipticity.