Nodal-Length Bounds for Surface Metrics with Isolated Non-Analytic Points

We prove a C_g sqrt(lambda) upper bound for nodal lengths on closed surfaces whose metrics are analytic away from finitely many isolated points and satisfy a summable quantitative analytic-scale condition near those points. The result includes an explicit smooth non-analytic torus family with Fourier coefficients exp(-|k|^alpha), 0 < alpha < 1. The theorem is a sufficient criterion for this structured class and does not resolve the corresponding upper bound for arbitrary smooth metrics.