A Scale-Dependent Brown–York Resolvent Functional for Quasilocal Energy: Curvature Moments and an Asymptotic Scale Law

This paper introduces a scale-dependent Brown–York resolvent functional for quasilocal energy. The construction identifies the Brown–York quasilocal energy as the zero-scale member of an analytic resolvent family built from the physical and reference shape operators of a closed spacelike two-surface. For nonzero scale, the construction is treated as a geometric, reference-calibrated quasilocal surface functional; it is not claimed to be the Brown–York Hamiltonian boundary charge. The Taylor coefficients of the family are the reference-subtracted power sums of the shape operators. The first coefficient is the Brown–York mean-curvature term. The second contains a reference-subtracted mean-curvature-square term together with an extrinsic-Gauss-curvature correction; for a Euclidean isometric reference, the Gauss equation relates this correction to the ambient sectional curvature, making precise the connection with the Willmore integrand and with the scalar mean-curvature-square term appearing in the time-symmetric Hawking mass. The second coefficient is not identified with either the Willmore functional or the Hawking mass itself. The paper proves Brown–York normalization, reference calibration, analyticity below the resolvent radius, and fixed-surface local positivity when the Brown–York energy is positive. It also establishes a large-sphere asymptotic scale law: under the stated asymptotically flat hypotheses, if t(r)/r → α with |α| < 1, the resolvent energy tends to M_ADM/(1−α)²; the usual ADM normalization is recovered throughout the sublinear sector t(r)/r → 0. The paper also explains why the usual Brown–York mean-curvature comparison does not by itself imply a general monotonicity theorem in the scale parameter, distinguishing the fixed-surface algebraic resolvent family from inverse mean-curvature flow. Round spheres in the time-symmetric Schwarzschild slice are evaluated in closed form, and the construction is positioned relative to Brown–York energy, Hawking mass, Anco's mean-curvature-flow family, and the classical ADM asymptotic limit. The Hamiltonian boundary-condition analysis for the nonzero-scale resolvent term is carried out in the companion note, "Hamiltonian Boundary Conditions for Brown–York Resolvent Energy: A Resolvent–Robin Boundary Datum" (doi:10.5281/zenodo.20650879).