Hamiltonian Boundary Conditions for Brown–York Resolvent Energy: A Resolvent–Robin Boundary Datum

This paper analyzes the boundary conditions under which the nonzero-scale Brown–York resolvent boundary term is functionally differentiable relative to the ordinary Brown–York Hamiltonian at the local boundary-jet level. The Brown–York resolvent functional introduced in the companion paper is a reference-calibrated quasilocal surface functional whose zero-scale member is the Brown–York Hamiltonian energy. Writing q_t(A) = tr[A(I − tA)⁻¹] and ψ_t(A) = q_t(A) − tr A, and holding the induced boundary metric, lapse, reference data, and resolvent scale t fixed, the nonzero-scale deformation contributes the residual boundary variation −(1/8πG) ∫ N δψ_t(S) dA = −(1/8πG) ∫ N tr([(I − tS)⁻² − I] δS) dA. This term is not controlled by ordinary Brown–York Dirichlet data because the shape operator S contains the first normal boundary jet of the spatial metric. Consequently, under the stated local boundary-jet assumptions, the resolvent deformation is not functionally differentiable on the unrestricted Brown–York Dirichlet tangent space for generic curved boundaries and nonzero t. On the central resolvent component, and on the support of a nonzero lapse, the ordinary Dirichlet obstruction vanishes only in the totally geodesic case. Outside that component, the full algebraic zero set allows each principal curvature to satisfy tκ_i ∈ {0, 2}. The paper further shows that fixing only the mean curvature k = tr S is generically insufficient: the resolvent weights the principal-curvature variations separately, leaving a trace-free obstruction unless the boundary is umbilic or lies in the exceptional spectral sector k = 2/t. At t = 0 the additional resolvent obstruction vanishes identically. The mixed resolvent–Robin condition fixes the induced boundary metric together with the scalar datum ψ_t(S), pointwise on the spatial boundary. On this restricted local boundary-jet phase space the deformation contributes no additional uncanceled boundary variation, and the candidate resolvent Hamiltonian is differentiable relative to the Brown–York Hamiltonian. Away from critical points, the fixed-ψ_t condition defines a regular codimension-one restriction of the local shape-operator jet space. Within the regular pointwise scalar boundary-datum class, ψ_t is locally minimal. At a given boundary jet, any regular scalar datum whose fixed-level tangent space removes the obstruction has differential proportional to dψ_t. If this implication persists on a neighborhood where both differentials remain nonzero, the alternative datum is a local reparametrization of ψ_t. Stronger tensorial conditions can also remove the obstruction but are not minimal scalar data. The paper also records the umbilic and Schwarzschild reductions. Within the umbilic configuration sector, fixing ψ_t is locally equivalent to fixing the mean curvature on an invertible branch; at an umbilic background with arbitrary nearby variations, this equivalence is a first-order statement. The paper additionally presents a formal enlarged boundary phase-space ansatz using an independent boundary coordinate φ_t, its conjugate momentum Π_t, and the prospective constraint φ_t = ψ_t(S). This enlarged construction is not established as part of the gravitational covariant phase space. No well-posedness theorem for the Einstein initial-boundary-value problem, no complete covariant Hamiltonian construction, and no new universal prescription for gravitational charges is claimed. This work is a companion to "A Scale-Dependent Brown–York Resolvent Functional for Quasilocal Energy: Curvature Moments and an Asymptotic Scale Law" (doi:10.5281/zenodo.20647336).