Brown-York Resolvent Energy: Dirichlet Obstruction and a Resolvent-Robin Boundary Condition

This paper analyzes the local boundary conditions under which the nonzero-scale Brown–York resolvent boundary term is functionally differentiable relative to the ordinary Brown–York Hamiltonian. The resolvent functional, introduced in the companion paper, is a reference-calibrated geometric surface functional whose zero-scale member is exactly the Brown–York quasilocal energy. With the induced boundary metric, lapse, reference data, and scale t held fixed, the deformation contributes a residual boundary variation proportional to δψ_t(S), where ψ_t(S) = tr[S(I − tS)⁻¹] − tr S and S is the spatial shape operator. Ordinary Brown–York Dirichlet data do not generally control this term, and fixing the mean curvature alone is insufficient outside the umbilic and exceptional sectors. The mixed resolvent–Robin condition — fixing the induced metric together with ψ_t(S) — removes the residual variation, and ψ_t is locally unique up to reparametrization among regular pointwise scalar data. The datum has exact geometric content: its deviation from the umbilic value at fixed mean curvature is an explicit resolvent-weighted measure of principal-curvature anisotropy, and for fixed t ≠ 0 the pair (tr S, ψ_t(S)) determines the unordered principal-curvature spectrum away from the exceptional branch t tr S = 2, where the datum degenerates to the constant −4/t. Under fixed, boundary-preserving assumptions the ordinary Brown–York shift sector is retained, and an axial shift yields the standard Brown–York angular momentum. No covariant Hamiltonian completion, deformed boundary stress tensor, or Einstein initial-boundary-value theorem is claimed. This work is a companion to "A Brown–York Resolvent Functional: Curvature Moments and Asymptotic Scaling" (doi:10.5281/zenodo.20647336). Version 3 retitles the paper (previously "Hamiltonian Boundary Conditions for Brown–York Resolvent Energy: A Resolvent–Robin Boundary Datum") and adds the exact anisotropy identity, extended spectral reconstruction, the exceptional degeneration, and the boundary-preserving shift-sector result.