Critical Schwarzschild Scalar Dynamics

We develop a machine-checked operator theory for the spherically symmetric massless scalar field in the Schwarzschild interior and classify the self-adjoint radial realizations permitted at the curvature endpoint. Starting from the scalar action in regular ingoing coordinates, the radial wave operator is transported by an exact terminal coordinate to a Schrödinger-type expression on the positive half-line whose singular term is the Hardy-critical inverse-square potential −1/(4x²).

The minimal and maximal differential operators are constructed independently and the adjoint problem is closed by weak second-derivative regularity. The horizon endpoint is limit-point and the curvature endpoint is limit-circle for the full Schwarzschild potential. The genuine positive and negative deficiency spaces both have complex dimension one, so the radial operator is not essentially self-adjoint and its self-adjoint realizations form a U(1) family.

Every maximal-domain field has a unique terminal expansion u(x)=√x(A+B log x)+o(√x), together with an independently controlled derivative asymptotic. These traces give the exact maximal Green form as the canonical Darboux pairing of A and B. Vanishing terminal Green flux alone does not select a unique member of the self-adjoint family.

The full Schwarzschild potential possesses an exact positive zero-energy solution and an associated nonnegative ground-state form. Finite full-potential form energy is equivalent to the no-log condition B=0. Requiring that criterion on an entire self-adjoint domain selects one realization, and completion of the original compact-core quadratic form identifies that realization with the Friedrichs extension. The selected operator is nonnegative and has trivial zero-energy kernel.

The result is deliberately test-field scoped: it classifies the terminal self-adjoint domain ambiguity and derives a canonical same-operator form selection. It does not claim a quantum-corrected metric, gravitational backreaction, or geometric singularity resolution.