Critical Schwarzschild Scalar Dynamics

This preprint develops a machine-checked operator theory for the spherically symmetric massless scalar field in the Schwarzschild interior and classifies the self-adjoint radial realizations permitted at the curvature endpoint. Starting from the scalar action in regular ingoing coordinates, the analysis derives the radial wave operator, passes through an exact terminal coordinate, and obtains a half-line Schrödinger operator on L2((0,∞),dx) whose singular term is exactly Hardy-critical,−14x2. The minimal and maximal operators are constructed independently and their adjoint relation is proved. The horizon endpoint is shown to be limit-point, while the curvature endpoint is limit-circle for the full Schwarzschild potential. The resulting deficiency indices are (n+,n−)=(1,1), so the radial operator is not essentially self-adjoint and admits a genuine U(1) family of self-adjoint terminal laws. Every maximal-domain field is shown to possess a unique terminal trace u(x)=x(A+Blog⁡x)+o(x), together with a controlled derivative asymptotic. The corresponding Green form is obtained explicitly in the (A,B)coordinates. Zero terminal Green flux is proved not to select a unique extension. The full Schwarzschild potential nevertheless provides additional intrinsic structure. An exact positive zero-energy solution yields a nonnegative ground-state form for which QM[u]<∞⟺B=0. Thus finite full-potential ground-state form selects the no-log realization. The paper further proves that this realization is exactly the Friedrichs extension obtained by closing the original compact-core quadratic form: Kno-log=KF. The resulting theorem package therefore separates three distinct questions: classification of the terminal self-adjoint ambiguity, identification of the physical terminal trace, and intrinsic selection of a distinguished realization. The work is deliberately restricted to the fixed-background scalar test-field problem. It does not claim curvature regularization, a quantum-corrected Schwarzschild metric, gravitational backreaction, or spacetime continuation through r=0. The complete theorem chain is formalized in Lean 4 against a frozen paper-scoped revision of the Concordia formal development.