Source-Interface C2–C3 Closure for a TEBAC Hilbert–Pólya Arithmetic Prime-Shift/GL(1) Heat Interface

This preprint presents Paper~D of the TEBAC Hilbert--P\'olya program. Its purpose is to prove source-interface C2--C3 closure for the fixed Paper~A arithmetic prime-shift/\(GL(1)\) heat interface. More precisely, the paper establishes weighted Hilbert--Schmidt completeness for the source prime-shift approximants and an exact CT defect operator-square representation for the CT kernel attached to the same source interface. The main result is a source-interface C2--C3 closure theorem: for each compact admissible seed interval \(J\), the source-level weighted Hilbert--Schmidt forcing holds uniformly on \(J\), and the CT defect kernel admits the positive representation \[ kCTˢʳᶜ(t) = Tr((Bˢʳᶜ)^* e^{-t Lsrc} Bˢʳᶜ) ≥ 0 almost every t>0. \] The term “closure” in this record is used only in the C2--C3 sense. The paper does not claim to prove the Riemann Hypothesis. It does not prove endpoint zero-mass, determinant comparison, the Hilbert--P\'olya operator realization, the squared Kre{i}n bridge, or the actual-to-Euler trace comparison. Those belong to separate companion components of the broader TEBAC RH/HP program. The paper is written to avoid dependence on RH, zero-location information, endpoint zero-mass, determinant comparison, or unproved C2--C3 geometric/analytic input. Its public claim is that Paper~D proves source-interface C2--C3 closure for the fixed Paper~A arithmetic prime-shift/\(GL(1)\) heat interface.