The Radial Scalar Primitive Quotient Theorem: Closing the Screen-Domain D5 Gap in the TSCT RAR Branch

The TSCT RAR theorem derives the empirical radial acceleration relation from a primitive scalar radial closure screen, conditional on six hypotheses. The Screen-Domain companion paper identified hypothesis D5 — that the first residual closure quotient is rank-1 — as algebraically open. This paper closes that gap with two corrections to the Screen-Domain paper. First, the correct target is not that the full annular screen algebra is rank-1 (that would overstate the claim) but that the scalar radial semigroup quotient is rank-1. Second, the no-slip lensing proposition is corrected: the dynamical RAR requires scalar quotient primitivity only; no-slip lensing requires the stronger full-screen condition. Under stationarity (D1), axisymmetry (D2), rotational support (D3), environmental decoupling (D4), radial source-universality, and first-order tensor trace-orthogonality, the only surviving primitive residual channel is the radial bridge-amplitude line. Hence R(1)r,scal ≅ ℂ·R_r. A scatter expansion quantifies RAR deviations as corrections tied to named violations of D1–D4, explaining bars (D2), mergers (D1), satellites (D4), and cluster failures (D3) in a unified framework. A replacement paragraph for the Screen-Domain paper's D5 section is provided.