The Screen-Domain Theorem: When Is a Gravitating System a Primitive Scalar Radial Closure Screen?

The TSCT Radial Acceleration Relation (RAR) theorem is a conditional result: given a primitive scalar radial closure screen, the galaxy rotation-curve formula gₒbs = gbar / (1 - exp (-sqrt (gbar/aH) ) ) follows in two lines of algebra. This paper specifies the domain conditions under which the primitive scalar hypothesis is satisfied — converting the question "does TSCT predict the RAR? " into "does TSCT predict that mature rotationally supported spirals are in the primitive scalar regime? ", and answering yes on five operationally specifiable grounds. Five screen-domain conditions are identified: (D1) Stationarity: the closure-defect operator ∂ₜ Δ̂ ≈ 0 on observational timescales. Satisfied by settled, non-merging systems. (D2) Axisymmetry: U_θ, Δ̂ = 0 for the angular rotation operator. Violated by strong bars and severe lopsidedness. (D3) Rotational support: v²_φ >> σ²ᵣ, σ²ᵦ. Violated by galaxy clusters and ellipticals. (D4) Environmental decoupling: the local closure pair does not coherently couple to external bridge structure. Partially violated by satellite galaxies in strong tidal fields. (D5) Rank-1 first residual: the first residual radial quotient R^ (1) ᵣ ≅ ℂRᵣ is scalar. The algebraic content that D1-D4 jointly render plausible. D1-D4 are operationally testable from kinematic data. D5 is the algebraic content; the paper argues that D1-D4 jointly make D5 the natural and minimal residual structure, though the algebraic proof remains an open task. The framework correctly classifies the empirical domains where the RAR holds tightly (mature isolated spirals) and where it does not (galaxy clusters, strong-bar systems, merging galaxies, satellite systems). Five concrete predictions follow: (1) clusters show RAR-like behaviour with an effective scale aeff = aH/κ²ₑff; (2) RAR scatter correlates with bar strength (m=2 Fourier amplitude) ; (3) merger remnants show transient RAR violations that heal as systems re-equilibrate; (4) satellite galaxies show RAR scatter correlated with host proximity; (5) no MOND-like effects are predicted in the Solar System regardless of acceleration regime, because the Solar System fails D4 (it is a subsystem within the Milky Way's screen, not a screen in its own right). The result is the operational completion of the RAR conditional theorem. Together, the RAR paper and this paper form a complete, falsifiable claim: the RAR holds where and because these five conditions hold. Each failure mode produces a structurally distinct effective law rather than a mere anomaly. Part of the TSCT publication series. Companion paper: The Radial Acceleration Relation as Primitive Closure Rendering. See also: Deposit D1 (Fibonacci Engine), Deposit D2 (Bridge Unification), Deposit D3 (Cosmology).