The Radial Acceleration Relation as Primitive Closure Rendering: A Conditional TSCT Derivation of Galactic Phenomenology from the IR Modular Horizon

The Radial Acceleration Relation (RAR) is one of the sharpest empirical regularities in galaxy dynamics: across thousands of rotation-curve points in rotationally supported spirals, the observed centripetal acceleration gₒbs tracks the baryonic Newtonian acceleration gbar via gₒbs = gbar / (1 - exp (-sqrt (gbar / g†) ) ) with characteristic scale g† ≈ 1. 2 × 10⁻¹⁰ m s⁻². This paper derives this relation from first principles within Two-Sided Closure Theory (TSCT), with no free parameters. The derivation uses four ingredients, all previously established in the TSCT stack. (1) The idempotent closure projector E² = E forces the semigroup evaluation e^-χQ = E + e^-χQ in exactly two lines of algebra, where Q = I - E is the residual projector. (2) The IR modular horizon of the cosmological closure pair supplies the acceleration scale aH = cH₀/2π ≈ 1. 08 × 10⁻¹⁰ m s⁻². (3) The Fisher-Study bridge metric, being quadratic in the bridge operator, forces the exposure parameter to be the square root χ = sqrt (gbar / aH) rather than gbar / aH. (4) The Metric Rendering Axiom identifies gbar = η (χ) gₒbs where η (χ) = 1 - e^-χ is the rendered fraction. The result gₒbs = gbar / (1 - exp (-sqrt (gbar/aH) ) ) follows immediately, with g† = aH = cH₀/2π containing no fitted parameters. The 10% offset between the predicted aH and the empirically measured g† lies within current uncertainties in H₀, distance calibration, and stellar mass-to-light ratios. Immediate consequences include: the deep-RAR limit gₒbs → sqrt (gbar · aH) ; the baryonic Tully-Fisher law v⁴ = GMb aH with normalisation fixed by aH; an apparent isothermal dark halo profile ρₐpp ∝ r⁻² for finite baryonic mass; a structural scaling law for galaxy clusters involving a multi-channel effective acceleration aeff = aH / κ²ₑff; and a no-slip lensing prediction gₗens = gdyn for primitive scalar radial screens. The result is not MOND. No modification of inertia or gravity is invoked. The interpolation function follows from the primitive survival law of the idempotent conditional expectation residual. The apparent dark halo is unrendered metric geometry, not a new particle species. Tier: P (proved) conditional theorem under the primitive scalar radial closure-screen hypothesis. Explicit kill conditions are stated in Section 11. Companion paper: The Screen-Domain Theorem identifies the physical systems that satisfy the primitive scalar hypothesis. Part of the TSCT publication series. See also: Deposit D1 (Fibonacci Engine), Deposit D2 (Bridge Unification), Deposit D3 (Cosmology).