A Shared-Marginal FRW–Vaidya Junction: First-Jet Closure and an Acceleration-Selected Energy-Condition Theorem

We match a spatially flat Friedmann–Robertson–Walker (FRW) spacetime to a generalized-Vaidya exterior on the child’s apparent horizon 𝑅𝐴 = 1/𝐻. Continuity of areal radius and Misner–Sharp mass makes the junction marginal on both sides, 𝑅Σ = 2𝑚. For the timelike interval−1 <𝑤tot <1/3, the Darmois conditions admit a non-comoving shell-free solution and fix all first-order exterior data: 𝑚,𝑅 = 6− 5𝛼 2(2− 𝛼) , 𝑚,𝑣 = 4𝛼(1− 𝛼) (2− 𝛼)2 , 𝑚,𝑢 =−𝑚,𝑣 , 𝑓,𝑅 = 4𝐻(𝛼− 1) 2− 𝛼 , where 𝛼 = 𝑅𝐴 = 3(1 + 𝑤tot)/2. Standard positive-energy transmission exists precisely for 0 < 𝛼 < 1, the accelerating era, and vanishes at the admissible boundary 𝛼= 1. With the normal directed from the child toward the exterior, that transmission is outward from the child. For 1 <𝛼<2, a shell-free exterior violates the null-stream energy condition. Allowing a thin shell does not repair it: shared marginality forces zero surface density, while bulk positive energy forces negative surface pressure and violates the shell energy conditions. The junction reproduces the Hubble-mass coefficient derived in the companion constraint paper, 8𝜋𝐺2 𝑐3 𝜌𝑚 + 4 3 𝜌𝑟 + 𝐾 , over its admissible domain. The match is a first-order on-tube result; 𝑚,𝑅𝑅 and the global exterior remain free