Poitou–Tate Descent and Finiteness of Sha[p∞] from Selmer Rank Bounds

This paper isolates the Gate-2 step in a modular Selmer descent program. Assuming an upper bound on Selmer rank and the existence of the corresponding number of independent rational points, we formally deduce finiteness of the p-primary Shafarevich–Tate group for elliptic curves over ℚ. The argument uses only standard Selmer theory, the Kummer map, and Poitou–Tate global duality. No Euler systems, Kolyvagin classes, or analytic input are used. The result is entirely formal once Selmer rank control is available. The Selmer rank bound assumed here is supplied by the Selmer-structure descent mechanism developed in the companion paper “Selmer-Structure Descent and Rank-2 Upper Bounds via Active Primes” (Zenodo DOI: https://doi.org/10.5281/zenodo.18344561). The present paper does not address the construction of rational points and assumes their existence as independent input. This work makes explicit a consequence of Selmer rank control that was previously inaccessible in the absence of Euler systems, thereby clarifying the arithmetic impact of Selmer-structure descent.