Finite Witness Detection of Selmer and Tate–Shafarevich Obstructions

We introduce a finite witness framework for detecting Selmer and Tate–Shafarevich obstructions associated to elliptic curves over Q. The framework replaces abstract cohomological descriptions with explicit finite witness structures equipped with a global closure invariant. We show that Selmer classes are precisely those admitting locally closing witnesses, while nontrivial Tate–Shafarevich elements correspond exactly to witnesses that fail to close globally. This yields a finite and computable classification of local–global obstructions within the Selmer-theoretic framework at fixed level, independent of analytic or asymptotic hypotheses beyond standard descent.