A Finite Poitou--Tate Detector Criterion for a Kummer--Selmer Bridge

This preprint is the first core module of the TEBAC determinant--Selmer program toward a modular analysis of the Birch--Swinnerton--Dyer problem over \(\mathbb{Q}\). Its scope is deliberately restricted: it does not claim a proof of the Birch--Swinnerton--Dyer conjecture. The paper proves a finite-dimensional detector criterion for a Kummer--Selmer bridge. Let \(S^{\mathrm{red}}\) be a reduced Selmer receptacle, and let \(\rho: K \to S^{\mathrm{red}}\) and \(\kappa: M \to S^{\mathrm{red}}\) be injective rational realizations of two finite-dimensional spaces \(K\) and \(M\). Given a finite detector family \(D = (d_1,\ldots,d_N) \subset (S^{\mathrm{red}})^\vee\) which separates \(U = \rho(K) + \kappa(M)\), one forms the response matrices \(A_K = \bigl(d_i(\rho(e_j))\bigr)_{i,j}\) and \(A_M = \bigl(d_i(\kappa(P_j))\bigr)_{i,j}\). The main theorem proves that if \(\operatorname{col}(A_K) = \operatorname{col}(A_M)\), then \(\rho(K) = \kappa(M) \subset S^{\mathrm{red}}\), and consequently \(K \simeq M\). The argument is purely finite-dimensional after the detector-separation hypothesis has been supplied. In the intended arithmetic application, the detector rows are to arise from admissible Poitou--Tate dual local conditions. The remaining arithmetic burden for subsequent BSD core modules is explicitly identified: construct the reduced Selmer receptacle, prove detector separation from finite local conditions, establish injectivity of the central and Kummer realizations, and prove the central image theorem without importing the BSD rank formula or leading-coefficient formula. This is a claim-safe preprint module. It is not a journal-refereed publication and is not a standalone proof of BSD. Broader BSD consequences are conditional on later core modules establishing the central image theorem, local N\'eron-symbol height calibration, residual Sha/Cassels--Tate/Tamagawa routing, and the central determinant identity.