Determinantal Kurihara–Semilocal Closure for $389.a1$ at $p=11$

The latest document in P5. Inheriting M_loc=[[1,2],[1,4]] (determinant 2) from document 08, this document formally constructs the finite anomalous norm-Bockstein operator B_N:V→W⊗I/I² (where V is the Selmer space, W is the direct sum of two local quotients, and I is the group ring augmentation ideal), giving the matrix [[X_397,2X_397],[X_991,4X_991]] relative to the basis. Pushing it to I²/I³ using exterior product multiplication, it precisely calculates the rank-2 determinant det(B_N)(P∧Q)=det(M_loc)(e_397∧e_991)⊗X_397X_991=2(e_397∧e_991)⊗X_397X_991—nonzero, primitive in the mixed augmentation direction. Meanwhile, the initial term on the modular form side (inherited from an earlier version) is 6X_397X_991; both fall exactly on the same mixed line 𝓛_397,991=F₁₁·X_397X_991, with a quotient of 3∈F₁₁×—the document explicitly states that this 3 "is not promoted to a canonical invariant," because changing the primitive root, generators, or determinant basis would rescale these coefficients by units; the only truly invariant facts are "generating the same line" and "being nonzero." The document invokes Chan-Ho Kim's semilocal theorem (under standard residual surjectivity, Manin constant, local-p-torsion, and Tamagawa hypotheses, the minimal-order nonzero mod-p Kurihara witness gives an isomorphism from the Selmer group to the local quotient) to verify that all hypotheses hold for 389.a1 at p=11, as well as the Castella-Sano 2026 refined nonvanishing theorem (reformulating the refined Kurihara conjecture in the form of Selmer complex determinants, closing the refined nonvanishing/integrality side independently of the complex rank-2 leading term issue)—three finite rank-2 facts are thus closed: P5-KUR-SEMILOC, P5-FIN-BocDET, P5-MIXEDLINE (all up to units). The document formally states that it does not prove that the finite operator B_N is exactly identical to the Bockstein regulator in the Burns-Kurihara-Sano or Nekovář Selmer complex determinant formalism under canonical normalization (P5-CANON-BocID remains OPEN), let alone that even with this identification, the complex rank-2 leading term comparison P5-CPLX-GPR is still unresolved.