Norm-Selmer Core-Vertex Certificate for $389.a1$ at $p=11$

Following up on the localization matrix M_loc=[[1,2],[1,4]] from document 07, this document formally connects it to the Selmer group structure. Using the Kummer exact sequence and the already closed Sha(E/ℚ)[11]=0, it first obtains Sel₁₁(E/ℚ)≅E(ℚ)/11E(ℚ)≅F₁₁²—every mod-11 Selmer class is represented by a global Mordell-Weil point. For each auxiliary prime ℓ, a norm local condition map λ_ℓ is defined, and relative to the basis P, Q, it precisely calculates λ₃₉₇(aP+bQ)=a+2b and λ₉₉₁(aP+bQ)=a+4b. Applying the condition for 397 alone cuts down one dimension (leaving F₁₁(Q-2P)), and applying the condition for 991 alone also cuts down one dimension (leaving F₁₁(Q-4P)). When both conditions are applied jointly, since the matrix determinant 2≠0, the intersection is exactly zero—the dimension sequence goes precisely 2→1, 2→1, 1∩1→0, which in terms of cardinality is 121→11→1. Theorem 4.1 thus proves that the combined map λ₃₉₇⊕λ₉₉₁ is an isomorphism, meaning the two anomalous directions together constitute a complete rank-2 killing set for the mod-11 Selmer group—this is exactly the finite Selmer skeleton that the localized determinant in document 07 was detecting. The document also calculates the transversality of the two rank-1 faces (wedge product coefficient 2, non-zero), and formally records this step as P5-NORM-COREVERTEX₁₁=CLOSED_EXACT. The document honestly draws the boundary: what remains to be done is no longer "whether the two anomalous directions detect the rank-2 arithmetic space" (already proven, and it is an isomorphism), but rather constructing the correct anomalous Bockstein/extended height comparison map to transport this pure Selmer determinant to I²/I³ where the finite Mazur-Tate initial element resides.