Anomalous Norm Localization for $389.a1$ at $p=11$

The research route pivots here: instead of attacking the complex-to-p-adic comparison head-on, it turns back to examine the arithmetic properties of the two auxiliary primes 397 and 991 used to construct the Kurihara witness. Exact point counting gives #E(F₃₉₇)=374=34·11 and #E(F₉₉₁)=1045=95·11—this is no coincidence; the Kurihara/Kolyvagin condition a_ℓ-ℓ-1≡0 (mod 11) is equivalent to 11|#E(F_ℓ), so primes that can provide a discrete derivative certificate are automatically anomalous primes in the classical non-anomalous Mazur-Tate height theory. The document explicitly marks a NO_GO: the Burns-Kurihara-Sano non-anomalous pairing formula requires that the chosen primes do not divide the relevant reduction orders, which both primes here violate, and the Bley-Macias Castillo computational framework is restricted to p=3 and cannot be directly applied. The document instead uses a new lemma (Proposition 4.1)—under a tamely totally ramified degree-p extension, the norm quotient of a good reduction curve is isomorphic to the mod-p quotient group of the reduced curve—to precisely calculate two one-dimensional localized quotients, forming the localization matrix M_loc=[[1,2],[1,4]], with determinant 2∈F₁₁× being nonzero. This proves that the two rank-1 norm obstruction planes intersect transversally ((Q-2P)∧(Q-4P)=2(P∧Q)), strictly establishing the isomorphism E(ℚ)/11E(ℚ)≅⊕E(ℚ_ℓ)/NE(L_ℓ) (Theorem 6.1), independent of any complex leading term comparison. The document additionally calculates the exact index [E(ℚ):E^S(ℚ)]=390830 and proves simultaneous reduction surjectivity (J_S=1). The document explicitly separates the left-hand side (modular form initial term coefficient 6) and the right-hand side (localization determinant 2), only stating that their quotient is 3∈F₁₁×, without claiming that this 3 is the Mazur-Tate comparison constant—the formal gap left is P5-ANOM-BocCOMP₁₁⁽²⁾ (anomalous Bockstein comparison), which is the starting point of document 08.