Lemma 8 ([Complementary local conditions]). If paired perfect local maps U→L, U'→L' (possibly only over a localized test ring after the limit) have a paired restriction nullhomotopy and induce L'/U'≃U^∨[-2] , the two Selmer complexes are …

Lemma 7 ([Finite models and matrix limits]). Consider diagrams of the global, local, and mapping-fiber cochain complexes above over finite local Artin coefficient rings of 2-power characteristic, with modules finite free (and compatible …

Corollary 6 ([The full-index discrepancy]). Under these assumptions, the single Hilbert-class trace satisfies X(E)+X(E^k) =v₂\!([E(K):ℤP]² c_E²\,∏_{ℓc_ℓ}(E)² #(E/K) ). Here [E(K):ℤP] is the full group index, including torsion.

Dualize G_c→C(U)→U and use the localization adjunction to identify the dual fiber. The comparison boundaries for nested local conditions with compatible orthogonality data are localization adjoints by the same triangles. The local equivalence can be checked on residue fields by perfectness; when the local pieces are split free in cohomology, the invariant restriction morphism is tested just in total degree two. We specify nullhomotopies or their existence from degree/amplitude or unramified factorizations when used; exact Selmer orthogonality on a DVR does not follow just from two generic dimensions.

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Lemma 8 ([Complementary local conditions]). If paired perfect local maps U→L, U'→L' (possibly only over a localized test ring after the limit) have a paired restriction nullhomotopy and induce L'/U'≃U∨[-2] , the two Selmer complexes are …

Lemma 7 ([Finite models and matrix limits]). Consider diagrams of the global, local, and mapping-fiber cochain complexes above over finite local Artin coefficient rings of 2-power characteristic, with modules finite free (and compatible …

Corollary 6 ([The full-index discrepancy]). Under these assumptions, the single Hilbert-class trace satisfies X(E)+X(Ek) =v2\!([E(K):ℤP]2 cE2\,∏ℓcℓ(E)2 #(E/K) ). Here [E(K):ℤP] is the full group index, including torsion.

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Lemma 8 ([Complementary local conditions]). If paired perfect local maps U→L, U'→L' (possibly only over a localized test ring after the limit) have a paired restriction nullhomotopy and induce L'/U'≃U^∨[-2] , the two Selmer complexes are …

Corollary 6 ([The full-index discrepancy]). Under these assumptions, the single Hilbert-class trace satisfies X(E)+X(E^k) =v₂\!([E(K):ℤP]² c_E²\,∏_{ℓc_ℓ}(E)² #(E/K) ). Here [E(K):ℤP] is the full group index, including torsion.