Local–Global Defects B, Tamagawa Numbers, and Norm-One Tori: Local–Global Identities for Norm-One Tori of Rank One

This paper establishes, at the elementary level of algebraic tori over numberfields, an identity relating Tamagawa numbers to local–global defects, and completes the construction of its self-contained components using norm-one tori ofquadratic extensions as the worked instance. The instantiation theorem reads asfollows: let K/k be a quadratic extension and T = R1K/kGm the norm-one torus;then Sha(T) is the trivial group and #Sha(T) = 1, whose proof is carried outvia an elementary path through the Hasse–Minkowski theorem for ternary forms;H1(k,X∗(T)) ∼ = Z/2Z, given by an exact computation of cohomology of the character lattice; τ(T) = 2 is output by the Ono–Sansuc identity at the point whereSha is trivial and the cohomological count equals 2. The first two items are selfcontained; the third is an output of an external theorem, and this dependencystratification is explicitly registered in the main text. The arithmetic invariants ofthe three quadratic fields Q(i), Q(√−5), Q(√5) are closed by deterministic exactinteger arithmetic: class numbers, numbers of roots of unity, and discriminants areobtained by reduced-form enumeration and reduced cycle walking; the closed formof L(1,χ) and the partial-sum readings are cross-checked within the strict truncation bound M/(N + 1); the local–global equivalence of norm forms over finiteprime samples holds digit by digit, and the modulo-residue condition at the ramified prime is explicitly witnessed as a guard mechanism. The numerical archivefunctions as an analytic consistency check and does not constitute an independentcomputation of τ(T). The Ono–Sansuc theorem for general tori is stated accordingto the literature, with a six-step proof skeleton registering its dependency positions; the corresponding statement for cyclic extensions of degree n is registered asproposition-level content, in which H1 ∼ = Z/nZ is self-contained, and the weakenable limit of this self-contained computation is the commutativity hypothesis: anyfinite Galois extension yields #H1(G,X∗(T)) = |Gab|, with the noncommutativecase bounded by the order of the abelianization; triviality of Sha is borne by theHasse norm theorem, and τ = n is an identity output. The whole paper is organizedby dependency stratification: self-contained proofs, citation-level inputs, identityoutputs, and cross-check-level readings each occupy their own place and may notsubstitute for one another.