Is the Brauer–Manin obstruction an instance of the anonymous fixed-point obstruction?

Speculative — open, with a candidate shared theorem to test.

The claim. The Brauer–Manin obstruction — a variety with points everywhere locally but none globally — is an instance of the anonymous obstruction (local data at every place, no global witness; the residue of a group action, [[N-012]]).

The candidate shared theorem (the gate). The Brauer–Manin pairing is a sum of local Brauer invariants that vanishes on global points by the product formula ([[EX-13]]) — reciprocity as the linking measurement. If that is the whole content, the two are one obstruction and this is an edge, not a resemblance. Discriminator: show the anonymous obstruction and the Brauer–Manin obstruction are equal (not merely analogous) through the product-formula pairing, theorem stated; genus 1, where Brauer–Manin detects all of Ш (Manin), is the test case. Open until written out.