local-to-global reciprocity as one object: the invariants at all places, combined, cancel to the identity

The product formula is one object that keeps arriving under different names. As absolute values: ∏_v |x|_v = 1 over all places of ℚ — and Ostrowski's theorem, the census of those places, is exactly its index set. As Hilbert symbols: ∏_v (a,b)_v = 1, Hilbert reciprocity, the product of the local quadratic symbols is trivial. As the Brauer exact sequence: 0 → Br(K) → ⊕_v Br(K_v) → ℚ/ℤ → 0, the sum of local invariants is zero. These are not three analogies but one theorem read at three degrees — the reciprocity of class field theory, of which the |x|-formula is the degree-one (norm) shadow and Hilbert reciprocity the degree-two (ℤ/2) shadow. The single fact: local invariants at every place, combined, cancel.

Its reach is its recurrence. It surfaced independently in two separate explorations — one disciplined, one purely speculative — each time under several of these names before the identity was seen. A structure found blind, twice, under different names, is one object rather than a coincidence; that convergence is the evidence, stronger than either sighting alone.

Calibration question: when a global fact is stated, ask what it says at each place and whether the local pieces are forced to cancel. If they are, you are looking at this object again — the product formula under another name — and the task is to find the degree at which it is being read.