the abelian-to-nonabelian transition read through commutativity-as-flatness — an abelian reciprocity is a flat symmetry-action of a group of characters, and turning the symmetry side nonabelian (curved) is the general step

Quadratic reciprocity reads two ways, and they are two layers of one answer, not competitors. Surface: flat-with-defects commutativity for primes — the Hilbert symbols multiply to one over all places, ∏_v (p,q)_v = 1, a symmetric form whose only defects sit at 2 and ∞; reciprocity as a fact about how local signs must agree. Depth: distributivity — Artin reciprocity makes one arithmetic structure (residues mod q, the ray class group) the symmetry group of another (the Galois group of an abelian number field), which is exactly "one structure acts as the symmetries of another." QR is the quadratic, abelian instance of that symmetry-action, and the product formula is what that instance looks like from the surface. The general step: Langlands is what the shape becomes when the symmetry side goes nonabelian — the Galois group is no longer captured by characters that form a group, and one-dimensional representations give way to higher ones. The shape is the abelian-to-nonabelian transition read through commutativity-as-flatness: a flat (abelian) reciprocity as the base case, curvature (nonabelianity) as the general step.

Calibration question: when a two-sided reciprocity or symmetry appears, do not stop at the surface reading (a product or sum forced to cancel) and do not dismiss the depth reading as a resemblance — ask whether there is a shared theorem making one structure the symmetry group of another. Here there is (Artin reciprocity), and the two readings stack. Telling a surface commutativity from a genuine symmetry-action — and seeing when they are one fact at two depths — is the reach.