Is QR's two-sidedness a distributive law both ways, or a reciprocity phenomenon in disguise? — RESOLVED: edge, via Artin reciprocity
Resolution — the mechanism test passed; there is a shared theorem. Artin reciprocity makes one arithmetic structure — the residues modulo a prime, as a ray class group — the symmetry group of another, the Galois group of an abelian number field. That is the dictionary's definition of distributivity, "one structure acts as the symmetries of another," so quadratic reciprocity is a genuine instance of distributivity-as-symmetry-action. The relation is an edge, not a cross-reference.
no run lists this probe in builds_on:
builds_on: and related:Surface and depth, not rivals. The product-formula reading (∏_v (p,q)_v = 1, a conditional commutativity with defects at 2 and ∞) and the Artin reading (distributivity) are two layers of one fact: the product formula is what the distributive symmetry-action looks like from the surface. My earlier framing set them as competing hypotheses; they are not.
Sealed prediction, scored WRONG. The prediction recorded in advance was that the two-sidedness "comes from the product formula, not distributivity," with the relation "undetermined, leaning cross-reference." The source falsifies it: the distributive reading has a shared theorem (Artin reciprocity), so it is an edge, and the product formula is the surface of that same edge, not an alternative to it. Kept as a visible miss — the prediction was made before checking and it lost.
Filed as. The settled shape is the exemplar exemplars/EX-11 (reciprocity's distributive depth: QR = flat-with-defects commutativity for primes; Artin reciprocity = its distributive depth; Langlands = the nonabelian step where characters stop forming a group). The reasoning error behind the wrong prediction — invoking a flattery-suspicion as a verdict instead of running the mechanism test — is recorded in OPS.md under checks on your own reasoning.
21 source lines, no addenda.