Quadratic Reciprocity from Three Primitives

The law of quadratic reciprocity is derived entirely from the three primitives — same,different, opposite — working inside the Tree of Continua C.The Legendre symbol pq ∈ {+1, −1} is the chiral involution φ(s) = −s applied tothe multiplicative group (Z/pZ)∗ : it asks whether p is a square mod q, and answers+1 (same as a square) or −1 (opposite to a square). The Gauss sum g(p) = a p ωp isa labeling of the cylinder sets of Z/pZ by the cyclotomic field Q(ωp ) ⊂ C, weighted bythe chiral involution. Both are periodic orbits in Per(C) — finite algebraic objects, exact,requiring no analysis.The identity g(p)2 = (−1)(p−1)/2 p is four lines of finite arithmetic in Q(ωp ). Quadraticreciprocity follows from computing g(p)q−1 in two ways inside Q(ωpq ) ⊂ Per(C).Every step is finite arithmetic on periodic orbits. The law of quadratic reciprocity isa theorem about how the chiral structures of Z/pZ and Z/qZ interact — forced by theopposite primitive, living entirely in Per(C).