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).
John Taylor crisptoast@tutanota.com (Fri,) studied this question.