Cyclotomic Square-Return Rigidity in the Pythagorean Angle Lattice: Double Packet Descent, Terminal Prime Support, and the Even-Class-Number Frontier

Square return of a primitive Pythagorean angle at an odd prime p is completely rigid when the class number of ℚ(ζ_p) is odd, and the remaining even-class-number cases are reduced to explicit 2-class data of real cyclotomic fields. Let W = u + iv be a primitive Gaussian integer with u > v > 0 and u ≢ v (mod 2), and let θ = arg(W²) be the corresponding acute angle of the primitive Pythagorean triangle (u² − v², 2uv, u² + v²). For an odd prime p we study the square-return condition sin(pθ)/sin θ ∈ ℚ^ײ, which is equivalent to the Gaussian equation W^p = κ(uM² + ivN²) with κ = ±1 and is intimately linked to Lucas sequences attached to W. We prove that square return forces p ≡ 1 (mod 8) and κ = 1. Over the real subfield F = ℚ(ζ_p)^+ of the p-th cyclotomic field the two coordinate conditions split into two packets of pairwise coprime square ideals; their quotient generates a quadratic extension of F that is unramified at every finite prime and is therefore controlled by Kummer theory and the class number of ℚ(ζ_p). Combined with Dummit’s parity theorem on unit signatures, this yields the main rigidity theorem: if the class number of ℚ(ζ_p) is odd, no primitive Pythagorean triple admits a square return at p. Below 1000 the only primes p ≡ 1 (mod 8) not covered by this odd-class-number criterion are 113, 337, 937 and 953. For these even-class-number primes we establish a dyadic splitting law, packet squareclass rigidity, an exact cyclic-code formula for the signature rank, the terminal-prime support law ℓ | MN ⇒ ℓ ≡ ±1 (mod 4p) obtained from a factorisation over ℚ(ζ₄ₚ), and relative class-group bounds for ℚ(ζ₄ₚ)^+/ℚ(ζ_p)^+. Parallel results are obtained for the metallic Pell family, and the remaining obstruction is isolated as the need for explicit 2-class data of real cyclotomic fields.