Pythagorean Angle Lattice: Division Content, Cyclotomic Decimation, Lucas Reduction, and Selmer Chamber Rigidity

The Pythagorean Angle Lattice yields an unconditional Lucas reduction of prime square events together with a dyadic Selmer chamber mechanism that excludes all such events for primes p ≡ 1 (mod 24) below 3200 and for p = 5209. We develop the arithmetic of the Pythagorean Angle Lattice (PAL) in three layers, each with an explicitly stated logical status. The first layer is intrinsic and unconditional. The lifted group of rational norm-one angles is free abelian, so every nonzero lifted angle has a division content, a unique division-primitive core, and a persistence–extinction filtration in which every finite division depth is dense while no nonzero angle survives all depths. Integer power maps induce the exact decimation map ρ_M(n)=n/(n,M), whose fibres give the unrestricted composition law Φ_d(X^M)=∏ρ_M(e)=d Φ_e(X) and the collision Dirichlet series 𝒦_M(s)=ζ(s)∏ℓ^a∥M ((a+1)−aℓ⁻ˢ). The same map reorganizes the cyclotomic factors of the PAL area response. The second layer is the area response R_D(∠)=sin(2D∠)/sin(2∠), and is also unconditional. We prove that for a primitive Pythagorean angle attached to V∈ℤ[i] with N(V)=c, R_p(∠)=L_p / c²⁽ᵖ⁻¹⁾, gcd(L_p,c)=1, where L_n is the Lucas sequence with roots V^4, V̄^4. Hence a prime rational-square event is exactly the assertion that a Lucas number is a perfect square. This yields, via Faltings applied to a hyperelliptic curve of genus 2p−3, the unconditional finiteness of square events for each fixed p. A complete elementary local sieve gives p≡1 (mod 24), the sharper congruence R_p(∠)≡p (mod 32), and a new fifth-prime condition: if 5∤c, then p is a quadratic residue modulo 25. The third layer is the Selmer mechanism, and it rests on exactly one arithmetic hypothesis, relative squareclass separation. We make the dyadic side effective: the local pairing on the first graded piece is (−1)^{TrF/ℚ(ab)}, the dyadic logarithm of a unit is extracted by a Frobenius power, and the Selmer target λ_p becomes computable. Two independent isotropy identities validate the computation, one of them reproducing the Gauss-carry parity U_p(1)=1 by an unrelated route. Executing the chamber principle then gives: for every prime p≡1 (mod 24) below 3200, and for p=5209, no relatively separated square event occurs. No unit-index, class-group, or circular-to-full-unit hypothesis is used.