The Pythagorean Angle Lattice: Exponent Walls, Elliptic Closure and Dyadic Division

Three arithmetic structures of the Pythagorean Angle Lattice — exponent walls in finite Weil representations, elliptic closure of spin commutators, and dyadic division via the tent map — are completely described, with explicit critical constants and local-global closure criteria. Let θ be the smaller acute angle of a primitive Pythagorean triangle (a, b, c) and put z = eiθ = (b + ia)/c. The difference of the two acute angles acts on rational phases by z ↦ i z⁻², while its half-angle gives the spin rotation q = (A + Bn)/(√2 c) with A = a + b, B = b − a, A² + B² = 2c². We study three arithmetic structures generated by these identities. First, reduction of z⁴ modulo prime powers produces finite rotations whose Weil operators possess unitary exponent intertwiners implementing r ↦ −2r; their dimensions and least finite orders are determined explicitly by the reduction order and its p-adic lifting exponent. Second, commutators of the spin rotations lead to the quartic Y² = t⁴ + (2 − 4λ²)t² + 1, λ = B₁B₂/(2c₁c₂), birational to the Legendre curve y² = x(x − 1)(x − λ²); a sharp local obstruction at 2 forces the even leg of a closing axis triple to be divisible by 8 (a local condition, necessary but not sufficient for closure), orthogonal axes never close, and closures occur either never or infinitely often. Third, folded doubling of Pythagorean angles is the tent map; its fold itinerary is the Gray code of the binary expansion of 4θ/π, and the Gaussian prime exponents determine the exact backward division depth. We obtain height-counting laws, a criterion for rational division of the difference angle, arbitrarily long quarter-turn avoidance, an Archimedean–dyadic density theorem and the exact critical constant lim inf R² |sin 8ϕ| = 2√2.