Dyadic Galois Theory of the Pythagorean Angle Lattice: Degrees, Galois Images and the 2-adic Coordinate of Depth

At dyadic depths the Pythagorean Angle Lattice is governed by explicit affine Galois groups, Cayley–Kummer normal closures, non-isomorphic parity fields with only rational dyadic rotations, and a universal 2-adic valuation law for real parts of Gaussian powers. Let U be the group of rational points of the unit circle, μ₄ = {±1, ±i} and 𝒜 = {Arg z : z ∈ U \ μ₄}. The Pythagorean angle lattice is PAL = {cot(α/D) : α ∈ 𝒜, D ≥ 1}; its angles are the arguments θ = Arg(W²) of squares of Gaussian integers W = u + iv, and W = 2 + i gives the (3,4,5) triangle. Dividing θ by D gives D division branches cot((θ + jπ)/D), the roots of an integral division polynomial of discriminant 2^{(D−1)(D−2)} D^D c^{2(D−1)}, c = u² + v², and exactly two of them lie in the layer PAL_D. We determine their arithmetic at dyadic depths. A composition law splits every even depth into two parity branches, and full dyadic degree holds at one, equivalently every, depth 2^{k+1} ≥ 4 exactly when c is not a square. For a primitive generator of opposite parity write W = i^{s₀} V^{2^t} with V not a square; then the Galois group of the depth-2^{k+1} splitting field is the affine group {j ↦ a j + 2b : a ≡ 1 (mod 4), b ≡ s₀ (a−1)/2 (mod 2^{min(t,k)})}, every branch has degree 2^{k−min(k,t,1+ν₂(j+s₀))}, the normal closure of each branch field is a Cayley–Kummer field, and the 2^κ stalls of the principal tower, κ = min(t,1+ν₂(s₀)), all occur at the start. For the 2^k generators of a squarefree norm built from k split primes, the depth-2^r coordinates generate an abelian extension of ℚ(ζ_{2^r}) with group C_{2^{r−1}} × C_{2^{r−2}}^{k−1}. When c is not a square, the two parity fields are never isomorphic from depth 8 on, each has automorphism group C₄ generated by the quarter-turn x ↦ (x−1)/(x+1), which together with its powers is the only rational dyadic rotation, and at an odd prime ℓ their discriminants have exponent n − gcd(n,ν_ℓ(c)) at every degree n = 2^k; at depth 8 they are cyclic quartic fields from a rational-parameter simplest quartic family, with explicit conductors 2^f M₄, f ∈ {0,2,3}. By Chebotarev’s theorem the primes at which a depth-2^{k+1} polynomial has a root have density 1/3 + 2/(3·4^k) when c is not a square, and the root and branch-pattern densities are given in every stratum. For principal coordinates at arbitrary depth we give the cyclotomic degree formula, a rationality criterion and an exact three-level test for relative degrees over cyclotomic fields. Finally, for the real parts ℱ_D of first-lobe Gaussian powers and D = 2^R m, R ≥ 2, ν₂(ℱ_D − 1) has the universal law 2^{R−k} (k ≥ R+1); at depth four the fixed-coordinate classification is complete, with a resonant tie case of density 1/56; doubling the depth shifts the valuation by an explicit amount; the odd part of the depth changes it with density λ = 1/(7·2^{R+ν₂(m−1)}) and a symmetric geometric jump law Pr(Δ = ±d) = λ/2^{d+1}; and the roots p_m of ℱ_{2^R m}(·,q) = 1 satisfy ν₂(p_m − p_n) = R + 2ν₂(q) − 1 + ν₂(m − n), a scaled isometry of ℤ₂^× onto a 2-adic ball.