Sign Cubes of the Pythagorean Angle Lattice: Winding Carries, Reed–Muller Cosets and Rounded Affine Reconstruction

Sign cubes of the Pythagorean Angle Lattice carry winding and dyadic-address data whose rounded affine structure yields seventeen four-prime carry states, Reed–Muller cosets, and a toric chamber count of residue patterns. 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 Arg(W²) of the squares of primitive Gaussian integers. For a squarefree hypotenuse M = p₁⋯pₖ with primes pⱼ ≡ 1 (mod 4), the 2ᵏ lattice angles of norm M are the signed sums ∑ ±θⱼ of the angles of Gaussian primes above the pⱼ, and they form a sign cube. We study the two pieces of rounding data that the lattice attaches to a sign cube: the winding, which selects the branch of every depth-D coordinate, and the dyadic address, which governs the principal dyadic division tower and its companion defect. Both are roundings of scaled versions of one signed-angle affine function on the Boolean cube. We prove that the parities of the rounded values of a strictly rounded affine function on an integer box determine the rounded integer array up to reflection and an even integer-affine shift, and that residues modulo q ≥ 3 determine it up to q times an integer-affine function. This identifies the number of residue patterns with a chamber count of a toric arrangement: P₂ = 2^{d−1}(R + 1) and P_q = q^d R. In every dimension the signs of the real parts of the 2ᵏ products determine the windings. For four primes the winding carries take exactly seventeen states, one of probability 1/3 and sixteen of probability 1/24, carried by twenty-four winding cells of equal volume; the carry statistic has an explicit Hoeffding decomposition with variances 1/63, 4/315, 13/420, 193/1260 and a closed two-coordinate Fourier spectrum. For five primes four times the top Walsh coefficient of the winding takes exactly the six values −1, −3/4, −1/2, −1/4, 0, 1/4, with an explicit law. For the companion defects we determine the frequency law, the multinomial moments through degree three, a Hamming-code law, the complete four-prime classification (exactly 33 of the 45 candidate triples occur) with its exact probability table, and the complete five-prime census R₅(4) = 101, derived from the toric count 647. At depth four each admissible top mask, and at depth five each fixed pattern of labels at least two, carries exactly one punctured Reed–Muller coset in every dimension; the same holds at every depth for at most seven primes, six-site obstructions need ten coordinate types, and at larger depths the number of cosets is unbounded. Every configuration defined by finitely many strict inequalities with nonempty interior is realized by Gaussian primes of distinct norms.