Critical Walls in the Pythagorean Angle Lattice: Affine Division, Depth Spectra, Galois Packets, Dyadic Wall Primes and the Arithmetic of the Equal-Leg Wall

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Critical walls of the Pythagorean Angle Lattice are completely described by affine content formulae, finite depth spectra, Galois packets of binary forms, dyadic wall primes with Ljunggren exceptions, and the arithmetic of the equal-leg wall. A primitive Gaussian integer W = u + iv with u > v > 0 and u ≢ v (mod 2) generates the primitive Pythagorean triangle (A, B, C) = (u² − v², 2uv, u² + v²) through W² = A + iB, and the Pythagorean angle lattice divides the angles 2 arg W and 2 arg W + π/2 by integers D. We study the arithmetic of this division. On the oriented rational angle group we give an exact content formula for all affine maps α ↦ m(π/2) + nα; in particular the acute-angle difference always has odd content. We prove that the set of depths at which a lattice value occurs is a finite set of multiples of a minimal denominator, truncated by an archimedean bound and determined by a quarter-turn parity condition. The quarter-turn walls q^D = ±i, q = W/¯W, are cut out by binary forms that decompose into τ(m) primitive Galois packets, where m is the odd part of D; the dyadic depths are exactly the irreducible ones, and Gaussian squaring commutes with the packet decomposition. On the dyadic tower the oriented wall values H_r = Re W2^r − Im W2^r and K_r = Re W2^r + Im W2^r have pairwise disjoint prime support, every odd prime divisor satisfies p ≡ (−1/p) (mod 2ʳ⁺²) and reconstructs the depth, and by Ljunggren’s theorem the only unit values beyond the quadratic level come from W = 3 + 2i, through the exceptional Ljunggren solution that also underlies Machin’s formula. At the quadratic level the wall is the equal-leg wall A = B; primitive sources satisfy A − B ≡ ±1 (mod 8), which decodes the three 2-adic gap types 1, 2, 8 of C − max(A, B), forces the sharp bound A − B ≥ 7 for the gap type 2, and identifies the nearest integral layers with the Pell equations x² − 2y² = ±1. We also determine the primitive spectral multiplicities and the Pell–Thue transition between the quadratic and higher walls.