Arithmetic–Spectral Criticality from the Pythagorean Angle Lattice: Gaussian Prime Sectors, Tensor Spectra, and Exact Spectral Collapse

The Pythagorean Angle Lattice reconstructs the spectrum {π²k²} and couples it to Gaussian split primes, producing an exact critical collapse at s = 2 where multiplicities become ω₊(m) and the counting and heat laws acquire a log-log enhancement: The Pythagorean Angle Lattice (PAL) is a hierarchy built from the acute angles of primitive Pythagorean triples and their integral angular subdivisions. Its definition and the portion of its structure required here are developed from first principles. For a primitive right triangle with legs 0 < a < b, hypotenuse c, acute angle θ opposite a, and governing parameter n = 2a/(c − b) = 2 cot(θ/2), the quarter-angle coordinate X = cot(θ/4) satisfies X² − nX − 1 = 0, X = (n + √(n² + 4))/2. Writing u = θ/2, the normalized PAL angular defect D_PAL(u) = 4(1 − u cot u) is shown to be a complete inverse-spectral datum within the natural class of positive self-adjoint operators with trace-class inverse. It reconstructs the normalized Fredholm determinant sin u / u and hence the simple spectrum {π² k² : k ≥ 1}. The discriminant n² + 4 of the same quarter-angle quadratic gives the intrinsic PAL congruence n² + 4 ≡ 0 (mod d), which at odd primes selects the Gaussian split sector p ≡ 1 (mod 4). Coupling the reconstructed archimedean spectrum to this arithmetic sector gives the positive self-adjoint family L_s = L_PAL ⊗ es A₊. Its spectrum consists, with tensor multiplicity, of the numbers π² k² p^s, k ≥ 1, p ∈ P₊. The coupling exponent s > 0 is a free parameter. We prove that it becomes critical exactly at s = 2, where it matches the quadratic homogeneity reconstructed from PAL and the two spectral-zeta convergence thresholds collide. At this critical value the tensor labels collapse multiplicatively, π² k² p² = π² (kp)², and multL_2(π² m²) = ω₊(m). Consequently, N₂(Λ) ∼ (√Λ)/(2π) log log Λ, K₂(t) ∼ (1/(4√π)) t−1/2 log log(1/t). We also prove the subcritical and supercritical laws. Algebraic collapse onto the same quadratic lattice also occurs at the higher even couplings s = 2j, but only j = 1 is critical: for j ≥ 2 the mean collapsed multiplicity is bounded by the convergent prime sum P₊(j). Finally, for the general exact spectrum C k^α p^s, we prove a homogeneity-matching principle with critical parameter s_c = α and a parallel collapse hierarchy at s = j α.