Pythagorean Angle Lattice: Möbius Precision, Spectral Addition, and Depth Reconstruction

Möbius precision operators on the divisor posets of the Pythagorean Angle Lattice yield an exact spectral-addition law whose traces and determinants reconstruct the complete prime-depth profile of the division content: Primitive Pythagorean triples determine rational norm-one Gaussian rotations and hence oriented rational angles. The Pythagorean Angle Lattice (PAL) is the division structure obtained by taking integral roots of these rotations inside the rational-angle group. For a nonzero PAL angle θ, its division content L(θ) records all integral divisibility, while the valuations a_p = v_p(L(θ)) form its prime-depth profile. This paper associates to that arithmetic data a canonical finite spectral tower. At level r we use the divisor poset D_r = Div(L^r), its incidence-zeta operator Z_r, and the Möbius precision Δ^{Mob}_{L,r} = (Z_r^* Z_r)^{-1}. The PAL decimation map ρ_{r,s}(d) = d/(d, L^{r-s}) induces a transfer operator T_{r,s}. We prove the exact covariance law T_{r,s} Δ^{Mob}_{L,r} T^*_{r,s} = Δ^{Mob}_{L,s}. Writing C_{r,s} = T_{r,s} T^*_{r,s} and R_{L;r,s} = C^{1/2}_{r,s} Δ^{Mob}_{L,s} C^{1/2}_{r,s}, we obtain the spectral-addition identity Tr R_{L;r,s} = E_L(r + s) = τ(L^{r+s}), E_L(x) = ∏_{p|L} (1 + v_p(L) x). Primewise, the operator is similar to a positive rank-one boundary perturbation of a Dirichlet–Neumann Jacobi matrix, with characteristic polynomial P_N + q P_{N-1}. We prove a rigidity theorem explaining why the canonical incidence normalization is precisely the normalization that yields the symmetric r + s trace law. Finally, the trace family, a normalized determinant response, and its finite jet all reconstruct exactly the unlabelled prime-depth multiset {v_p(L) : p | L}. The construction is canonical relative to counting measure, but it deliberately forgets prime labels and the division-primitive PAL core. Thus PAL division content acquires a decimation-covariant incidence geometry whose spectral data give an exact arithmetic classification of depth profiles.