Dyadic Modular Dynamics on the Pythagorean Angle Lattice: Normal Forms, Parabolic Closing, Transfer Operators, Barnes Spectral Reconstruction, and Character Gradings

Dyadic modular dynamics on the Pythagorean Angle Lattice yields exact normal forms, parabolic closing, transfer operators, and a Barnes spectral reconstruction that realizes the arithmetic Laplacian as a graded supertrace with pole cancellation and character transport: We develop a dyadic modular and operator-theoretic extension of the Pythagorean Angle Lattice program. The starting point is the three-branch half-angle dynamics A(q) = q/(1 + 2q), S(q) = (1 + q)/(3 + q), C(q) = (1 − q)/(3 − q), together with the primitive parabolic half-step B(q) = q/(1 + q), B² = A. After conjugation by B, the positive branch semigroup admits an exact Markov partition and a free-monoid normal form. Grouping maximal parabolic runs yields a unique signed countable coding by pairs (a, ϵ) ∈ ℤ≥0 × {0, 1}. Resolving the width-two parabolic branch A = B² by independent width-one half-steps produces a two-index state space (a, m) whose representation multiplicity is exactly ⌊(k − 1)/2⌋. This gives an exact transfer-operator realization of the Barnes double zeta ζ₂(2s, 3 | 2, 1) = (1/2)ζ(2s − 1) − (1/2)(1 + 2^{−2s})ζ(2s), which coincides with the arithmetic spectral zeta of the Pythagorean arithmetic Laplacian. The same structure produces an exact half-step grading Γ_B, a graded Barnes supertrace, and cancellation of the leading two-dimensional spectral pole. We also derive a coordinate-cocycle mechanism by which reflection antisymmetry at the 3–4–5 base point isolates the quadratic Dirichlet character χ_{−3} up to explicit elementary corrections. In parallel, we establish two-sided parabolic closing families around rational Pythagorean points, their quadratic periodic halos, universal k^{−1} positional and k^{−2} multiplier laws, explicit discriminant packets, and local square-divisibility character transport. The resulting framework separates three structures that had previously appeared entangled: intrinsic dyadic determinant characters, primitive half-step parity, and rational-base-point Dirichlet specialization. Several statements concerning squarefree-value asymptotics, nuclearity/Fredholm determinants, and a complete group-envelope classification are recorded separately as open problems rather than asserted as established theorems.