Orbital Functional Geometry: Orbits, Spectra, and the Geometry of Analytic Functions

We introduce the Orbital Space O, constructed from the equation eP(x)(az+b) = f(x). A point of O is a closed orbit — the trajectory of the solution z as x makes one full turn around a root of P in ℂ. We develop the complete theory: metric structure (Hausdorff distance), foliation, symmetry group (D∞), classification into 2ⁿ⁺¹ classes, encoding theorem (zeros and poles readable from orbits), orbital spectroscopy, differential and integral calculus in three directions, orbital Laplacian with continuous spectrum λ = 2/a², and flatness of the intrinsic geometry.