Previous papers in this program recover the asymptotic orbit of each ray separately: a fixed direction produces a geometric attractor G () and a first rational fingerprint () P₁,. The next structural question is whether a whole bundle of neighboring rays carries a coherent local geometry. This paper gives a positive answer at the orbit level. We work in a transport-compatible smooth critical-point class: along each admissible ray the coefficient ratios have a first-order transport law \ a₍+a₍ = () ^ (1+P₁, () n) + (n^-2), \ with a positive C² edge map (). This is narrower than a full contour-integral theorem for arbitrary smooth singular varieties, but it is exactly the orbit-side regime in which geometric compatibility can be formulated and proved. The first main theorem is an integrable edge-map criterion. We show that the ray family comes from a local support potential precisely when the recovered logarithmic edge field is curl free: \ ᵢⱼ () =ⱼᵢ (). \ In that case there exists H with ⱼ () =e^ⱼ H (), unique up to an additive constant. Thus the leading tail orbit determines a local potential geometry on ray space. The second main theorem is a canonical curvature-fingerprint decomposition. From the Hessian B () =² H () we form the universal quadratic transport polynomial \ Q₇, (): = 12₉=₁ᵈ B₉₉ () ⱼ (ⱼ-1) + ₁ ₈<₉ ₃ B₈₉ () ᵢⱼ. \ Then every first fingerprint splits uniquely as \ () [P₁, = () L_ + () Q₇, + () R_, \] where L_ is determined by one-step probes, Q₇, is the universal curvature contribution, and R_ is a reduced same-scale residual satisfying R_ (0) =R_ (eⱼ) =0. Hence the n^-1 level itself separates into geometric transport, curvature-generated mixed poles, and intrinsic same-scale shape. The third main theorem is a ray-bundle normal-form and rigidity theorem. At the level of first-order orbit jets, transport-compatible families are classified exactly by triples (H, u, R) consisting of a support potential, a one-step amplitude vector, and a reduced residual family. The canonical class R 0 is rigid. In particular, the linear-hypersurface model \ (1-₁ z₁--d zd) ^- \ is first-order rigid among all transport-compatible families with the same support potential and the same one-step amplitudes. The fourth main theorem is a quantitative finite-bundle tomography theorem. From finitely many coefficient probes on a finite ray stencil we recover, the Hessian entries B₈₉, the reduced mixed residuals R_ (eᵢ+eⱼ), and the diagonal residuals R_ (2eⱼ) with deterministic biases (N^-2) for the edge map and (h²+N^-1) for curvature and reduced shape, where N is the radial horizon and h is the ray-stencil mesh. Thus a finite amount of orbit data already separates product-type transport, pure curvature coupling, and genuinely new same-scale shape.
Mohammad Abu-Ghuwaleh (Sat,) studied this question.