Geometric Tomography of Smooth Critical-Point Transport from Renormalized Tail Orbits: Integrable Edge Maps, Curvature Fingerprints, and Ray-Bundle Rigidity

Previous papers in this program recover the asymptotic orbit of each ray separately: a fixed direction ν produces a geometric attractor Gρ(ν) and a first rational fingerprint ρ(ν)[P1,ν]. 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 \[ {anν+β}{anν} = ρ(ν)^β(1+{P1,ν(β)}{n})+(n⁻²), \] with a positive C² edge map ρ(ν). 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: \[ ∂_ilogρ_j(ν)=∂_jlogρ_i(ν). \] In that case there exists H with ρⱼ(ν)=e∂ⱼ H(ν), 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(ν)=∇² H(ν) we form the universal quadratic transport polynomial \[ QH,ν(β) := 12∑ⱼ₌₁^d Bⱼⱼ(ν)β_j(β_j-1) + ∑1≤ i<j≤ d Bᵢⱼ(ν)β_iβ_j. \] Then every first fingerprint splits uniquely as \[ ρ(ν)[P1,ν] = ρ(ν)[L_ν] + ρ(ν)[QH,ν] + ρ(ν)[R_ν], \] where L_ν is determined by one-step probes, QH,ν is the universal curvature contribution, and R_ν is a reduced same-scale residual satisfying R_ν(0)=R_ν(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 \[ (1-σ_1 z_1-⋯-σ_d z_d)-θ \] 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 ρ, the Hessian entries Bᵢⱼ, the reduced mixed residuals R_ν(eᵢ+eⱼ), and the diagonal residuals R_ν(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.