Smooth Minimal-Point Rigidity for Renormalized Tail Orbits: Uniform Cone Transfer, Support-Potential Atlases, and Quantitative Obstruction Detectors

The preceding papers in this program built an orbit theory for renormalized Taylor tails, then extended it to several variables, directional edge maps, same-scale shape separation, and local ray-bundle tomography. The remaining structural question is the sharp one: which first-order orbit jets actually come from the classical smooth minimal-point regime of multivariate singularity analysis? This paper gives a complete first-order answer. We start from the standard smooth-point coefficient template on a cone of directions. More precisely, we assume the usual Pemantle--Wilson / Raichev smooth-point asymptotic expansion in a form uniform over a compact cone and over bounded lattice perturbations. We do not reprove the contour theorem itself; rather, we take that analytic-combinatorial input as known and transfer it into the language of renormalized tail dynamics. Our first main theorem is a uniform cone transfer law. If \[ a_α=[z^α]F(z), T_α^F(w)=∑β∈^d{aα+β}{a_α}w^β, \] and if a compact cone of directions admits a unique strictly minimal smooth critical point map ζ(ν), then \[ Tnν^F(w) = Gρ(ν)(w) +1/nρ(ν)[Pˢᵐ1,ν](w) +(n⁻²), ρ(ν)=ζ(ν)⁻¹, \] uniformly on compact w-polydiscs. The first fingerprint is forced to have the canonical form \[ Pˢᵐ1,ν(β) = u(ν)\!·\!β +12 β^ ∇^2Λ(ν)\,β, Λ(ν):=-ν· log ζ(ν), u=∇log A, \] where A(ν) is the standard smooth-point amplitude. Our second main theorem is a support-potential atlas theorem. The logarithmic edge field is exact: \[ ∇Λ(ν)=log ρ(ν)=-log ζ(ν). \] Hence the raywise critical-point data recovered from the orbit glue on overlaps, giving a global support-potential atlas on any simply connected direction domain. The third theorem is the central rigidity statement of the paper. In the more general transport-compatible jet class developed earlier, one may decompose any first fingerprint as \[ P1,ν=L_ν+QΛ,ν+R_ν, \] where QΛ,ν is the universal curvature polynomial and R_ν is the reduced same-scale residual. We prove that for every genuine smooth minimal-point family one has \[ R_ν≡ 0. \] Thus the reduced same-scale residual is not merely absent in examples: it is a structural obstruction. Conversely, a local first-order orbit jet is smooth-point compatible if and only if the reduced residual vanishes and the one-step amplitude field is exact. Our fourth theorem is a quantitative finite-cone detector. From finitely many coefficient ratios on a finite set of rays and lattice probes one reconstructs ρ, ∇²Λ, and u with explicit deterministic biases \[ {ρ}-ρ=(N⁻²), {B}-∇^2Λ=(N⁻¹), {u}-u=(N⁻¹), \] and one obtains obstruction scores on higher probes that are (N⁻¹) in the true smooth-point class. Persistent larger defects therefore rule out smooth minimal-point origin. This paper turns the previous multivariate orbit theory into a sharp compatibility theorem with the classical smooth-point regime: it identifies exactly what the orbit of a genuine smooth minimal point can look like at first order, and it provides finite, quantitative diagnostics for detecting when one has left that class.