Second-Order Smooth Minimal-Point Rigidity for Renormalized Tail Orbits: Cubic Support Jets, Quartic Normal Forms, and Finite-Cone Compatibility Detectors

The previous paper in this program identified the exact first-order image of the classical smooth minimal-point regime inside the broader multivariate orbit theory of renormalized Taylor tails. The present paper lifts that characterization to the next genuinely nonlinear level. We prove a complete second-order smooth-origin theorem. We begin from a standard smooth strictly minimal critical-point cone input, but now with a coefficient template accurate to one order deeper. From that analytic-combinatorial input we derive a uniform renormalized-tail expansion \[ Tnν^F(w) = Gρ(ν)(w) +1nρ(ν)[P1,ν](w) +1{n^2}ρ(ν)[P2,ν](w) +(n⁻³), \] on compact w-polydiscs. The first fingerprint keeps the known quadratic normal form \[ P1,ν(β)=u(ν)·β+12β^ B(ν)β, B=∇^2Λ, u=∇log A, \] while the second fingerprint admits the canonical quartic decomposition \[ P2,ν(β) = 12 P1,ν(β)^2 +v(ν)·β +12β^ U(ν)β +16 T(ν)[β,β,β], \] where U=∇²log A, v=∇ b, and T=∇³Λ is the cubic support jet. In particular, every genuine smooth minimal-point family forces all reduced second-order same-scale residuals to vanish. This gives the central rigidity theorem of the paper: after removing the universal square term 12P₁², no free quartic or exotic cubic shape survives in the true smooth-point class. Conversely, we prove a second-order compatibility criterion. A local orbit jet (ρ,P₁,P₂) comes from a synthetic smooth-point template if and only if the first-order residual vanishes, the one-step amplitude field is exact, the reduced second-order residual vanishes, the extracted quadratic and cubic pieces agree with ∇ u and ∇ B, and the linear second-order field is exact. The last part of the paper turns the theory into a quantitative diagnostic scheme. Using finitely many coefficient ratios on a finite family of rays and lattice probes, we recover ρ, u, B, v, U, and T with explicit deterministic asymptotic biases, and we construct fourth-order obstruction scores that are forced to be (N⁻¹) inside the true smooth-point class. This yields a finite-cone compatibility detector for second-order smooth origin. The paper therefore upgrades the previous first-order rigidity result into a full second-order normal-form theorem: it identifies exactly what the renormalized-tail orbit of a genuine smooth minimal point can look like up to order n⁻², and it isolates the precise finite defects that certify departure from that classical regime.