All-Orders Smooth Minimal-Point Rigidity for Renormalized Tail Orbits: Bell-Cumulant Closure, Hierarchical Compatibility, and Finite-Cone Recovery

The previous papers in this program identified the exact first-order and second-order images of the classical smooth strictly minimal critical-point regime inside the broader multivariate orbit theory of renormalized Taylor tails. The present paper closes that smooth-origin side of the theory at arbitrary finite order. We prove an all-orders rigidity theorem. We begin from an abstract order-M smooth-point coefficient template on a cone of directions. Such templates are precisely what one obtains from classical smooth-point multivariate singularity analysis when the coefficient asymptotics are pushed to depth M, but the present paper does not rederive that contour machinery. Starting from that template, we prove a uniform logarithmic ratio expansion \ a₍+a₍ = () ^ +₌=₁M K₌, () nᵐ + (n^-M-1), \ where the logarithmic orbit cumulants satisfy the exact linear jet law \ K₌, () = 1 (m+1) !^m+1 () [^m+1 + =₁^m1!^₌- () ^. \] Thus the reduced same-scale data at order m are not arbitrary: they are completely determined by one support potential and by the lower scalar transport fields ₀, , ₌-₁. Passing from logarithmic cumulants to ordinary fingerprints yields a universal Bell-polynomial closure law \ P₌, =ₘ (K₁, , , K₌, ), \ which extends the second-order quartic normal form to the full hierarchy. The core theorem of the paper is a complete hierarchical compatibility criterion. Given a truncated orbit jet (, P₁, , PM) on a simply connected direction domain, we characterize exactly when it comes from a smooth-point template of order M: after passing to logarithmic cumulants, each cumulant must have degree at most m+1, its top homogeneous piece must equal the symmetrized mth derivative of, each degree-one piece must be exact, and every intermediate homogeneous tensor must be the gradient of the previous lower-order one. Equivalently, the entire hierarchy is generated by a single exact edge field together with a ladder of scalar transport potentials. The last part turns the theory into a finite diagnostic scheme. From finitely many logarithmic ratio evaluations on a finite scale window and on finitely many rays, we reconstruct the truncated cumulants, interpolate the compatible model, and build validation scores on independent probes and neighboring rays. Inside the smooth-point class these scores obey explicit deterministic bounds of size \ (hN+N^-1+N NM), \ where hN is the ray-mesh size and N is the observation noise. We state this explicitly as a local finite-horizon asymptotic detector, not as a globally conditioned numerical algorithm under arbitrary noise. The paper therefore gives the first all-orders normal-form theorem for renormalized tail orbits in the smooth minimal-point regime. It identifies exactly what the orbit can look like at every finite depth, isolates the full obstruction hierarchy, and provides a quantitative finite-cone method for testing smooth origin.