The previous paper in this sequence gave the exact all-orders normal form and finite-order compatibility criterion for smooth minimal-point renormalized tail jets, but it left open the decisive next question: does every compatible hierarchy come from an actual exact orbit? This paper answers that question in the intrinsic orbit category of analytic germs. We start from a compatible hierarchy on an open cone: a 1-homogeneous support potential and transport fields ᵣ of degree -r, generating logarithmic cumulants \ K₌, () = 1 (m+1) !^m+1 () [^m+1 + =₁^m1!^₌- () ^. \] First, for every M we prove an exact finite-order realization theorem. After a conic cutoff and a linear background extension, the coefficient array \ a_= (M () ) \ defines an analytic power series whose exact ratio cocycle and renormalized tail orbit realize the prescribed order-M hierarchy. Second, we prove an infinite-order Borel cone synthesis theorem. Given a full compatible tower (, ₀, ₁, ), we construct a locally finite phase \ _ (x) = (x) +ₑ₀ᵣ (|x|) ᵣ (x) \ with rapidly delayed cutoffs, and show that it has the full asymptotic expansion generated by the tower on every compact subcone. This yields an actual analytic germ realizing the entire infinite hierarchy. Third, we classify the resulting realizations exactly. Two analytic realizations have the same full hierarchy on a cone if and only if their admissible cocycles differ by a flat exact coboundary \ r, ₉=r, ₉u+₄䲛u_, \ where the gauge u is subexponential and orbit-flat to every algebraic order. Hence the compatible hierarchy is a complete invariant of exact renormalized-tail orbits modulo flat gauge. Finally, we package the construction as a finite-jet synthesis map and prove quantitative stability under perturbation of the input tensors. Thus the hierarchy theory is upgraded from a compatibility and detector theory to an exact completion theory. What remains open is the narrower geometric question of realizing these hierarchies inside specific meromorphic smooth minimal-point model classes.
Mohammad Abu-Ghuwaleh (Tue,) studied this question.