This paper provides the constructive foundation for Recursive Generation Theory (RGT). Building upon the meta-theoretic justification established in the first paper of the series 1, this work turns to the following three progressive questions: Do mathematical objects satisfying the RGT postulates exist? If so, what are the physically meaningful realizations? What structural relations hold among different realizations? First, the paper presents an exact mathematical statement of the RGT postulates, formalizing the three constraints — recursivity (C1), discrete spiral symmetry (C2), and low-energy projection validity (C3) — as independently verifiable criteria. The weak form of constraint C2 obtains an operational criterion through the quantitative parameter ₀, with ₀ = 0. 1 serving as a working definition; its dynamically derived value is left for the mode-locking mechanism to deliver in subsequent work. Second, by constructing an analytically solvable minimal model R_ (a five-dimensional complex linear recursive system), this paper rigorously proves **Proposition E**: there exists at least one mathematical object satisfying all constraints C1–C3. The proof is completely analytic, relying on neither numerical experiments nor external references, and thus provides a closed logical premise for all subsequent derivations that depend on the existence of R. Proposition E simultaneously provides the closed-form solution of the stationary distribution _ CN (0, ²/ (1-²) I₅), a rigorous proof of the O (N^-1/2) convergence rate, and the analytic verification that the low-energy effective equation converges to an Ornstein–Uhlenbeck diffusion equation after coarse-graining projection. With the existential guarantee in place, this paper constructs two physically richer candidate models — the **lattice projection model** and the **graph recursion model** — and verifies their constraint satisfaction status item by item. The lattice model independently satisfies C1–C3 through transfer matrix analysis and existing rigorous diffusion limit theorems; the graph recursion model satisfies C1, and C2 and C3 receive numerical support, with C2 satisfaction relying on a Z₅ coloring constraint. This paper establishes a **local equivalence lemma** that characterizes the approximate equivalence of the two model classes with a quantitative error bound of O (N^-1/3) under the same low-energy parameters (D, ), and on this basis states **Universality Class Hypothesis H** (the two model classes belong to the same universality class in the coarse-grained low-energy limit). All conclusions are organized strictly according to epistemic certainty. The paper concludes with a systematic listing of unresolved open problems.
Lin Sun (Thu,) studied this question.