Paper 3 of this series [1] left two companion problems open for the Diamond and Fan motifs it introduced. For the Diamond, whose exact quasi-equilibrium reduction gives a closed-form, generic, supercritical ip bifurcation at λ flip = ϵη/[2(η 2 − ϵ 2)], Open Problem D.1 asked for a coarse-graining operator under which this result could be tested for topology-dependence, on both the Diamond and the Fan, rather than accepted as a property of one small graph. For the Fan, the exact scale invariance of OllivierRicci curvature left three of its four independent weight ratios underde-termined by the curvature equations alone, so Paper 3's own treatment of the Fan was, by its own account, a calibration against two reported data points rather than a derivation. This paper reports genuine, but dierent, progress on each. For the Diamond, we build the recursive lattice obtained by replacing every edge, at every generation, with a fresh copy of the same ve-edge motif, and establish two exact local curvature laws that hold at every generation. Combining them gives a closed-form renormalized ip threshold and cubic normal-form coecient at general coarse-graining depth d, which reduce exactly to the bare Diamond's own values at d = 4 and ow smoothly to a nite, nontrivial xed point, (λ flip , c 3) → (ϵ/η, −4/3), as d → ∞: the ip is coarse-graining-stable, and the generation step is exactly a MigdalKadano coupling recursion on c 0 := 8/d, with c * 0 = 0 its unique xed point. The universal endpoint ξ * = 2 found in Paper 3 for the dissipative extension is shown to be universal across the entire generational hierarchy as well. Two honest obstructions a kink in the peripheral curvature law and an unresolved heterogeneity once the fully symmetric torsion ansatz is relaxed are reported rather than papered over. For the Fan, we resolve Paper 3's scale-invariance obstruction with a dierent separation of variables: using the curvature equations for exactly what they determine the weight ratios and the middle torsion as functions of the one remaining free torsion T s and deferring the absolute scale entirely to the torsion equations. Eliminating the scale between the latter gives a single equation in T s ; at the test value η/ϵ = 1/2, and after a log-reparametrization removes a spurious numerical artifact, this equation has a genuine root, located to machine precision. In the regime containing this root, all four curvatures admit exact closed forms unavailable to Paper 3 veried against the underlying optimal-transport computation to machine precision; combining two of them in the limit T m → ∞ turns the numerical coincidence into a proof that η/ϵ → 1/2 as α → 0. 1 Read together, the two results are asymmetric in a way worth stating plainly: the Diamond's single free weight admits a genuine multi-generation renormalization-group trajectory; the Fan's four coupled weight classes admit, for now, an exact reduction of the bare motif itself the prerequisite the Diamond's own program needed before any coarse-graining could be built, not yet that extension itself.