Why it is a question
García-Pérez, Boguñá and Serrano derive ⟨k⟩′ = r^ν⟨k⟩ with ν = 2/(γ−1) − 1 or 2/β − 1, so that the flow runs to a complete graph if γ < 3 or β < 2 and to a ring otherwise. That derivation treats the coarse-grained graph as an S¹ model with renormalized parameters at every separation. THEORY-190 shows the closure holds only asymptotically: between neighbouring supernodes, where most links lie, the union of member links exceeds the S¹ form, by up to 86% in a worked example. Whether the flow and its phase boundaries survive that excess, the record cannot say. Until 2026-10-20 this was part of THEORY-190’s promote_when.
What would answer it
- An exact ν(γ, β, r), or a bound on it, for the coarse-grained graph itself rather than for its first-order S¹ approximation, confirming or correcting the boundaries γ = 3 and β = 2.
- An exact computation of ⟨k⟩′ for small r on the actual union of member links, compared with the source’s series for r = 2.
What would not answer it: more real networks whose layers collapse, or the source’s first-order flow again. The question is about what the model does under the transformation.