We construct a hierarchical quantum graph Gn on the compact three-sphere S3 whose vertex count follows a generalised Fibonacci recursion and whose radial scale obeys Rn = lPlφn. On this graph we introduce a Bethe-ansatz-inspired hierarchical wave-function ansatz (FBA) whose quasi-momenta and scattering phases are set by the golden-ratio geometry. We prove asymptotic orthonormality of FBA states, derive block-recursive structure of the adjacency matrix, establish an asymptotic spectral gap of the graph Laplacian, and show that the con- nectivity density Cn → φ−1 is a topological invariant of the hierarchy. The framework is compatible with Bell-inequality violation via the Hopf-bundle structure of S3 as analysed in companion work. Phenomenological consequences include a CMB spectral tilt ns ≈ 0.965 and an effective equation of state w ≈ −1 without a cosmological constant. Keywords: Bethe ansatz; Fibonacci graph; fractal hierarchy; three-sphere; golden ratio; quantum nonlocality; compact cosmology.
Preece et al. (Mon,) studied this question.