FB(S³)R: Hierarchical Bethe-Type States on Fibonacci Graphs over S³

In this work we construct a hierarchical family of Bethe-type spectral states generated by recursively defined finite Fibonacci graphs embedded over the compact manifold S³. Instead of modifying the operator setting or introducing phenomenological parameters, we explore a structurally minimal principle: spectral hierarchy can emerge from recursive algebraic compatibility alone. The key idea is conceptually simple but structurally powerful: recursive coupling enforces spectral organisation. Starting from finite adjacency operators defined on Fibonacci graph levels, each successive level is generated through a block-recursive construction preserving internal spectral consistency. The fundamental recursive structure is governed by operators of the form: Aₙ₊₁ = ( Aₙ Bₙ ; Cₙ Dₙ ) which induces recursive relations for characteristic polynomials Pₙ(λ) = det(λ I − Aₙ) allowing spectral information to propagate coherently across hierarchy levels. This mechanism produces a controlled spectral architecture in which eigenvalue structure, resolvent behaviour, and zero-mode multiplicities remain explicitly tractable. The hierarchy does not rely on continuum limits; instead, structured spectral organisation appears as a consequence of algebraic compatibility between successive finite operators. A central structural clarification introduced in this work is that nontrivial hierarchical spectral structure can arise entirely within finite operator systems. The recursive extension preserves spectral coherence while expanding the admissible configuration space in a controlled manner. Within this framework, Fibonacci growth provides a minimally redundant combinatorial scheme for extending adjacency operators without introducing uncontrolled degeneracies. Each hierarchical level enlarges the spectral structure while preserving compatibility constraints inherited from previous levels. Propagation of spectral constraints is governed by resolvent relations Rₙ(λ) = (λ I − Aₙ)⁻¹ ensuring that stability of the hierarchy can be analysed explicitly through operator norm bounds and multiplicity constraints for admissible zero modes. Conceptually, the work contributes to a broader programme exploring how discrete geometry and recursive algebraic structure constrain admissible spectra. In this perspective, recursive structure → spectral compatibility → hierarchical organisation emerges as a natural structural principle for finite operator systems defined over S³. The results provide a compact and extensible framework in which recursion, graph structure, and spectral constraints interact in a mathematically transparent way, offering a basis for further investigation of organised spectral hierarchies in finite systems.