FB(S³)R: Compact S³ Cosmology, Quasi-local Diffeomorphism Charges, and Discrete φ-Level Embedding

The work develops a cosmological framework with a compact spatial section S³, where both geometry and dynamics are encoded through quasi-local diffeomorphism charges and a discrete φ-level embedding. The central claim is that topology acts as a physical regulator, not just a background condition, directly shaping the spectrum of observables and the structure of gravitational degrees of freedom. A key technical component is the spectral analysis of operator families Aₙ acting on field modes, with eigenvalue structure determined through det(λ I − Aₙ). The compactness of S³ enforces a discrete spectrum λ, eliminating the need for artificial ultraviolet cutoffs. This discreteness is not imposed but emerges naturally, yielding a self-regularizing theory in which ultraviolet and infrared sectors are simultaneously controlled. In this sense, S³ geometry replaces conventional renormalization prescriptions with a geometric quantization mechanism. The introduction of quasi-local charges provides a refined notion of conserved quantities adapted to compact manifolds. These charges are constructed from diffeomorphism generators but remain sensitive to finite regions, encoding local curvature and field data while respecting global constraints. Crucially, their algebra reflects a nontrivial closure structure, where locality and topology are inseparably intertwined. A central innovation is the discrete φ-level embedding, which organizes field configurations into stratified layers labeled by φ. Rather than a continuous parameter, φ defines a discrete hierarchy of embeddings, leading to a quantized layering of geometric data. This structure induces a selection rule on admissible modes and constrains transitions between sectors, effectively acting as a topological selection principle. The interplay between these ingredients reveals a coherent picture: S³ compactness → spectral discreteness → quasi-local charge algebra → φ-level stratification. Together, they define a framework in which gravitational dynamics is encoded not only in differential equations but also in the algebraic and topological structure of the configuration space. Beyond its formal construction, the model suggests a reinterpretation of cosmological observables. Quantities traditionally viewed as continuous may instead arise from underlying discrete spectra, with measurable consequences tied to the eigenvalues λ and their distribution. The appearance of expressions like det(λ I − Aₙ) is therefore not merely technical, but indicative of a deeper principle: cosmology as a spectral theory on compact manifolds. This perspective opens a pathway toward unifying geometric, algebraic, and spectral methods in cosmology, where discreteness is not an approximation but a fundamental feature emerging from topology itself.