This strict revised version presents a technically strengthened analysis of the FB (S³) R model in the limiting regime a → ℓP, where the classical FLRW description approaches the Planck boundary and standard perturbative quantum-field methods face ultraviolet divergence. The work no longer treats the Planck limit as an ontological “core” or as a completed quantum-gravity theory. Instead, it formulates a precise effective framework: a compact S³ spatial sector, a self-adjoint Wheeler–DeWitt boundary at a = ℓP, bounded curvature under explicit semiclassical assumptions, and spectral zeta regularisation of compact vacuum modes. The main improvement of this revision is the replacement of broad claims by conditional mathematical statements. The topology is now restricted to the smooth, closed, orientable, simply connected sector, where the Poincaré–Perelman theorem selects Σ ≃ S³. This makes the topological claim precise: the paper studies a well-defined compact sector rather than asserting that all cosmological topology is uniquely fixed. A second major refinement concerns the Planck-scale curvature analysis. The Kretschmann scalar is treated in the closed-FLRW form, and the result is stated as a conditional boundedness theorem: K (tP) ≤ C / ℓP⁴, provided the regularised scale factor, H, and Ḣ remain finite at the Planck boundary. This is explicitly presented as a semiclassical regularity result, not as a complete derivation of quantum gravity. The vacuum-energy section has also been substantially strengthened. Instead of assigning a fixed universal value to vacuum energy, the paper introduces a scale-covariant compact spectral family with ωₖ (a) = νₖ / a and obtains the general compact-spectrum scaling ρᵥacʳeg (a) = Cζ / (2π²a⁴). For the conformally shifted scalar reference spectrum, this gives ρᵥacʳeg = 1 / (480π²a⁴). The result is therefore a finite regularised vacuum contribution, not a prediction of the observed late-time dark-energy density. The role of the golden ratio has been clarified. φ is not a fundamental physical constant and is not inserted as an independent dynamical parameter. It appears as the Fibonacci asymptotic ratio φ = lim Fₙ₊₁ / Fₙ, functioning as a coarse-graining invariant of the compact S³ spectral hierarchy. The Fibonacci structure organises the spectrum; it does not replace the exact spectral zeta regularisation. A further key improvement is the Wheeler–DeWitt treatment. The effective configuration space is now a ∈ [ℓP, ∞), and the sub-Planck region is not assigned physical probability support. The Planck boundary is handled through a self-adjoint boundary condition, for example cos α Ψ (ℓP) + sin α ℓP Ψ′ (ℓP) = 0, ensuring zero probability flux through the boundary and conservation of the physical norm. The Noether analysis is now explicitly framed through Noether’s second theorem. Diffeomorphism invariance on compact S³ yields Bianchi-type differential identities, while full-slice Hamiltonian/Noether boundary charges vanish because ∂S³ = ∅. At the same time, the revised version correctly notes that quasi-local charges on internal subdomains are not excluded, avoiding the overstatement that compactness eliminates all meaningful energy notions. Overall, this revision turns the earlier Planck-limit manuscript into a stricter compact-topology framework. Its central contribution is the coherent combination of compact S³ geometry, conditional bounded curvature, finite spectral-zeta vacuum scale, self-adjoint Wheeler–DeWitt boundary dynamics, Fibonacci spectral coarse-graining, and Noether-II constraint structure. The result should be read as a mathematically disciplined Planck-boundary model within the FB (S³) R programme, not as a replacement for ΛCDM and not as a complete theory of quantum gravity.
Preece et al. (Tue,) studied this question.