FB(S³)R: A Noether-Consistent Framework for Quantum Cosmology on R × Σ with Compact Topology and Discrete φ-Embedding

This work presents a topology-explicit, variationally consistent framework for quantum cosmology on spacetimes of the form: M = ℝ × Σ, where the spatial hypersurface Σ is compact and boundaryless. In this setting, diffeomorphism invariance is treated via Noether’s second theorem: the associated Noether currents reduce (on shell) to superpotentials and define boundary charges (Iyer–Wald type). Consequently, for compact ∂Σ = ∅ topologies—most notably the spherical sector Σ ≃ S³—the diffeomorphism Noether charges vanish globally, leaving the Noether-II identity structure as the invariant content. A central contribution is an invariant discrete “embedding level” defining a φ-indexed scale ladder. The golden ratio φ is fixed by an invariance requirement of unambiguous refinement and composition of discrete scale levels, formulated via Zeckendorf-canonical representation. Two mathematically consistent realizations are developed:(i) a smooth embedding with a continuous scale coordinate, and (ii) a discrete floor-embedding that requires explicit junction conditions to maintain distribution-free Euler–Lagrange equations and on-shell current conservation across level transitions. Compact topology implies discrete mode spectra, yielding falsifiable signatures in cosmological observables, including low-ℓ structure in the CMB and infrared features in the stochastic gravitational-wave background. A likelihood-based verification protocol is provided, designed for implementation in standard cosmological pipelines (CLASS/CAMB with Planck likelihoods and related datasets), enabling direct confrontation with observations.