FB(S³)R: Benchmark residual invariants in compact S³ cosmology

Framework proposes a benchmark-grade invariant cosmology on compact S³ geometry, where the dark sector is treated not as a speculative particle model, but as a rigorously reconstructible residual tensor target derived directly from Einstein geometry and observable matter content. Instead of introducing phenomenological dark-energy assumptions, the construction defines a reproducible mapping from invariant quantities to observational likelihoods for CMB, BAO, supernovae, lensing, growth and topology analyses. The central object is the residual stress tensor Tᴰᵤᵥ = c⁴/(8πG) Gᵤᵥ − Tᵛⁱˢᵤᵥ, interpreted as an invariant “data target” rather than a microscopic ontology. In this approach, compact spatial curvature with topology M ≃ I × S³ generates a discrete spectral structure where Laplacian eigenvalues follow λₙ = n(n + 2)/A², producing finite-volume covariance kernels and nontrivial low-mode effects absent in flat ΛCDM limits. The first nonzero spectral gap appears at λ₁ = 3/A², directly linking global curvature radius A to observable harmonic suppression and mode discretization. A key contribution of the framework is the transition from conceptual cosmology to a fully specified invariant-to-data pipeline. Observable spectra are reconstructed through compact transfer operators of the form Cˣʸₗ = Σₙ Aₙ Δˣₗₙ Δʸₗₙ, where admissible covariance coefficients satisfy positivity and convergence conditions Aₙ ≥ 0 and Σₙ Aₙ Ξₙ,A(0) < ∞. This converts topology and residual curvature from philosophical interpretation into directly falsifiable likelihood objects embedded in Bayesian inference and posterior predictive residual analysis. The work further integrates Noether-II gauge constraints, quasi-local Brown–York and Misner–Sharp energies, compact zero-mode conditions, and hyperspherical transfer-function technology into a unified predictive ledger. Importantly, the paper does not claim that topology alone explains cosmic acceleration or CMB anomalies; instead it defines a strict benchmark architecture capable of testing such hypotheses under reproducible numerical standards. The proposed benchmark object B = (I, Θ, π, T, N, V, R) formalizes invariant definitions, priors, transfer operators, covariance models, validation suites and public release manifests as mandatory components of scientific reproducibility. Within this formulation, compact S³ cosmology emerges not as an alternative metaphysical narrative, but as a transfer-grade geometric inference framework connecting curvature, topology, spectral discreteness and residual stress-energy directly to cosmological observables. The resulting programme establishes a falsifiable pathway for evaluating compact-universe scenarios against precision cosmological datasets while preserving full covariance, gauge consistency and coordinate independence.