Compact Spherical FLRW Residuals: Closure, Local ΛCDM Degeneracy, and Amplitude-Selection Boundaries

Compact spherical FLRW residuals are analyzed through a covariant Einstein-residual framework in which the effective dark-sector tensor is defined after a fixed visible-sector subtraction. The work proves a strict homogeneous closure theorem: within the locally FLRW residual class, the Einstein residual reduces to the minimal pressureless-current plus fixed-vacuum branch if and only if the closure conditions d (−pD) /d ln a = 0 and da³ (ϵD + pD) /d ln a = 0 hold simultaneously. In this case the residual uniquely takes the form ϵD (a) = ϵw, 0 a⁻³ + ϵq and pD (a) = −ϵq, producing exact local degeneracy with standard ΛCDM at the homogeneous background level. The construction does not claim a derivation of observed dark-sector abundances and does not claim observational detection of compact topology. Instead, it establishes an explicit amplitude-selection boundary: compact spherical topology can constrain global volume, quotient spectra, covariance structure, and observer-position dependence, yet topology alone cannot determine Ωc, Ω_Λ, or Λₑff. The paper carefully separates residual closure from microscopic origin and from phenomenological abundance selection, preventing the usual overextension of topological cosmology claims. A central result is the proof that the minimal homogeneous branch is locally indistinguishable from cold dark matter plus a cosmological constant with the same curvature and amplitudes. Any physical distinction must therefore arise only through quotient-projected spectra, compact covariance kernels, perturbative null tests, or additional microscopic amplitude-selection mechanisms. The work introduces a compact closure atlas based on the variables Aq (a) = −pD (a) and Aw (a) = a³ (ϵD + pD), showing that the entire separately conserved homogeneous residual sector reduces to a one-function flow in residual space, while the ΛCDM-equivalent branch appears as the constant nonnegative submanifold. The paper further develops a unified geometric and field-theoretic interpretation combining a conserved topological-current sector with a fixed four-form vacuum sector. The current branch is associated with compact spherical topology and winding structure, while the vacuum branch emerges from a standard four-form stress tensor satisfying Tq μν = −ϵq g_μν. Importantly, the analysis proves that topological winding alone does not generate dust behavior: the pressureless branch requires a coarse-grained Brown-current effective description with linear energy density. Likewise, discrete flux quantization supplies only an additional amplitude-selection constraint rather than a direct prediction of cosmological abundances. The broader significance of the framework lies in its conceptual clarity: it provides a mathematically controlled way to distinguish between background degeneracy, topological consistency, and genuine topology evidence. Rather than proposing a new expansion history, the work establishes rigorous null conditions under which reconstructed Einstein residuals may consistently be interpreted as a protected current plus fixed vacuum sector on compact spherical FLRW backgrounds. The result is therefore not a replacement for ΛCDM, but a sharply defined geometric residual formalism connecting compact topology, homogeneous stress-tensor closure, quotient projection, and cosmological perturbation theory within a convention-explicit covariant framework.