Compact S³/Γ Topological Residual Cosmology: Invariant Closure, Protected Sectors, and Perturbative Null Tests

Compact S³/Γ Topological Residual Cosmology proposes a disciplined way to treat the dark sector not as an assumed substance, but as an invariant residual target reconstructed after the metric and visible sector are fixed: Tᴰμν: = c⁴Gμν/ (8πG) − Tvisμν. The central idea is that compact positive-curvature FLRW cosmology supplies protected geometric and topological slots — finite volume, discrete spherical spectra, quotient structure S³/Γ, winding labels, and compact four-form sectors — but does not by itself determine the observed amplitudes of dark matter or dark energy. In this framework, topology gives the allowed invariant structure; amplitude selection must come from a separate physical rule, likelihood, or completion mechanism. The paper separates two questions that are often conflated: closure and origin. The minimal current/form branch closes the homogeneous residual only when the reconstructed dark sector has the exact form ϵD (a) = ϵw, 0 a⁻³ + ϵq and pD (a) = −ϵq. Equivalently, the two diagnostic invariants Aq (a): = −pD (a) and Aw (a): = a³ϵD (a) + pD (a) must be constant and nonnegative. This gives a sharp null-test structure: dpD/d ln a = 0 and da³ϵD + pD/d ln a = 0. If these conditions fail, the minimal homogeneous branch is falsified under the adopted visible-sector subtraction. A key result is the distinction between protected sectors and numerical abundances. The topology of S³/Γ can protect winding/current labels such as Q ∈ Z and flux labels such as N ∈ Z, while the actual values of Ωw, Ωq, or Λeff remain undetermined without an amplitude selector. This prevents the model from becoming a vague “topology explains everything” claim. The paper also shows that positive curvature selects the S³ cover, while simple connectedness is required to select the manifold S³ itself; without that assumption, spherical quotients S³/Γ remain admissible. The perturbative sector is equally restrictive. A fixed four-form branch carries no independent scalar perturbation, while the current branch must reduce to a pressureless geodesic component unless extra sectors are explicitly added. Thus the model must pass not only background tests, but also compact-mode perturbative null tests: no residual pressure, no residual anisotropic stress, no non-CDM sound speed, and no hidden extra perturbative degrees of freedom. The relevant bookkeeping is not ordinary flat Fourier space, but the compact spherical spectrum, with modes such as λn = n (n+2) /A² and quotient-projected Γ-invariant harmonics. The strongest scientific value of this framework is therefore its falsifiability. It does not claim that the Universe is proven to be globally S³, nor that topology alone derives dark matter or dark energy. Instead, it defines a compact residual-closure programme: Tᴰμν → Aq (a), Aw (a) → background null tests → perturbative null tests → topology likelihood → amplitude selection. If the residual satisfies these invariant tests, compact topological residual cosmology remains a viable conditional-origin framework. If any test fails, the minimal branch is ruled out rather than rescued by unstated assumptions.