Invariant Residual–Transfer Completion: A Dynamical Audit Framework for Dark-Sector and Topology Claims

Invariant Residual–Transfer Completion extends the earlier compact spherical FLRW residual programme into a broader invariant audit and transfer-completion framework for dark-sector and topology claims. The previous version established the homogeneous Einstein-residual closure boundary: the reconstructed residual reduces to the ΛCDM-degenerate pressureless-current plus fixed-vacuum form when d (−pD) /d ln a = 0 and da³ (ϵD + pD) /d ln a = 0 hold simultaneously. The present work generalizes this result into a full invariant completion hierarchy linking residual reconstruction, perturbative transfer structure, covariance prediction, and observational admissibility. The central object is the convention-audited residual Tᴰ, λ_μν = c⁴ (G_μν + λg_μν) / (8πG) − Tᵛⁱˢ_μν, making explicit that no dark-sector reconstruction is convention-free until both the visible subtraction and Einstein-side λ convention are specified. The compact closure atlas Aq = −pD and Aw = a³ (ϵD + pD) is retained from the earlier framework but upgraded into a dynamical completion diagnostic: constant nonnegative Aq and Aw still identify the exact dust–vacuum normal form, while any atlas drift requires an explicit exchange current within the same residual convention. A major extension of this version is the explicit separation of physical claim layers: residual reconstruction ≠ homogeneous closure ≠ source action ≠ amplitude selection ≠ transfer evolution ≠ covariance prediction ≠ likelihood evidence ≠ blind validation. This structure is encoded in the audit chain R ≺ H ≺ S ≺ A ≺ T ≺ C ≺ L ≺ B, where each layer requires its own invariant data and completion conditions. The framework therefore blocks hidden upgrades such as interpreting background degeneracy as microphysical origin, compact topology as an abundance selector, or covariance deformation as observational detection. Compared with the earlier compact FLRW residual work, this release adds the missing transfer-completion layer. Compact topology enters observables only through projected eigenspaces, quotient covariance, observer-dependent sky maps, transfer kernels, perturbative evolution, and likelihood comparison. Topology alone does not determine Ωc, Ω_Λ, Λₑff, or any dark-sector abundance. The framework also introduces a finite diagnostic vocabulary for incomplete cosmological proposals, including convention defects, closure defects, exchange-current defects, transfer residuals, quotient-covariance anisotropy, observer dependence, benchmark unfairness, and blind-validation failure. The result is a falsifiable invariant audit calculus for compact-topology, modified-gravity, protected-sector, and dark-residual cosmology proposals. This version does not claim detection of compact topology, does not derive observed dark-sector abundances, and does not replace ΛCDM. Its purpose is stricter: to define the invariant data required before a cosmological framework can be consistently upgraded from residual reconstruction to physical source realization, from homogeneous degeneracy to perturbative completion, and from topological consistency to observational topology evidence. Dark Sektor Residual (custom GPT) Invariant residual-closure evaluator for dark-sector hypotheses, perturbative consistency, conservation laws, operator completeness, and observational viability across both compact and open cosmological frameworks, including S³ and local R³ limits, through adversarial scientific analysis.