Einstein-Residual Closure Standard: Invariant Dark-Sector Audits, Normal Forms, and Topology-Likelihood Boundaries

Einstein Residual Closure Standard develops a formal invariant standard for evaluating dark-sector closure claims, compact-topology reconstructions, and residual cosmology proposals within Einstein-form gravity. Building on the earlier residual-transfer completion framework, the paper upgrades the programme from a general audit methodology into a stricter closure-standard formalism centered on residual admissibility, normal forms, topology-likelihood boundaries, and perturbative completion requirements. The central object is the convention-explicit Einstein residual Tᴰ, λ_μν = c⁴ (G_μν + λg_μν) / (8πG) − Tᵛⁱˢ_μν, defined only after both the visible-sector subtraction and Einstein-side λ convention are fixed. The work emphasizes that residual reconstruction, source realization, amplitude selection, perturbative transfer structure, covariance prediction, and observational evidence are distinct mathematical layers and cannot be promoted into one another without additional invariant data. A major result of this version is the formulation of a full Einstein-residual closure standard based on invariant admissibility criteria, closure-defect diagnostics, normal-form classification, and topology-likelihood separation principles. The framework introduces a finite audit structure for incomplete cosmological proposals, including convention defects, closure defects, exchange-current defects, transfer residuals, quotient-covariance anisotropy, benchmark unfairness, and blind-validation failure. The paper retains the compact closure atlas Aq = −pD and Aw = a³ (ϵD + pD), showing that constant nonnegative atlas entries uniquely identify the exact dust–vacuum normal form ϵD (a) = ϵw, 0 a⁻³ + ϵq, pD (a) = −ϵq. However, the present work extends this beyond homogeneous reconstruction by defining explicit admissibility boundaries for perturbative transfer structure, covariance evolution, and topology-dependent observational claims. A central theme of the framework is the strict separation between compact-topology consistency and actual topology evidence. Compact topology enters observables only through projected eigenspaces, quotient covariance, transfer kernels, observer-dependent sky maps, perturbative evolution, and likelihood comparison. Topology alone does not determine Ωc, Ω_Λ, Λₑff, or any dark-sector abundance, and background degeneracy alone does not constitute topology detection. The companion paper, A Residual First Standard for Dark Sector Closure and Compact Topology Tests, provides a shorter operational version of the framework focused on residual admissibility, closure logic, and compact-topology test structure. Together, the two papers define a reproducible invariant audit standard for evaluating dark-sector, modified-gravity, compact-topology, and protected-sector cosmology proposals. This release does not claim detection of compact topology, does not derive observed dark-sector abundances, and does not replace ΛCDM. Its purpose is stricter: to establish a formal residual-closure standard specifying which invariant structures, perturbative sectors, transfer operators, covariance predictions, and likelihood mappings must exist before a cosmological proposal can be considered physically closed, observationally testable, or perturbatively complete. 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.