Operator–Spectral Classification of Cosmological Asymmetries: Symmetry-Protected Modes vs Mixing-Induced Observables

This work introduces an operator–spectral diagnostic framework for classifying cosmological observables according to their dynamical status: symmetry-protected invariants versus mixing-induced, epoch-dependent quantities. The criterion is based on whether an observable corresponds to a kernel element (zero eigenvalue) of the governing evolution operator, or instead arises from nonlinear combinations of eigenmodes with distinct eigenvalues. Applied to ΛCDM cosmology, the framework shows that the dark energy density \rho_\Lambda is symmetry-protected, residing in the kernel of the cosmological evolution operator, while the commonly cited ratio \Omega_\Lambda / \Omega_m \approx 2.2 is not an eigen-observable and therefore cannot represent a fundamental invariant. This reframes the cosmological coincidence problem: the apparent “specialness” of the present-day ratio is structurally ill-posed, as such ratios necessarily evolve over cosmic time. The classification is supported by spectral analyses of independent observational contexts, including the radial acceleration relation (RAR) and the CMB power spectrum, and leads to falsifiable predictions regarding the constancy of \rho_\Lambda, the evolution of \Omega_\Lambda / \Omega_m, and the absence of special physics at the present epoch. The framework is diagnostic rather than explanatory and is intended as a reusable tool applicable beyond cosmology to any system governed by linear evolution operators.