Quantum–Kinetic Dark Energy (QKDE): An effective dark energy framework with a covariantly completed time-dependent scalar kinetic normalization

A minimal, effective dark-energy framework—Quantum–Kinetic Dark Energy (QKDE)—is developed in which the scalar kinetic normalization carries a slow background time dependence through a covariantly completed clock field χ such that K = K(χ) > 0, while the Einstein–Hilbert metric sector remains unmodified. The effective action S = ∫ d⁴x √−g [½ M²ₚₗ R + K(χ)X − V(ϕ)] admits a diffeomorphism-invariant completion; working in unitary gauge χ = t reproduces the background equations employed numerically in this work. Within the EFT–DE description this corresponds to αₖ = (K̇ ϕ̇²)/(H² M²ₚₗ) > 0 with αᴮ = αᴹ = αᵀ = αᴴ = 0, so tensors are luminal and the Planck mass is constant. Within this effective framework, a closed first-order background system in e-fold time is obtained; scalar perturbations propagate with c²ₛ = 1, satisfy Φ = Ψ, and source linear growth through the unmodified metric Einstein equation. The scalar-field equation takes the form of an exchange equation with the clock sector, while the total energy–momentum tensor is covariantly conserved. All observable signatures therefore enter solely through the expansion history H(a) and the induced growth D(a). Two kinetic normalizations are treated in detail: (i) a curvature-motivated form K = 1 + αR / M², for which an iteration-free algebraic identity for K′/K is derived; and (ii) a phenomenological running K = 1 + K₀ (1 + z)ᵖ. A reproducible numerical pipeline is provided together with a Fisher setup based on exact variational (sensitivity) equations for distances, H(z), and fσ₈(z). Stability and admissibility reduce to K(χ) > 0 and a nonvanishing algebraic denominator in the curvature case. The framework yields sharp, falsifiable null predictions on linear scales: μ(a, k) = Σ(a, k) = 1, η(a, k) = 0, c²ᵀ = 1; any statistically significant deviation lies outside the effective QKDE baseline. The framework is interpreted as an effective, unitary-gauge cosmological description arising from a covariantly completed theory, rather than as a manifestly covariant scalar–tensor model written directly in fixed time slicing.