Covariant beyond-Horndeski closure of the bounce sector in a cyclic early-dark-energy cosmology

We construct an explicit covariant beyond-Horndeski completion of the high-density bouncesector of a cyclic cosmological trajectory that contains a low-density double-exponential early-darkenergy(EDE) episode. The target background was previously obtained as an effective numericalcycle with expansion, a transient EDE fraction, curvature-driven turnaround, contraction, and asecond bounce. Here we address the main theoretical limitation of that construction: the highdensitybounce was imposed through an effective modified Friedmann equation. Adopting thecovariant beyond-Horndeski reconstruction of Mironov, Rubakov, and Volkova, we write an explicitaction with F(π,X) = f0(π) + f1(π)X + f2(π)X2, G4(π,X) = 1/2 + g40(π) + g41(π)X, andF4(π,X) = f40(π) + f41(π)X. The reconstructed solution matches the final contracting high-densitystate of the effective trajectory at ρ/ρc = 0.961533 with |ΔH| = 2.8×10−17, reproduces the measuredbounce slope ˙HB = 0.669005, and shifts the bounce position by only 3.16 × 10−4 e-folds. The twoindependent background equations are satisfied with maximum residuals 5.9×10−16 and 2.7×10−15.The same covariant solution crosses H = 0 and generates the expanding branch without prescribinga separate post-bounce trajectory. Away from the regular γ-crossing neighborhood, the scalarperturbation coefficients satisfy min GS = 3.070, minFS = 2.165, and 0.165 < c2s< 0.962, while thetensor sector is positive and luminal. We further examine covariant replacements for two constitutiveingredients of the effective model: an ekpyrotic scalar with w > 1 for relative shear suppressionand a field-dependent kinetic normalization Z(χ) for contraction-dependent scalar damping. Thesereplacements remove the need to interpret the original friction coefficients as fundamental laws,but a microscopic derivation of energy transfer into the primordial radiation sector remains open.The result is therefore a covariant and perturbatively stable closure of the bounce sector, not yet amicroscopic completion of the entire cyclic matter sector.