A unified cosmographic reconstruction of the Hubble expansion: Bayesian constraints on dark energy and modified-gravity diagnostics

We introduce the adaptive expansion-index (AEI) parametrization, a compact and physically regularized representation of late-time expansion histories, and present it as the central methodological contribution of this work. Rather than parametrizing H(z), q(z) or w(z) directly, we model the logarithmic expansion index ε(z) ≡ dln H/dln (1 + z) = 1 + q(z) with a three-parameter saturating profile that interpolates smoothly between a matter-like regime at high redshift and a de Sitter attractor in the future. The model integrates analytically to E(z)=[((1+z)3ν+κ )/(1+κ )]ε ₘ/(3ν ) and contains spatially flat ΛCDM exactly as the interior point (εm, ν) = (3/2, 1), so that departures from the concordance model are tested within a nested parameter space. By construction the expansion rate is positive, finite-time future singularities are absent, the transition redshift is analytic, and the deceleration, jerk, snap and lerk parameters follow from an exact recursion. Because the ansatz encodes assumptions of smoothness, monotonicity, a single transition and regular asymptotics, the reconstruction is parametric rather than model-independent, and its parameters are best read as coordinates within this representation. We constrain the model with 34 cosmic-chronometer measurements, 13 DESI DR2 BAO distance ratios with full covariance, and Pantheon+ supernovae with SH0ES calibration, treating the sound horizon rd as free. We find no statistically compelling departure from ΛCDM, while the H0 discrepancy persists as a shift in rd. Quantities tied to high-redshift, asymptotic or extrapolated behaviour, including the cosmic age, higher-order cosmography, the effective dark-energy equation of state and illustrative f(T), f(Q) and f(R) reconstructions, are more parametrization-dependent and are presented as applications of the framework.