Cosmographic admissibility and weak-field recovery in an analytic infrared-completed f ( Q ) gravity model

We construct and test an analytic infrared-completed Formula: see text gravity model through an admissibility chain that combines cosmographic, background-observational, and local solar-system diagnostics. The analysis is carried out in a clearly specified symmetric teleparallel setting: the cosmological evolution is evaluated in the spatially flat FLRW coincident branch, while the solar-system sector is formulated separately in the local weak-field and approximately spherically symmetric branch. The model is an analytic deformation of STEGR whose correction sector starts at order Formula: see text near Formula: see text and saturates at large Formula: see text, thereby preserving the leading low-Formula: see text normalization and suppressing large deviations during the matter-dominated regime. Its coefficients are fixed algebraically by Formula: see text and the present deceleration parameter Formula: see text, so the background branch contains no hidden fitting freedom once Formula: see text are specified. We derive the modified Friedmann equation, the regularity function Formula: see text, and the calibration formulae, and we map the admissible region under branch continuity, Formula: see text, Formula: see text, present acceleration, and matter-era recovery. Beyond the deceleration parameter, we compute the jerk hierarchy Formula: see text and the Formula: see text diagnostic, finding for the fiducial branch Formula: see text and Formula: see text. We also benchmark the calibrated branch against cosmic-chronometer Formula: see text data and supernova- and BAO-style distance residuals relative to Formula: see textCDM. Finally, we evaluate the local non-metricity scale at solar-system radii, recover the Newtonian limit at Formula: see text, compare the resulting PPN envelope with the Cassini/Shapiro and perihelion-level scales, and report the solar-system-filtered admissible region. The resulting framework provides a compact and reproducible admissibility pipeline linking analytic Formula: see text model building, extended cosmography, background data diagnostics, and local weak-field consistency.