Geometric–Thermodynamic Background Constraints from Pantheon+ and DESI 2024 - A First Observational Layer of a Relativistic Consistency Program.

We perform a fully late–time geometric reconstruction of the cosmic expansion history using only low–redshift observables, combining the Pantheon+ standardized supernova sample with the DESI 2024 baryon acoustic oscillation (BAO) measurements. The supernova likelihood is analytically marginalized over the absolute magnitude offset, removing any dependence on the Hubble constant, while the BAO likelihood is profiled over the nuisance combination α ≡ r_d H_0 / c, ensuring complete independence from early–time physics and cosmic microwave background priors. Within a minimal holographic event–horizon deformation of the background geometry, characterized by a single parameter c, we obtain joint constraints Ω_m0 = 0.2685 ± 0.0148 and c = 1.15 ± 0.12, together with a precise late–time determination of α = 0.03287 ± 0.00016, providing a direct constraint on the product r_d H_0 without early–universe calibration. The analysis demonstrates that supernova and BAO distance measurements already form an observationally closed geometric system at late times, enabling a self–contained reconstruction of the background expansion history. This preprint constitutes the background layer (Stage C1) of a staged relativistic consistency program aimed at isolating the physical origin of cosmic acceleration.