Latent-Space Gaussian Processes for Dark-Energy Reconstruction from Observational $H(z)$ Data

Using the 37-point cosmic-chronometer subset of observational Hubble parameter (OHD) data, we develop a Bayesian Gaussian-process framework to reconstruct the normalized dark-energy density $f(z)$ and equation of state $w(z)$, focusing on how the choice of latent space affects the inference. We compare a Gaussian-process prior placed directly on $f(z)$ with the conventional latent-$H$ formulation, and also test a log-$f$ branch that enforces $f(z)>0$. We further analyze OHD-like mock data generated from fiducial $\mathrm{\Lambda }$CDM and mildly evolving $w_0{w}_a$ models, using both the observed redshift distribution and a higher-quality high-redshift setup. For real OHD, leave-one-out cross-validation shows no strong predictive preference between latent-$f$ and latent-$H$ reconstructions. The inferred $f(z)$, $w(z)$, and $Om(z)$ remain consistent with $\mathrm{\Lambda }$CDM across the tested external priors, while apparent $Om(z)$ trends are prior sensitive and not robust evidence for dark-energy evolution. Residual differences between the two latent constructions are small, sign mixed, prior dependent, and mainly confined to the weakly constrained high-redshift tail. We therefore interpret the real-data results primarily as a methodological assessment. In mock tests, the framework responds to injected mild evolution in the reconstructed dark-energy quantities and $Om(z)$, with detectability depending on method and data coverage. Improved high-redshift OHD reduces the discrepancy between latent constructions and makes the $Om(z)$ response more consistently detectable. The latent-$f$ approach is therefore a viable alternative to latent-$H$, while current constraints are limited mainly by sparse high-redshift OHD and dependence on external priors.