Probing the Hubble Tension with an Infinite-Future Condition on the Hubble Parameter

We study the impact of imposing an infinite-future condition on Gaussian-process (GP) reconstructions of $H(z)$ from 37 cosmic-chronometer measurements. Implementing the asymptotic limit $H(-1)=0$ expected in constant-$w$CDM with $w>-1$ as a pseudo-point at $z=-1$ with tunable uncertainty $\sigma_{-1}$ lowers the GP-inferred Hubble constant from $H_0=68.71\pm6.08$ to $H_0\simeq(64.67$--$65.86)\pm(4.45$--$4.85)~{\rm km\,s^{-1}\,Mpc^{-1}}$. The resulting $H_0$ remains within $\sim0.32$--$0.61\,\sigma$ of the \textit{Planck} $\Lambda$CDM value, while the separation from representative local distance-ladder measurements increases to $\sim1.45$--$1.83\,\sigma$. A scan over $\sigma_{-1}$ shows that the shift is governed by the effective pseudo-point weight, interpolating between the hard-condition and data-dominated limits. Finally, constant-$w$CDM Markov Chain Monte Carlo (MCMC) fits to the same $H(z)$ data with $H_0$ fixed to the GP-inferred values show that $\Omega_m$ is only weakly affected, whereas lower $H_0$ shifts $w$ toward less negative values; allowing curvature broadens constraints and remains consistent with $\Omega_k=0$.