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- wCDM with $w>-1$ as a pseudo-point at $z=-1$ with tunable uncertainty ${\sigma }_{-1}$ lowers the GP-inferred Hubble constant from $H_0=68.71\pm 6.08$ to H 0 ≃ (64.67–65.86) ± (4.45– $4.85)\mathrm{k}m{s}^{-1}Mp{c}^{-1}$. The resulting H 0 remains within  ∼ 0.32–0.61  σ of the Planck ΛCDM value, while the separation from representative local distance-ladder measurements increases to  ∼ 1.45–1.83  σ. 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- wCDM Markov Chain Monte Carlo (MCMC) fits to the same H( z) data with H 0 fixed to the GP-inferred values show that Ω m is only weakly affected, whereas lower H 0 shifts w toward less negative values; allowing curvature broadens constraints and remains consistent with ${\Omega }_k=0$.