We present a new theoretical framework for the halo mass function (HMF) that accurately predicts the abundance of dark matter halos over an exceptionally wide range of masses and redshifts, based on a generalised Press─Schechter model with triaxial collapse (GPS+). The HMF is formulated mainly as a function of the variance of the linear density field, with a weak explicit mass dependence and no explicit redshift dependence, which is able to naturally reproduce the correct normalisation and high-mass behaviour without requiring an empirical fitting. Using the UCHUUN-body simulation suite under Planck cosmology, combining six simulations with up to 300 realisations, we measured the HMF over 6.5 ≤ log(M200m/[h−1 M⊙]) ≤ 16 and 0 ≤ z ≤ 20 with reduced cosmic variance. Over this full domain, we find that GPS+ matches the simulations to within 10─20%, performing similarly to the Sheth─Tormen model at z ≲ 2, but with substantially results at higher redshifts. In the latter case, the Sheth─Tormen model can deviate by 70─80%, while GPS+ will remain within ∼20%. Finally, we show that the halo mass definition is key: M200m yields a nearly universal, weakly redshift-dependent HMF, whereas adopting the evolving virial overdensity from (Bryan, G. L. & Norman, M. L. 1998, ApJ, 495, 80) ends up degrading the agreement at low redshifts and high masses.