We present a new theoretical framework for the halo mass function (HMF) that accurately predicts the abundance of dark matter haloes across an exceptionally wide range in mass and redshift. Building on a generalised Press & Schechter model and triaxial collapse (GPS+), we predict the HMF in terms of the variance of the linear density field, with only a weak explicit dependence on halo mass and no explicit dependence on redshift. The GPS+ model naturally provides the correct normalization and high-mass behaviour without requiring empirical fitting. We calibrate and validate the GPS+ model using the Uchuu N-body simulation suite, which combines large cosmological volume and high mass resolution under Planck cosmology. Using six simulations with up to 300 realizations, we obtain precision HMF measurements spanning halo masses in the range 6.5<=log(M200m/[h^-1^M_{sun}_])<=16 over 0<=z<=20, with reduced cosmic variance. Across this full domain, the GPS+ model reproduces the simulated HMF with deviations typically below 10-20%. Comparison with the Sheth-Tormen (ST) model shows similar performance at z<~2, but markedly improved agreement at higher redshifts, where ST can deviate by 70-80% while our model remains within ~20%. Finally, we assess the impact of the halo mass definition: adopting the evolving virial overdensity of Bryan & Norman (1998ApJ...495...80B) worsens agreement at low redshift and high masses, whereas M200m yields a more universal, nearly redshift-independent HMF.