  The luminosity of the Tip of the Red Giant Branch (TRGB) is instrumental for
the construction of the distance ladder, and its accurate modelling is key for
determining the local Hubble parameter. In this work, we present an extensive
quantitative analysis of the TRGB luminosity, accounting for virtually all
input physics that affect it: chemical composition, opacity, diffusion, nuclear
reaction rates, electron screening, neutrinos, convection efficiency, boundary
conditions and mass loss, amongst others.
  Our analysis is based on a newly produced grid of $\sim 3\times 10^6$ TRGB
models, evolved from pre-main sequence up to the helium ignition at the TRGB,
and covering a wide range of metallicity (Z = 0.0001-0.02) and initial mass (M
= 0.8-1.4 $M_{\odot}$). Through a Monte-Carlo analysis, we study the systematic
variation of the TRGB luminosity due to the combined effect of all above input
physics, and show that a maximum theoretical uncertainty of about $1.6 \%$ is
still present on the current generation of models, dominated by systematics of
radiative opacity. Results are also provided in several photometric bands.
  As a by-product of our analysis, we demonstrate robust evidence for the
linear response of the Tip luminosity to individual changes of input physics,
which can significantly simplify future analyses. A comparison of our results
with other stellar evolution codes shows excellent agreement, while our grid of
models is available upon reasonable requests.
