  The brightness of the tip of the Red Giant Branch is a useful reference
quantity for several fields of astrophysics. An accurate theoretical prediction
is needed for such purposes. Aims. We intend to provide a solid theoretical
prediction for it, valid for a reference set of standard physical assumptions,
and mostly independent of numerical details. We examine the dependence on
physical assumptions and numerical details, for a wide range of metallicities
and masses, and based on two different stellar evolution codes. We adjust
differences between the codes to treat the physics as identical as possible.
After we have succeeded in reproducing the tip brightness between the codes, we
present a reference set of models based on the most up to date physical inputs,
but neglecting microscopic diffusion, and convert theoretical luminosities to
observed infrared colours suitable for observations of resolved populations of
stars and include analytic fits to facilitate their use. We find that
consistent use of updated nuclear reactions, including an appropriate treatment
of the electron screening effects, and careful time-stepping on the upper red
giant branch are the most important aspects to bring initially discrepant
theoretical values into agreement. Small, but visible differences remain
unexplained for very low metallicities and mass values at and above 1.2 Msun,
corresponding to ages younger than 4 Gyr. The colour transformations introduce
larger uncertainties than the differences between the two stellar evolution
codes. We demonstrate that careful stellar modeling allows an accurate
prediction for the luminosity of the Red Giant Branch tip. Differences to
empirically determined brightnesses may result either from insufficient colour
transformations or from deficits in the constitutional physics. We present the
best-tested theoretical reference values to date.
