Goodness-of-fit test for isochrone fitting in the Gaia era. Statistical assessment of the error distribution

[...] This paper presents a rigorous derivation of a goodness-of-fit statistics for colour-magnitude diagrams (CMD). We discuss the reliability of the underlying assumptions and their validity. We derived the distribution of the sum of squared Mahalanobis distances of stellar data and theoretical isochrone for a generic set of data and models. We applied this to the case of synthetic CMDs constructed to mimic real data of open clusters in the GAIA sample. We analysed the capability of distinguishing among different sets of input physics and parameters that were used to compute the stellar models. We generated synthetic clusters from isochrones computed with these perturbed quantities, and we evaluated the goodness-of-fit with respect to the unperturbed isochrone. We show that when $r$ magnitudes are available for each of the $N$ observational objects and $p$ hyperparameters are estimated in the fit, the error distribution follows a $\chi^2$ distribution with $(r-1)N - p$ degrees of freedom. We show that the linearisation of the isochrone causes negligible deviation from this result. We investigated the possibility of detecting the effects on stellar models that are induced when varying convective core overshooting efficiency, $^{14}$N$(p,\gamma)^{15}$O reaction rate, microscopic diffusion velocities, outer BCs, and colour transformation. The results suggest that it is possible to detect the effect induced by only some of the perturbed quantities. [...] A variation in the convective core overshooting efficiency was detectable only for photometric errors of 0.003 mag and only for the 1 Gyr case. The effects induced by the outer boundary conditions and the bolometric corrections are the largest. [...] As a last exercise, we addressed the validity of the goodness-of-fit statistics for real-world open cluster CMDs, contaminated by field stars or unresolved binaries. [...]