  The purpose of this letter is to remove the arbitrariness of the ad hoc
choice of the algebra and its representation in the noncommutative approach to
the Standard Model, which was begging for a conceptual explanation. We assume
as before that space-time is the product of a four-dimensional manifold by a
finite noncommmutative space F. The spectral action is the pure gravitational
action for the product space. To remove the above arbitrariness, we classify
the irreducibe geometries F consistent with imposing reality and chiral
conditions on spinors, to avoid the fermion doubling problem, which amounts to
have total dimension 10 (in the K-theoretic sense). It gives, almost uniquely,
the Standard Model with all its details, predicting the number of fermions per
generation to be 16, their representations and the Higgs breaking mechanism,
with very little input. The geometrical model is valid at the unification
scale, and has relations connecting the gauge couplings to each other and to
the Higgs coupling. This gives a prediction of the Higgs mass of around 170 GeV
and a mass relation connecting the sum of the square of the masses of the
fermions to the W mass square, which enables us to predict the top quark mass
compatible with the measured experimental value. We thus manage to have the
advantages of both SO(10) and Kaluza-Klein unification, without paying the
price of plethora of Higgs fields or the infinite tower of states.
