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

Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model