Discrete spectral triples converging to dirac operators

We exhibit a series of discrete spectral triples converging to the canonical spectral triple of a finite dimensional manifold. Thus we bypass the non-go theorem of Gökeler and Schücker. Sort of counterexample. In [2] Connes gave the most general example of non commutative manifolds defined from finite spectral triples, and such discrete manifolds were classified independently by Krajewsky [5, 6] and Paschke and Sitarz [7] the same year. But almost the same time, Gökeler and Schücker [3] proved that the naive discretizations of Dirac Operators were not able to fulfil the axioms of non-commutative manifolds. This caused a fading in the attempts to illuminate NCG from the point of view of lattice theories. Later attempts to show NCG operations in lattices have preferred to avoid the full axiomatic required by K-theory and Reality . But the subsequent works on lattices, from the successful show of Lüscher with the index theorem, to the insistent studies of Jian Dai or M. Requardt, to name some ones, should invite us to try to develop the full geometric set-up. In [3], the non-go principles also ruled out some non-naive formulations. At that time, inspired on some previous speculations, I risked to suggest privately the use of a doubling of the representation of the algebra, thus a function A(x), x in one dimension, could be discretized to an operator A(i) with representation: A|v >= 