A tale of two attoms (new hypotheses should not be half-hearted but doubled)

This post is a continuation of a more than 1 year oder one written in French. To target a wider audience I have switched to the lingua franca of contemporary science literature since then.

Quantum History starts with Atoms of matter confirmed by Perrin...
Following wikipedia let's recall that Jean Perrin (30 September 1870 – 17 April 1942) was a French physicist who, in his studies of the Brownian motion of minute particles suspended in liquids, verified Albert Einstein’s explanation of this phenomenon and thereby confirmed the atomic nature of matter. He was awarded the physics Nobel prize in 1926 for having ‘put a definite end to the long struggle regarding the real existence of molecules’. Perrin reviewed and explained in a very pedagogical way his (and former) studies to establish the reality of atoms and molecules in a book entitled Atoms published in French in 1913 and in English in 1916. Here is an extract from the foreword:

Atoms Jean Perrin 1916

Quantum Higgstory leads to the two "attoms" of spacetime uncovered by Chamseddine, Connes et Mukhanov 
Following parallel lines to the former paragraph one could say that the periphysicist what I am is a French blogger (born 15 November 1972) who, in posts about the zitterbewegung of chiral fermions through the Higgs boson vacuum, tries to understand and publicize the so called Connes' construction of the standard model that leads to a spectral noncommutative unification of the four fundamental forces. This latter ambitious program has recently inspired a bold hypothesis : the existence of quanta of geometry which happen to come in two kinds. This might be a first step towards a theory of quantum gravity and it appears to be connected to the mimetic dark matter proposed by Chamseddine and Mukhanov. Unlike Perrin, I have a modest talent to popularization so I would just propose the two quanta of geometry to be dubbed attoms and provide serious interesting bibliography and relevant extracts to support my moonshine talkings. Needless to say that the experimental confirmation of attoms of spacetime will be difficult. So I do not pretend to be able to provide any proof of the attoms hypothesis today but simply indulge myself to imagine a possible narrative describing this fascinating hypothesis of theoretical and mathematical physics rooted in already validated experiments (I have Higgs bosons and cosmic microwave background in mind).

So the tale would start following the explanation of the "gauging" of the Higgs boson thanks to noncommutative geometry as explained below by nobody less than R. Brout  one of the fathers of the (LA)BEH(GHKW) mechanism:  

R. Brout Aug 1999

Personally I would like to parallel the role of the Brownian motion for the proof of atoms with the unloved or just naive enough to be dangerous zitterbewegung interpretation of the BEH mechanism and its possible connection with attoms. To explain very briefly how the noncommutative geometrization of physics leads to them, let's quote their fathers:

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics.. Physical applications of this quantization scheme will follow in a separate publication.
(Submitted on 4 Nov 2014)

The separate publication will have to meet great expectations. Said it differently Chamseddine, Connes and Mukhanov and their coworkers and followers are still writing the story. Here is an excerpt from the last development:
... we conclude that in noncommutative geometry the volume of the compact manifold is quantized in terms of Planck units. This solves a basic difficulty of the spectral action [1] whose huge cosmological term is now quantized and no longer contributes to the field equations... 
One immediate application is that, in the path integration formulation of gravity, and in light of having only the traceless Einstein equation ..., integration over the scale factor is now replaced by a sum of the winding numbers with an appropriate weight factor. We note that for the present universe, the winding number equal to the number of Planck quanta would be ∼ 1061 [9]
(Submitted on 8 Sep 2014 (v1), last revised 11 Feb 2015 (this version, v4))

Time will tell if this 1061 number of Planck quanta will play for the attoms hypothesis the one Avogadro constant played for the demonstration of atoms. But even if it doesn't its name is easily found: the Eddington-Dirac-Jordan constant of course! 

//addendum 31 october 2015
With hindsight I propose a better name for the number of spacetime at(t)oms in the present universe : Archimedes observable! The reading of the last words in the famous Sand Reckoner by one of the oldest known physicist-mathematician shows indeed an approximate coïncidence too beautiful to stay unquoted:
 ... it is obvious that the multitude of sand having a magnitude equal to the sphere of the fixed stars which Aristarchus supposes is smaller than 1000 myriads of the eighth numbers*.
King Gelon, to the many who have not also had a share of mathematics I suppose that these will not appear readily believable, but to those who have partaken of them and have thought deeply about the distances and sizes of the earth and sun and moon and the whole world this will be believable on the basis of demonstration. Hence, I thought that it is not inappropriate for you too to contemplate these things.
3rd century B.C
1000 myriads of the eighth numbers = 1063