A new way towards the Theory of Everything: Universe conjectured as a 3D Newtonian lattice and Matter conjectured as topological singularities of this lattice
Is the Universe a Lattice?
One fundamental problem of modern physics is the search for a theory of everything able to explain the nature of space-time, what matter is and how matter interacts. There are various propositions, as Grand Unified Theory, Quantum Gravity, Supersymmetry, String and Superstring Theories, and M-Theory. However, none of them is able to consistently explain at the present and same time electromagnetism, relativity, gravitation, quantum physics and observed elementary particles.
Here, one explains that, by developing a complete theory of the deformation of solid lattices using Euler’s coordinates [1,2], this one can be used for the description of the spatiotemporal evolution of the Universe instead of the general relativity. In this way, it is suggested that the Universe could be a massive elastic three-dimensional lattice described by using Euler’s coordinates in the absolute space of an observer situated outside the Universe, and that fundamental building blocks of Ordinary Matter could consist of topological singularities of this lattice, namely diverse dislocation loops, disclination loops and dispiration loops [3,4,5]. One finds then, for an isotropic elastic lattice obeying Newton’s law, with specific assumptions on its elastic properties, that the behaviors of this lattice and of its topological defects display “all” known physics. Indeed, this theory contains intrinsically and allows one to deduce directly the various formalisms of electromagnetism, special relativity, general relativity, gravitation and quantum physics. It allows also one to give simple answers to some longstanding questions of modern cosmology, as the universe expansion, the big-bang and the dark energy. But it appears above all a completely new scalar charge, the curvature charge, which has no equivalence in the modern physical theories, which creates a very small deviation to the equivalence principle of Einstein between inertial mass and gravitational mass, and which allows one to give very simple explanations of the weak asymmetry observed between matter and anti-matter, the origin of the weak interaction force, the formation of galaxies, the disappearance of antimatter from the universe, the formation of gigantic black holes in the heart of the galaxies and the nature of the famous dark matter. Moreover, studying lattices with axial symmetries, one was able to identify a lattice structure whose topological defect loops coincide exactly with the complex zoology of elementary particles, and which allows one to explain quite simply the asymptotical behavior and the nature of the strong interaction force.
One fundamental problem of modern physics is the search for a theory of everything able to explain the nature of space-time, what matter is and how matter interacts. Since the 19th century, physicists have attempted to develop unified field theories , which would consist of a single coherent theoretical framework able to account for several fundamental forces of nature. For instance:
– Grand Unified Theory  merges electromagnetic, weak and strong interaction forces,
– Quantum Gravity , Loop Quantum Gravity  and String Theories [16-23] attempt to describe the quantum properties of gravity,
– Supersymmetry [10-15] proposes an extension of the space-time symmetry relating the two classes of elementary particles, bosons and fermions,
– String and Superstring Theories [16-23] are theoretical frameworks incorporating gravity in which point-like particles are replaced by one-dimensional strings, whose quantum states describe all types of observed elementary particles,
– M-Theory [24-32] is a unifying theory of five different versions of string theories, with the surprising property that extra dimensions are required for its consistency.
However, none of them is able to consistently explain at the present and same time electromagnetism, relativity, gravitation, quantum physics and observed elementary particles. Many physicists believe now that 11-dimensional M-theory is the theory of everything. However, there is no widespread consensus on this issue and, at present, there is no candidate theory able to calculate the fine structure constant or the mass of the electron. Particle physicists expect that the outcome of the ongoing experiments – search for new particles at the large particle accelerators and search for dark matter – are needed to provide further input for a theory of everything.
Four years ago, a theory was proposed [1,2], which lays methodically the foundations of an original approach by Euler coordinates of the solid lattices deformations, using only the Newton’s law and the two first principles of the thermodynamics as fundamental physical principles.
The concept of tensor dislocation charges and tensor disclination charges within a lattice was also introduced in details. This new charge concept allows one to quantify the linear topological singularities, which can appear at the microscopic scale of a solid lattice, such as dislocations and disclinations. But localized topological singularities have also been described, such as twist disclination closed loops presenting a scalar charge of rotation, responsible for a divergent field of rotation vectors, analogous to the electrical charge responsible for a divergent electrical field, or edge dislocation closed loops presenting a scalar charge of curvature, responsible for a divergent field of curvature vectors presenting some analogy with the space curvature of general relativity.
Numerous analogies appeared between this eulerian theory of deformable media and the theories of electromagnetism, gravitation, special relativity and general relativity, reinforced by a possible solution of the famous paradox of electron field energy. These analogies were surprising and remarkable but, by far, not perfect. It was then tantalizing to analyze much more carefully these analogies and to try to find how to perfect them. The purely qualitative description, step by step, of the main results recently obtained [3,4,5] in this search is the subject of this paper, in which it is suggested that Universe could be a massive elastic 3D-lattice, and that fundamental building blocks of Ordinary Matter could consist of topological singularities of this lattice, namely diverse dislocation loops and disclination loops. We find, for an isotropic elastic lattice obeying Newton’s law, with specific assumptions on its elastic properties, that the behaviours of this lattice and of its topological defects display “all” known physics, unifying electromagnetism, relativity, gravitation and quantum physics, and resolving some longstanding questions of modern cosmology. Moreover, studying lattices with axial symmetries, represented by “colored” cubic 3D-lattices, one can identify a lattice structure whose topological defect loops coincide with the complex zoology of elementary particles, which could open a promising field of research.
Eulerian deformation theory of newtonian lattices
When one desires to study the solid deformation, one generally uses lagrangian coordinates to describe the evolution of the deformations, and diverse differential geometries to describe the topological defects contained in the solid.
The use of lagrangian coordinates presents a number of inherent difficulties. From the mathematical point of view, the tensors describing the continuous solid deformation are always of order higher than one concerning the spatial derivatives of the displacement field components, which leads to a very complicated mathematical formalism when the solid presents strong distortions (deformations and rotations). To these mathematical difficulties are added physical difficulties when one has to introduce some known properties of solids. Indeed, the lagrangian coordinates become practically unusable, for example when one has to describe the temporal evolution of the microscopic structure of a solid lattice (phase transitions) and of its structural defects (point defects, dislocations, disclinations, boundaries, etc.), or when it is necessary to introduce some physical properties of the medium (thermal, electrical, magnetic or chemical properties) leading to scalar, vectorial or tensorial fields in the real space.
The use of differential geometries in order to introduce topological defects as dislocations in a deformable continuous medium has been initiated by the work of Nye  (1953), who showed for the first time the link between the dislocation density tensor and the lattice curvature. On the other hand, Kondo  (1952) and Bilby  (1954) showed independently that the dislocations can be identified as a crystalline version of the Cartan’s concept  of torsion of a continuum. This approach was generalized in details by Kröner  (1960). However, the use of differential geometries in order to describe the deformable media leads very quickly to difficulties similar to those of the lagrangian coordinates system. A first difficulty arises from the complexity of the mathematical formalism which is similar to the formalism of general relativity, what makes very difficult to handle and to interpret the obtained general field equations. A second difficulty arises with the differential geometries when one has to introduce topological defects other than dislocations. For example, Kröner  (1980) has proposed that the existence of extrinsic point defects could be considered as extra-matter and introduced in the same manner that matter in general relativity under the form of Einstein equations, which would lead to a pure riemannian differential geometry in the absence of dislocations. He has also proposed that the intrinsic point defects (vacancies and interstitials) could be approached as a non-metric part of an affine connection. Finally, he has also envisaged introducing other topological defects, as disclinations for example, by using higher order geometries much more complex, as Finsler or Kawaguchi geometries. In fact, the introduction of differential geometries implies generally a heavy mathematical artillery (metric tensor and Christoffel symbols) in order to describe the spatiotemporal evolution in infinitesimal local referentials, as shown for example in the mathematical theory of dislocations of Zorawski  (1967).
In view of the complexity of calculations in the case of lagrangian coordinates as well as in the case of differential geometries, it seemed that it would be better to develop a much simpler approach of deformable solids, but at least equally rigorous, which has been published in [1,2], in which one develops a new and original eulerian approach of the deformation of solids. The deformation of a lattice is characterized by distortions and contortions. A vectorial representation of the tensors, presenting undeniable advantages over purely tensorial representation thanks the possibility to use the powerful formalism of the vectorial analysis, allows to obtain the geometro-compatibility equations of the lattice which insure its solidity, and the geometro-kinetics equations of the lattice, which allow one to describe the deformation kinetics.
Regarding the description of defects (topological singularities) which can appear within a solid, as dislocations and disclinations, it is a domain of physics initiated principally by the idea of macroscopic defects of Volterra  (1907). This domain experienced a fulgurant development during the twentieth century, as well illustrated by Hirth  (1985). The lattice dislocation theory started up in 1934, when Orowan  , Polanyi  and Taylor  published independently papers describing the edge dislocation. In 1939, Burgers  described the screw and mixed dislocations. And finally in 1956, Hirsch, Horne et Whelan  and Bollmann  observed independently dislocations in metals by using electronic microscopes. Concerning the disclinations, it is in 1904 that Lehmann  observed them in molecular crystals, and in 1922 that Friedel  gave them a physical explanation. From the second part of the century, the physics of lattice defects has grown considerably.
In [1,2], the dislocations and the disclinations are approached by introducing intuitively the concept of dislocation charges by using the famous Volterra pipes  (1907) and an analogy with the electrical charges. With Euler coordinates, the concept of dislocation charge density appears then in an equation of geometro-compatibility of the solid, when the concept of flux of charges is introduced in an equation of geometro-kinetics of the solid.
The rigorous formulation of the charge concept in the solids makes the essential originality of this approach of the topological singularities. The detailed development of this concept leads to the appearance of tensorial charges of first order, the dislocation charges, associated with the plastic distortions of the solid (plastic deformations and rotations), and of tensorial charges of second order, the disclination charges, associated with the plastic contortions of the solid (plastic flexions and torsions). It appears that these topological singularities are quantified in a solid lattice and that they have to appear as strings (thin tubes) which can be modelized as unidimensional lines of dislocation or disclination, or as membranes (thin sheets) which can be modelized as two-dimensional boundaries of flexion, torsion or accommodation.
The concept of dislocation and disclination charges allows one to find rigorously the main results obtained by the classical dislocation theory. But it allows above all to define a tensor of linear dislocation charge, from which one deduces a scalar of linear rotation charge, which is associated with the screw part of the dislocation, and a vector of linear flexion charge, which is associated with the edge part of the dislocation. For a given dislocation, both scalar and vectorial charges are perfectly defined without needing a convention at the contrary of the classical definition of a dislocation with its Burger vector! On the other hand, the description of the dislocations in the eulerian coordinate system by the concept of dislocation charges allows one to treat exactly the evolution of the charges and the deformations during very strong volumetric contractions and expansions of a solid medium.
Cosmic lattice and cosmological expansion
In [1,2], numerous analogies appeared between the eulerian theory of deformable media and the theories of electromagnetism, gravitation, special relativity and general relativity, reinforced by a possible solution of the famous paradox of electron field energy. The existence of analogies between the theories of continuum mechanics and solid defects and the theories of electromagnetism, special relativity and gravitation has already been the subject of several publications, from which the more famous are certainly those of Kröner [37,38]. Excellent reviews in this physics field have also been published, in particular by Whittaker  (1951) and Unzicker  (2000). But none of these publications has gone as far as the approach published in [1,2] concerning these highlighted analogies. These analogies were so surprising and remarkable that it was tantalizing to analyze them much more carefully and to try to find how to perfect them, what was done and published in [3,4,5].
By choosing a particular imaginary lattice with a free energy per unit volume which depends linearly and quadratically on its volume expansion, and quadratically on its shears and on its local rotations, this lattice presents a pure transversal waves propagation only if these waves are circularly polarized, and does not present longitudinal waves when the volume expansion of the lattice is smaller than a critical value. In this case, the longitudinal waves propagation is replaced by the appearance of non-propagating local modes of longitudinal vibrations of the volume expansion of the lattice.
In the presence of a gradient of the scalar expansion field with a spherical symmetry, the transversal wave rays are bended, and this curvature can reach a point of no return if the gradient is sufficient, leading to a surprising analogy with the photons sphere of a black hole.
Adding an initial expansion kinetic energy to this lattice, it can expand from a singular point in space. This phenomenon presents first a very quick expansion, which is followed by a slowing down, then a re-acceleration of the expansion. Under certain conditions on the elastic free energy of the lattice, this expansion can be followed by a reverse cycle of contraction. These behaviors present a disturbing similarity with the cosmological theories of Big Bang and Big Crunch and, in the case of this lattice one finds a very simple explanation of the dark energy of the astrophysicists.
From these analogies with the non-propagation of longitudinal waves and the wave rays curvature of general relativity, as well as with the cosmological expansion of universe, this lattice has been called the Cosmic Lattice.
Maxwell’s equations and special relativity
The evolution equations of such a lattice when the field of expansion is homogeneous and constant are absolutely similar to the Maxwell’s equations of electromagnetism . This analogy includes not only the two Maxwell’s equation couples, but also all the electromagnetism phenomenologies, such as the dielectric properties of matter, the diamagnetic, paramagnetic and ferromagnetic properties of matter, as well as the electric charges and currents. And all these phenomenologies are associated with topological singularities moving inside the lattice. In particular, the rotation charge of a twist disclination loop is the perfect analogy of the electrical charge.
The localized topological singularities (for example the closed loops of edge dislocation and twist disclination) meet a relativistic dynamics inside the lattice, which results in the appearance of the Lorentz transformation. The twist disclination loops presenting a scalar rotation charge are also submitted to a force, which is perfectly analogous to the Lorentz force acting on the electrical charges.
In fact, the cosmic lattice behaves as the ether with regard to moving clusters of topological singularities. These moving clusters present a contraction of length and a time dilation, which depend on their velocity with regard to the lattice, similar to the effects of the special relativity, but which are perfectly real and measurable by an observer situated outside the lattice. This allows this external observer to explain very simply the famous twin paradox of the special relativity.
Local observers, which would be constituted from moving clusters of topological singularities, have no physical way to measure their own velocity with regard to the lattice, which leads them to postulate the classical special relativity in their reference frame, with all the paradoxes which are associated with its use.
Newton’s gravitation and general relativity
The presence of a localized cluster of topological singularities leads to a divergent perturbation of the scalar volume expansion in its vicinity, which depends at the same time on the lattice distortion energy induced by the cluster, on its scalar curvature charge and on its scalar rotation charge.
One can also imagine macroscopic topological defects of the lattice, as a hole in the lattice with a vacancy nature or a lattice portion inclusion with an interstitial nature. These two defects are topologically perfectly symmetrical, but their properties are very different inside the lattice. They present in fact very interesting analogies, respectively with the black holes and the neutron stars of the astrophysics.
It appears several interaction properties between the various topological defects of the lattice through their own expansion field, but with a large domination of the effects due to the distortion energy of the singularities. This leads to the existence of a dominant interaction force between two topological singularities, which presents, at great distances, a perfect similarity with the Newton’s gravitational interaction force between two massive bodies.
The measuring rods and clocks peculiar to the reference frame of a local observer (himself constituted of a cluster of lattice topological singularities) seem always to him as perfectly immutable, which leads him to postulate the invariance of the transversal waves propagation velocity in his frame, when an observer situated outside the lattice can measure enormous variations of the length of the measuring rods, of the time measured by the local clock and of the wave propagation celerity in the frame of the local observer as a function of the local variations of the scalar expansion of the lattice.
It is possible to find a metric translating the behaviors of the measurement rods and the clocks in the vicinity of a spherical cluster of topological singularities, which becomes perfectly identical to the Schwarzschild metric of general relativity at sufficient distance of the cluster. As a consequence, the curvature of transversal wave rays in the vicinity of a cluster are qualitatively and quantitatively identical to the light rays curvature calculated by general relativity in the vicinity of massive celestial objects.
This metric becomes somewhat different from the Schwarzschild metric in closed proximity of the cluster of topological singularities. In the case of a black hole, if the radius of the Schwarzschild sphere (the point of no return) remains the same as the radius of Schwarzschild in general relativity, the radius of the photon sphere becomes identical to the radius of the Schwarzschild sphere and the radius of infinite time dilation becomes null in the case of the cosmic lattice, which is much more satisfactory to the spirit than a photon sphere radius corresponding to 3/2 of the Schwarzschild radius and an infinite time dilation radius corresponding to ½ of the Schwarzschild radius as calculated by the Schwarzschild metric in general relativity.
This metric leads also the external observer to formulate simple and decoupled equations in order to describe the geometric curvature of the lattice and the time gap between the local clocks in the vicinity of a cluster of topological defects. For the local observers, the same physical quantities are described by the very complex equations of space-time curvature of the general relativity.
Weak interaction force and cosmological evolution of matter
It appears also a very short range interaction force, corresponding in fact to a capture potential, between the rotation charge of a twist disclination loop and the curvature charge of an edge dislocation loop, which leads to the formation of a stable topological loop of dispiration. This force presents a surprising similarity with the weak interaction force of the standard model of elementary particles, whereas there is no charge analogous to the curvature charge in all the modern physical theories.
The existence of this curvature charge which has no analogous in the modern physical theories leads to surprising effects. Indeed, the interstitial or vacancy nature of the topological loops of edge dislocation involved in the loops of composed topological singularities is associated to the nature of particle or antiparticle, with respectively a weak gravitational component of repulsion or attraction, which is subtracted or added to the main attractive gravitational component related to the distortion energy of the singularity. This effect creates a very small deviation to the equivalence principle of Einstein between inertial mass and gravitational mass, and it explains perfectly the weak asymmetry existing between matter and antimatter, with an attractive gravitational component very slightly higher for the antimatter. Moreover, only the pure interstitial edge dislocation loop, corresponding to the neutrino, presents really antigravity. This gravitational effect of the curvature charge allows one to explain very simply several phenomena which are badly or even not explained in the scenario of cosmological evolution of matter, as the precipitation of matter and anti-matter in the form of galaxies, the disappearance of anti-matter from the universe, the formation of gigantic black holes in the center of the galaxies, and the nature of the dark matter of the astrophysicists.
The formation and the cooling of the cosmic microwave background, the Hubble’s law and the redshift of the galaxies are also very well explained in this scenario of cosmological evolution of matter.
Quantum physics and particles spin
The existence of localized perturbations of the volume expansion instead of longitudinal waves propagation if the scalar expansion of the lattice is smaller than a critical value allows one to find mathematically and exactly the Schrödinger equation of the quantum physics for the description of the spatiotemporal behavior of the topological expansion perturbations associated to the moving topological singularities. As a consequence, one can explain the quantum wave function associated with a topological singularity as the local amplitude and phase of the volume expansion fluctuations, in other words as the gravitational fluctuations associated with this singularity.
This interpretation of the quantum wave function is perfectly matching the Bohm’s interpretation of quantum physics, and allows also one to give simple explanations to the superposition and the exclusion principles, as well as to the existence of bosons and fermions.
The loops of topological singularities do not admit a static solution for the field of gravitational perturbations (volume expansion perturbations) in their immediate vicinity. This means that the loops have to spin around one of their axes in order to find a dynamic solution of the expansion perturbation field. It appears then a compulsory quantification of their angular momentum, perfectly similar to the spin of elementary particles, and one shows that this rotation motion implies a magnetic momentum in the case of the twist disclination loop, analogous to the electrical charge, and that it does not violate the laws of special relativity.
On the other hand, there can exist transversal waves packets propagating inside the lattice, which are submitted to the condition that they present a non-null helicity so that their energy is conserved. These wave packets present a kind of “malleability”, a plasticity of their extension in space, without loss of their identity. This implies all the properties of non-locality, of linear momentum, of wave–particle duality, of entanglement and of decoherence of the photons of modern physics.
Standard model of elementary particles and strong interaction force
One can imagine a “colored cubic lattice”, with specific properties of stacking and of rotation of three types of “colored” planes, which allows one to construct a set of topological singularities perfectly similar to the various elementary particles of the standard model, with singularities corresponding to the leptons and singularities corresponding to the quarks. The singularities corresponding to the quarks have mandatory, for topological reasons, to combine as pairs (baryons) or as triplets (mesons) of loops, bound by a force generated by a ribbon of “colored” stacking faults. This binding force presents in fact all the asymptotic properties of the strong force of the standard model.
In this colored lattice, one can also perfectly identify topological singularities corresponding to the W and Z bosons, as well as to the gluons.
On the other hand, the fact that it is possible to replace an edge dislocation loop by an edge disclination loop, with a rotation angle of 90° or 180°, inside a composed topological loop of dispiration allows one to explain simply the existence of three families of particles in the standard model.
Finally, the massive entity associated with each cell of the cosmic lattice could correspond to the Higgs particle of the standard model, with the consequence that the nature of this massive entity is completely different from the nature of the other particles, as these ones correspond to topological singularities of the lattice.
It is remarkable that the description, using Euler’s coordinates in an absolute space-time frame, of a massive “colored” cubic 3D-lattice containing loop topological singularities and having very particular elastic properties allows one to find analogies with all observed natural physical phenomena.
It appears that this theory is the first and only (i) to combine all known physics in a very simple manner, including electromagnetism, relativity, gravitation and quantum physics, (ii) to give a simple meaning to the local space-time and the quantum behavior of topological singularities, (iii) to find a completely new scalar curvature charge which creates a very small deviation to the equivalence principle of Einstein between inertial mass and gravitational mass, and which allows very simple explanations of the weak asymmetry observed between matter and anti-matter, the weak interaction force, the formation of galaxies, the disappearance of antimatter, the formation of gigantic black holes in the heart of the galaxies and the famous dark matter, (iv) to propose simple explanations to well-known problems of modern cosmology, as the universe expansion, the big-bang and the dark energy, and (v) to propose a model of “colored” cubic 3D-lattice whose diverse microscopic loop singularities correspond exactly to each of the elementary particles of the three families of particles of the standard model, and which allows to give a simple structural explanation to the strong interaction force.
 G. Gremaud, “Théorie eulérienne des milieux déformables – charges de dislocation et désinclinaison dans les solides”, Presses polytechniques et universitaires romandes (PPUR), Lausanne (Switzerland) 2013, 751 pages, ISBN 978-2-88074-964-4
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 G. Gremaud, “Universe and Matter conjectured as a 3-dimensional Lattice with Topological Singularities”, Amazon, Charleston (USA) 2016, 650 pages, ISBN 978-2-8399-1934-0
 G. Gremaud, “Universe and Matter conjectured as a 3-dimensional Lattice with Topological Singularities”, July 2016, Journal of Modern Physics, 7, 1389-1399, DOI 10.4236/ jmp.2016.712126
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Detailed author’s references
 Livre: G. Gremaud, “Théorie eulérienne des milieux déformables – charges de dislocation et désinclinaison dans les solides”, Presses polytechniques et universitaires romandes (PPUR), Lausanne 2013, 751 pages
ISBN 978-2-88074-964-4, disponible sur PPUR
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 Book: G. Gremaud, “Eulerian theory of newtonian deformable lattices – dislocation and disclination charges in solids” , Amazon, Charleston (USA) 2016, 312 pages
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 Livre: G. Gremaud, “Univers et Matière conjecturés comme un Réseau Tridimensionnel avec des Singularités Topologiques”, Amazon, Charleston (USA) 2016, deuxième édition, 664 pages
ISBN 978-2-8399-1940-1, disponible sur Amazon, CreateSpace et Kindle
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 Book: G. Gremaud, “Universe and Matter conjectured as a 3-dimensional Lattice with Topological Singularities”, Amazon, Charleston (USA) 2016, second edition, 650 pages
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 Article: G. Gremaud, “Universe and Matter conjectured as a 3-dimensional Lattice with Topological Singularities”, July 2016, Journal of Modern Physics, 7, 1389-1399 , DOI 10.4236/jmp.2016.712126
 Article: G. Gremaud, “Maxwell’s equations as a special case of deformation of a solid lattice in Euler’s coordinates”, September 2016, arXiv :1610.00753 [physics.gen-ph]
 Livre: G. Gremaud, “Univers et Matière conjecturés comme un Réseau Tridimensionnel avec des Singularités Topologiques”, première édition, 2015,
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 Book: G. Gremaud, “Universe and Matter conjectured as a 3-dimensional Lattice with Topological Singularities”, first edition, 2015,
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 Illustrated summary (V10): “Universe and Matter conjectured as a 3-dimensional Lattice with Topological Singularities”, Lausanne, March 2017, 41 pages
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 Summary of the theory: “Could the Universe be a massive elastic 3D-lattice and ordinary matter consist of topological singularities?”, Lausanne, November 2014
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