The natural question to ask now could be what role $\sin \left[\Phi \right]$ plays in Equation (5a), or Equation (5a) itself through that factor. To answer this, we must think about the process we are dealing with. By Equation (5) we are not calculating the energy of a particle but the energy of particle formation from a wave packet (from the SWP itself from its constituents). A formation that carries with it an energy and that obeys a certain process whose energy balance for a closed system is null. A process which, in short, is a process of sudden densification, such as that which necessarily occurs in collisions for this purpose, in which all the energy collapses, going from being photonic, if that is the case, to corpuscular, to a dense object with a kinetic development, which may be null for some relative system.

Consequently, whether we are talking about the formation of the particle derived from a previous annihilation (in which all the energy becomes of the electromagnetic species) and subsequent recombination, or about a first formation (from that same electromagnetic species), there will always exist from some relative system a term (5a) which will be nothing but an energy surplus or energy differential between the starting electromagnetic energy (characterised for each inertial reference system by its frequency) and the one made available for the corpuscular formation. A surplus of kinetic energy that will have to be returned in the annihilation processes (becoming tangible for all the relative systems in which it is pertinent to do so), through its extinction through condition (13) of the phase factor, as we have already seen to the generational changes in [1] , and we will detail now, as the only way to carry it out in a sudden way.

The term ${\bar{E}}_k$ of Equation (5a) is, therefore, an extension or generalization of the relativistic dynamic expression $E_k$ for this circumstance, as reflected in Equation (8) itself, shown by a wave term. Extension that is there, but which is not shown because its application in corpuscular phenomenology is non-existent (although it is present in all of it) except for the aforementioned annihilation

processes ( $\sin {\left[\Phi \right]}_{\upsilon }=0$) or for those in which matter eventually manifests itself

as a wave, either because of its duality (matter-wave) or because of the very process of material formation we are dealing with, which necessarily involves this duality. Generally speaking, in this attempt to interpret and reconcile the two physical realities, the term wave would only apply to those processes in which the dynamical parameters cannot reside circumstantially in the corpuscle or cannot manifest themselves through it.

On which it is worth reiterating that we speak of applicability and not of presence, since in all corpuscular phenomenology kinetic energy is represented by ${\bar{E}}_k$, although we only perceive its corpuscular component $E_k$ (which according to the equations do not exist by itself), and that, consequently (we nuance), it is not that ${\bar{E}}_k$ is an extension or generalisation of $E_k$, but that $E_k$ is a functional restriction of ${\bar{E}}_k$. A restriction that we can associate to

$\sin {\left[\Phi \right]}_{\upsilon }=1$ (although it does not necessarily have to be so, as we shall see) for

this case, that is, for the one ${\bar{E}}_k=E_k$ and is given (shaping all our reality) after the process of formation (and its subsequent evolution), in which the dynamic parameters can reside in the corpuscle (which is why it does not do so explicitly in the phase factor).

Following on from the above, we can think that, in fact, we could have at some time $\sin {\left[\Phi \right]}_{\upsilon }=1$ for the formed particle, since, whether it comes from a previous annihilation or change of generation ( $\sin {\left[\Phi \right]}_{\upsilon }=0$) or not (a process of initial formation or a dual state with $0<\sin {\left[\Phi \right]}_{\upsilon }<1$), at some point in the material creation the pro-corpuscle could present itself without kinetic energy ( $E_k=0$), densified but alien to any wave activity (equivalent to $\sin {\left[\Phi \right]}_{\upsilon }=1$). In other words, if we say that at some time all the factors of Equation (5a) are undifferentiated within the integral of the equation, we are saying that $\sin {\left[\Phi \right]}_{\upsilon }$ does not exist as such or equivalently that $\sin {\left[\Phi \right]}_{\upsilon }=1$.

What we cannot consider as true, given the dependence of the phase factor $\sin {\left[\Phi \right]}_{\upsilon }$ with $\upsilon$ ( ${\left[\Phi \right]}_{\upsilon }=\left[\Delta k\left(\upsilon t-x\right)\right]$), is that it remains $\sin {\left[\Phi \right]}_{\upsilon }=1$ for the subsequent evolution, which would even make the subsequent generational changes ( $\sin {\left[\Phi \right]}_{\upsilon }=0$) unfeasible. From which we conclude, since the energy integral evolves as if it were so ( $\sin {\left[\Phi \right]}_{\upsilon }=1$), that in this case, the phase factor is not applicable ( $\sin {\left[\Phi \right]}_{\upsilon }\in \left]0,1\right[$) (because it is in another phase, as we have already mentioned) or is not effective in the calculation of the energy value (in fact we do not need it to know $E_k$), which would have been fixed ( ${\bar{E}}_k=E_k$) through the value $\sin {\left[\Phi \right]}_{\upsilon }=1$ previously reached after the formation of the corpuscle (which is what marks really this applicability).

That is to say, $\sin {\left[\Phi \right]}_{\upsilon }\left(\in \left]0,1\right[\right)$ is not applicable, after the formation of the particle, because the particle is already configured (another phase) or because it represents, in the best case, a sort of inappreciable perturbation or an average null perturbation on the base state $\sin {\left[\Phi \right]}_{\upsilon }=1$ reached. It follows, on the other hand, that the evolution of the variable $\upsilon$ for $\sin {\left[\Phi \right]}_{\upsilon }>0$, that is to say, for all of $\upsilon <{\upsilon }_p$, ceases (definitively) to have any repercussion or to be significant, and consequently, we can consider the notational suppression (which we had even remarked) of the said variable $\sin \left[\Phi \right]\equiv \sin {\left[\Phi \right]}_{\upsilon }$ to be appropriate, although it continues to play its role.

Notwithstanding this simplification, the phase factor is still there, it is not extinguished and has its importance beyond the energy computation. A kinetic energy without a phase factor, as a principle of corpuscular promotion and extinction, is, in fact, less feasible, and less assumable for the understanding of physical processes (that is a way towards questioning the current physics paradigm), than a phase factor without a energy element associated clearly. A phase factor that, in the worst case, would always allow us to establish the connection between the two species, build that energy element from its absence, and understand the changes it involves, which is ultimately what we have done in all the previous development [1] .

In addition to the above, the term (5a) does not speak exclusively of the kinetic energy of the particle, which implies a massive coefficient as a reference, but of that same energy as a coefficient or multiplying factor, in such a way that the same as in Equation (3) we had for the wave function Ψ an envelope

$\psi =\sin \left[\Delta k/2\left(\upsilon t-x\right)\right]/\left(\upsilon t-x\right)$ or body of waves, which represented an amplitude or measurable magnitude on an elementary wave ${\text{e}}^{\pm i\left(k_{0}x-w_{0}t\right)}$, here we have, energetically speaking, the group constituted in dynamic particle (assimilated by $E_k$), also as amplitude or measurable magnitude, on the sine part of the pulse $\sin \left[\Phi \right]=\sin \left[\Delta k\left(\upsilon t-x\right)\right]$, that is, on the form of the pulse itself but without

packing, taken as an elementary wave or energy carrier.

If we look at this is what we have achieved by the SWP with respect to the formation of a wave packet type, the construction of a carrier (coming out of the energy integral), which is not just any carrier but an (energy) carrier containing the group velocity (and not the phase velocity, as usual) and the momentum of the wave packet, that is, of the material particle, which is why this carrier, which we have called the phase factor, should be called the phase factor of the material object or wave of the matter, which more clearly stated would be nothing but a matter wave, since it embodies this functionality and fulfils all the formal requirements, supporting the dynamical characteristics of the particle.

Although a wave packet is made up of a wave body or modulated amplitude and its carrier wave (two factors, therefore), the energy of the carrier has no impact on the energy balance (all the energy is concentrated in that amplitude). In our case, although our carrier is for an amplitude, which represents energy (supported on a particle), it is a wave, i.e. we are also talking about two elements

( $E_k\times \sin \left[\Phi \right]$) with a multiplicative relation, in which, likewise, the carrier ( $\text{sin}\left[\Phi \right]$) represents an invaluable summand in the total energy balance because all the energy is concentrated in the other summand ( $E_k$).

Consequently, we have a phase factor as a carrier which, like any carrier, does

not affect the energy value of the particle, and therefore any $\text{sin}\left[\Phi \right]\ne 0$ state is inapplicable, so that, being $\text{sin}\left[\Phi \right]=1$ transparent, $\text{sin}\left[\Phi \right]\in \left]0,1\right[$ appears, at best, as a negligible energy perturbation on the newly constituted material state, capable, however, of collapsing or undoing the modulation it supports (as any carrier would do), when it itself, for $\text{sin}\left[\Phi \right]=0$, collapses.

The formation of the corpuscle, therefore, which, as we said, corresponds in reality to the energetic transit of the other two terms, does not annihilate totally the wave nature of the first of them, but leaves the oscillating part, that elementary wave (carrier), which ordinarily does not manifest itself (all our dynamics is sufficiently understood without it), as a sample or vestige (matter wave), outside the spatial limits of the particle, but associated with it. And, in particular (about the differentiated phenomenology of application), as a manifestable wave register (wave-particle duality) of its corpuscular kinetic parameters, as an authentic carrier or catalyst of the physical reality of the particle, of its true and hidden wave nature. This is the least restricted functionality of the phase factor $\text{sin}\left[\Phi \right]$.

Wave-particle duality which takes place, either by means of the transfer of these parameters to the phase factor (which in these circumstances will fulfil $\text{sin}\left[\Phi \right]\in \left]0,1\right[$) or their simple application (if they already contain them), as a consequence of the circumstantial loss of the unequivocal corpuscular character of a first stage in the process (which may remain unfinished) of deformation (inverse formation), which entails the impossibility of expressing itself corpuscularly or the necessity of expressing itself in another way. This leads to the conclusion not only of the relevance of $\text{sin}\left[\Phi \right]\in \left]0,1\right[$ in the evolution of the phase in the corpuscular state (already validated) but also of the same in the process of corpuscular formation and deformation (wave-corpuscle duality).

Understanding, however, that in these circumstances of duality, although the state is not entirely corpuscular, the kinetic treatment of Equation (8) continues to be valid because the process continues to be carried out in the integral limits (phase) established for it in Equation (5a), contrary to what happens in other processes of inverse formation in which these limits are lost due to the total loss of the corpuscular character (decays), or in which other limits are taken (decreasing generational change). The latter processes in which Equation (8) is still valid, but for a different energy balance, as a consequence of the reconversion of almost all the rest energy into kinetic energy, per the relation between the new rest mass and the previous one, now inertial, as occurs (in the inverse sense) in the increasing generational change [1] . Processes that, on the other hand, bring us face to face with the idea of the principle of relativity and its fulfilment and importance.

Indeed, the conservation of the principle of relativity, as fulfilled by $E_k$ in Equation (8), is indispensable and can be crucial in itself to understand what

happens with respect to the wave factor $\text{sin}\left[\Phi \right]$ and to characterise its behaviour mathematically, that is, to represent mathematically this dual behaviour, both in terms of the discontinuous change between $\text{sin}\left[\Phi \right]=0$ and $\text{sin}\left[\Phi \right]=1$ and in terms of the materiality and kinetic changes of the associated energy, which are the two states variables that interest us really for the formation, while $\text{sin}\left[\Phi \right]\in \left]0,1\right[$ belongs to the stable state of matter (including that of duality).

For this, we have to place ourselves in the idea that for inertial systems there is no intrinsic kinetic energy, but rather a relationship between systems as a function of their relative speed, just as potential energy is a function of their height difference. Consequently, Equation (5a) cannot change as a function of the different speeds (between $\upsilon ={\upsilon }_p$, which fulfils $\text{sin}\left[\Phi \right]=0$, and the subsequent $\upsilon =0$) anymore or in a different way than it does for the different observers.

Continuing with this idea, we can consider a starting situation or collapse in

which $\text{sin}\left[\Phi \right]=0$ is fulfilled for a reference system associated with a pro-corpuscle, which, once constituted as a corpuscle, evolves, due to the energy surplus, with $\text{sin}\left[\Phi \right]\ne 0$ for an infinitesimal variation of $\upsilon$ with respect to its original system. For a particle in its original system, we will have passed from a situation in which it is instantaneously $\text{sin}\left[\Phi \right]=0$ to one in which, de facto, it is always $E_k={\bar{E}}_k$ in Equation (8), even though the phase factor $\text{sin}\left[\Phi \right]$ may evolve, from $\text{sin}\left[\Phi \right]=1$ to $\text{sin}\left[\Phi \right]\ne 1$, as explained above.

The sine function is not capable of making this behaviour possible by itself. We cannot explain this jumping behaviour (which obeys boundary conditions), in fact, or emulate it utilising any function. The appropriate way to represent this behaviour is by means of an improper function such as the Dirac delta or impulse function, that is, by this type of distribution, which we can also associate directly with the mass, since this is the one that changes its situation really:

$\begin{array}{l}{\bar{E}}_k=\left(\frac{\bcancel{\hslash }2}{\pi a^3\Delta k}\right)\sin {\left[\Phi \right]}_{\upsilon }{\displaystyle \underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\text{d}\upsilon }\triangleq \left(\frac{\bcancel{\hslash }2}{\pi a^3\Delta k}\right){\displaystyle \underset{\upsilon =0}{\overset{\upsilon <{\upsilon }_p}{\int }}\delta \left(\upsilon \right)\text{d}\upsilon }{\displaystyle \underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\text{d}\upsilon }\\ =\left(\frac{\bcancel{\hslash }2}{\pi a^3\Delta k}{\displaystyle \underset{\upsilon =0}{\overset{\upsilon <{\upsilon }_p}{\int }}\delta \left(\upsilon \right)\text{d}\upsilon }\right){\displaystyle \underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\text{d}\upsilon },\\ \delta \left(\upsilon \right){[}_{\text{0}\text{for}\upsilon \ne 0}^{\infty \text{for}\upsilon =0}\text{verifying}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\delta \left(\upsilon \right)\text{d}x}=1.\end{array}$ (14)

We are not saying, and hence the notation used ( $\triangleq$), that the function

$\text{sin}\left[\Phi \right]>0$ and the Dirac delta are equivalent, but rather that circumstantially

the function, together with the physical constraints, can behave or be defined as such, for any final speed ${\upsilon }_f<{\upsilon }_p$.

A distribution that we can express in another way to extend the behaviour of

the equation to the state $\text{sin}\left[\Phi \right]=0$.

$\begin{array}{l}{\bar{E}}_k\triangleq \left[\frac{\bcancel{\hslash }2}{\pi a^3\Delta k}\left(1-{\displaystyle \underset{\upsilon >0}{\overset{{\upsilon }_p}{\int }}\delta \left(\upsilon -{\upsilon }_p\right)\text{d}\upsilon }\right)\right]{\displaystyle \underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\text{d}\upsilon }\\ =\left(\frac{\bcancel{\hslash }2}{\pi a^3\Delta k}\Gamma \left[\Phi \right]\right){\displaystyle \underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\text{d}\upsilon }=m_R{\displaystyle \underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\text{d}\upsilon }.\end{array}$ (15)

In which, similarly, we are not saying (according to the equal delta) that the

original function $\text{sin}\left[\Phi \right]$ and the distribution $\Gamma \left[\Phi \right]$ are equivalent, but that $\text{sin}\left[\Phi \right]$ together with the physical conditioners can behave or be defined as such, which is nothing but as a filter function that only reaches two states, that of pass, for $\Gamma \left[\Phi \right]=1$, and that of non-pass or annihilation when $\Gamma \left[\Phi \right]=0$, for

$\upsilon ={\upsilon }_p$. State of annihilation from which it will derive, according to the phase concept and the application of certain conditions [that expressed in Equation (13), in a recurring manner], all phenomenology of matter, that is, its creation and transformation, in correspondence with the phase changes associated with these processes, which we will deal with in a study that requires this one, in which we will detail and quantify for each of the classes of particles what has already been advanced in [1] . Phase changes for which the principle of relativity is not violated either, since what the preceding expression says is that there is an energy relation between two differentiated states (before and after the phase change) whose transit occurs suddenly, both of which can have a certain additional kinetic energy depending on the reference system. Not different from what happens in any elementary particle transition between the initial and final products.

Although the theoretical model developed could at some point present some kind of limitation (for what is not within its reach, it has it obviously), what it makes clear so far is that it provides a framework associated with the process of construction of matter and its wave carrier (phase factor) with which to explain and contrast many phenomenologies associated with this construction, which is thus presented as an interpretative key or as a kind of starting point, which it will rightfully be if from it we are capable of tackling various questions of physics.

Questions of physics that are and will be mainly of an underlying reality and difficult transit, that which necessarily starts from a basic and deep starting point and which is usually of little interest or forgotten as it generates a less mechanistic or pragmatic physical knowledge as was once the case with quantum entanglement, which we are going to deal with next. A knowledge that is nonetheless the true knowledge, the true physics, the one we will have to strive for if we want to know our deep reality, the one that leads to the physical principles, which we can only reach from a dissection or prospective treatment such as the one we are dealing with here. A less mechanistic treatment which, nevertheless, and in a strict or literal sense, is more physical than the one developed on the basis of a mathematical artifice, which does not speak of reality but of a scheme of it.

In accordance with this dissection, we have established a wave symmetrisation process that gives rise to a free particle and with it to a matter wave (phase factor) that represents a memory of the primordial wave state (wave packet) and a register of the kinetic variables and, if desired, as such a wave, of a quantum state concerning certain observables, through a suitable dimensional extension and a suitable representation or SU symmetry group on it. A phase factor which, as we have seen in the previous development, and as shown in the generational changes [1] , far from being an element at the mercy of the particle, is to a large extent, occasionally and for relevant and/or transit issues, the causal element: we can assure that physics is much more difficult or of impossible solutions when the cause with effect is inverted.

Having said this, and following this prospect, it is not difficult to realise that this process of symmetrisation is in itself a process of entanglement of wave packets which gives rise to the aforementioned particle and a single or non-interlaced wave (phase factor). A single wave that is susceptible of establish an entanglement process with the phase factor of another particle, which gives rise to a dual phase factor, or directly a single shared one (in the case of the generation of pairs), and with this to a combination of states of these particles or quantum entanglement of the same, as described by Schrödinger in 1935 in connection with the critique of quantum mechanics formulated by EPR [2] , which, although unsuccessful, highlighted the paradox or theoretical vacuum that persists to this day, and which now concerns us.

A dual factor phase that would be wherever the two particles are situated, whether they are close to each other or at the far ends of the universe, because it is nothing but a plane wave of certain characteristics which, like its two forming phase factors, is the promoter of any change or alteration of the material whole.

To which we should add that it is not only that the plane wave can be at the far ends of the universe, but that it is quite possible (we can consider it as true), according to what we will develop in the following section about the phases, that this distance does not exist or is null in the system of light (proper length). This is not strange in the relativistic environment, which opens up a host of possibilities (which we will not discuss here) concerning the true nature of that plane wave and of light as such.

Apart from this, which requires a more specific development, the measurement or detection of a particle at one of the distant extremes would, according to this scheme, do nothing more than act on the dual phase factor, which is in charge of materialising the relevant changes in each of the particles (or preserving the coherence of states), that is, assigning and, if necessary, collapsing to the corresponding states instantaneously. The action is instantaneous because the impact on the particles of what happens in the dual phase factor is not produced by a continuous function in time but is a sudden change of state representable by an improper function. That is, it is not produced by transmission of an electromagnetic nature, even if such a nature is present, but through an alteration of the wave construction process (symmetrisation) or of the waves of the already constructed object, where the phase factor is a “wave by-product” of that construction and therefore an element bi-univocally and indissolubly related to the forming waves, in the same way as the two members of some equality ( $1+1-1+1=2$), in which we cannot alter the list of summands (a state as a consequence of the sum of waves) without altering the final result (the resulting wave), because they are equivalent members.

Consider in this respect that in origin we are dealing with wave packets that are constructed by the superposition of waves, which is nothing more than the sum (integral) of waves in the form represented in Equation (1), in which the carrier is part of its result, as we saw in Equation (3). A result that will have no energetic impact on the particle through Equation (5) but a memory of that energetic state and, by extension, of its other properties (as the spin), in such a way that, on one side (member of the equality) the particle may have one representation of that spin (which we then notice when it collapses) and from another a function that accounts for it.

For ease of understanding, and without loss of generality, we can imagine that the aforementioned dual phase factor (entanglement of phase factors) is established by the direct connection of the phase factor 1 (F1) of particle 1 (P1) and the phase factor 2 (F2) of particle 2 (P2), as shown in Figure 1(a) . For the entanglement of this dual phase factor, which in this case we can consider shared,

we can add a pair of superposed states, $\psi =\alpha |{\uparrow }_1{\downarrow }_2\rangle +\beta |{\downarrow }_1{\uparrow }_2\rangle =\alpha {\varphi }_a+\beta {\varphi }_b$,

characteristic of quantum entanglement, which would give rise, as shown in Figure 1(b) , to the splitting of the phase factor, according to the two possibilities or combinations of states for a null value of the quantum variable in question, such as spin (spin=0), and for a single direction, which allows us to simplify the study we intend with a clear scheme and free of other quantum particularities.

From here, and in this context that we propose, when we detect state $|{\uparrow }_1\rangle$ in P1, as a consequence of the measurement process, it is not that a transmission mechanism of information is produced so that P2 collapses to state $|{\downarrow }_2\rangle$, it is

that the function $\psi$ has collapsed to state ${\varphi }_a$ as a consequence of the observation, at the same time that it discards or extinguishes the branch ${\varphi }_b$ (see Figure 1(c) ). Which on the one hand gives rise to the value measured in P1 and, on the other hand, establishes an instantaneous correlation of states for P2. In this case, and continuing with the small numerical analogy of the construction process, on $\text{P1}\equiv 1+1-1+1=\left[2\equiv \left(\psi \right)\right]=1+3-2\equiv \text{P2}$ one cannot modify (fix) the result 2 of $\psi$ (resulting from the sum of waves) without modifying (fixing) in an analogous way P1 and P2 (the list of summands), and vice versa.

Being purist with phenomenology, and according to the mentioned coherence between the parts (derived from the construction process), we would be in a continuous change of states of the two particles and in a continuous alternating establishment and extinction or back and forth of the paths ${\varphi }_a$ and ${\varphi }_b$ according to the probability determined by the wave function, in such a way that it is the measurement of the observable that freezes the system to the state that results in the measurement, that involves the whole system and that at that instant has that configuration. An alternation or back-and-forth as depicted in Figure 1(d) , where the sum of the (p) legs has a length $\alpha$ and the sum of the (q) legs has a length $\beta$, which is but discrete expressions of the probability of encountering one state or the other, it being when the measurement reaches the system at a leg (p), corresponding to the state $|{\uparrow }_1\rangle$ that we have taken as an example, that the system de facto passes to the configuration shown in Figure 1(c) .

The solution to the question of quantum entanglement is to understand that there is no action at a distance between two bodies when they have some form of connection at a distance, and that the action between them, in this case, can be instantaneous despite a (quasi) infinite distance, in the same way, that the action on two bodies attached to the ends of an ideal rope is instantaneous when we pull the centre of the rope downwards, since the impulse in this (ideal) case is instantaneous. Or, to put it even better (leaving aside this ideal character), when we pull the two ends of the string downwards (and here is the key), as happens with the branch, Figure 1(c) , which is extinguished from the phase factor, since it is all of it which has the information and operates at the ends, so that there is no distant communication between the objects (particles) but two objects close to their respective sources of information, there being consequently no superluminal communication and no break in the principle of locality.

A simile that helps us support or conclude in a more formal way that the way to transfer information for these purposes is, essentially and without alternatives, through transversal action (and not longitudinal action between extremes), with which the entire length of the phase factor is affected at the same time. Which is not difficult to implement through the participation of another similar wave with a phase shift (that modulates) or an inverse one (that cancels), or through the collapse condition seen in Equation (13) and used for generational changes of particles, among other possibilities, known or not, derived fundamentally from the idea of construction, to transmit information. The question is not so strange actually, since this is what we do when we measure in the act of observation: act with a wave (photon). The difference is that the wave not only acts on the observed particle but also on a system of solidarity items, which seems natural.

Everything illustrated serves to exemplify how the mechanism of identification and transmission of information from a quantum state to the extremes is effectively carried out by simply extinguishing or disabling the combination of non-representative states (or the persistence of representative states at the same extremes) through transversal action, although not to clearly show what makes it possible. In effect and as we said, the transmission of a quantum state is, through the process presented (as would be required of any theory that seeks to resolve the EPR paradox), more effective, simultaneous, and faster (as well as immediate), than that which would derive from the speed of light (without violating causality), but it is (and this is what is important) the real character of the phase factor what makes this possible. That is, not being a physical entelechy is what makes this possible or effective, making it possible for the alteration of the states (as physically compiled through the superposition of waves) to have physical consequences in the particles (correlation), and making there exist a pre-established end-to-end physical longitudinal path on which to exert a transversal action, a channel on which the observation wave can act. This, on the other hand, endorses the legitimate and real character of this phase factor and that of the theoretical framework that promotes it, which is ultimately what establishes the connection between this factor and the particle and that which is ultimately established between particles.

This, which makes phenomenology possible and explains it conceptually, highlights the undeniable need to take the development of our explanations down a notch, to bring them much closer to the subjects and the nature of the objects. With this, any attempt to explore these phenomenologies must be carried out through this undulatory framework, of which I am providing the bases, or by connecting any other with it, as it is the only one that contains the information and can provide us with it, it being understood as unfruitful, therefore, to go deeper into quantum entanglement, or to deal with what has already been said, from the quantum formalism without establishing this connection. It is not up to us now to establish it, nor is it in the pretensions of this work, which is more interested in laying those foundations and in developing, in this case, everything that Equation (5) suggests to us.

Now it is a priority to establish the energetic correspondence of the different terms and particularly the second distinction between (5a) and (8) for the relativistic kinetic energy, which is determined or characterized by its mass $m_r$,

$\begin{array}{c}{m}_r=\frac{\bcancel{\hslash }b}{\pi a^3}=\frac{2\hslash \times \left[T\right]}{\pi a^3\Delta k}=\frac{2h\times \left[T\right]}{\pi a^3\left(2\pi \right)\Delta k}=\frac{h\times \left[T\right]}{{\pi }^2{a}^3\left(2\pi \Delta \kappa \right)}\\ =\frac{h\times \left[T\right]\left(\Delta \lambda /2\pi \right)}{V}=\frac{h\times \left[T\right]\Delta ƛ}{V},\end{array}$ (16)

as a concretisation or representation of a sort of massive coefficient of the wave group related to its amplitude, which is determined by the dimensional value a of the particle (proper size) and by $\Delta k=2b^{-1}$, associated with the normalisation constant A explicit in Equation (6). Massive coefficient that, according to the last member of Equation (16), we can understand as related to a volumetric density (of some energy representation $\hslash \left[T\right]$) defined by $\Delta ƛ$ for a volume V proportional to ${\pi }^2$, that is, for the volume of a toroid (as already anticipated in Figure 1 ) that fulfils the relation $2r^{2}R=a^3$, given that in this case:

$V=2\pi R\times \pi r^2=2{\pi }^2{r}^{2}R={\pi }^2{a}^3,$ (17)

as a physical and geometric structure resulting from the symmetrisation or entanglement process of two groups of waves facing each other by its opposite or complementary movement as represented in Equation (1b).

A massive coefficient that we see that is conformed in Equation (16) by variables of two different natures, a corpuscular one, which is geometrical and energetic, and $\Delta ƛ=\Delta \lambda /2\pi$ which is purely undulatory. A distinction that will have physical consequences that we will deal with later, which also has a mathematical representation, given that it is because of this differentiated or heterogeneous and non-miscible nature that we have been able to maintain the $\pi$ value associated with the elements of one species and the other differentiated, that is, what has allowed (and obliged) us to use $\Delta k$ and not $\Delta \kappa$ as a variable of wave evolution (despite $\Delta k=2\pi \kappa$ being fulfilled). Having, consequently, the possibility of taking out the value $1/2\pi$ as a factor or integrating it in $\Delta \lambda$ and reaching an angular measure $\Delta ƛ$ of the variable, in correspondence with the angular representation $\Delta k$ of the wave number, which is also the same that we have used from the beginning and which gives rise to the normalisation constant.

According to the definition of mass as a volumetric density of $\Delta ƛ$ expressed by Equation (16), it is also evident that the higher the coefficient $m_r$ the smaller the volume $V\left(a\right)$ has to be for all particles sharing the same normalisation constant, since in the same particles $\Delta ƛ$ is a fixed and unique value characteristic of the wave group, because it is fulfilled (in what is a relationship of constants):

$2\hslash \Delta ƛ=\hslash b$. (18)

From Equation (16) we also obtain $m_r\times V=m_r\left({\pi }^2{a}^3\right)=\pi b\hslash$, which shows that for a single characteristic initial value of the wave group (its width), defined by the constant b [or, as we explained in Equation (5), its corresponding and unique normalisation constant ( $b\leftrightarrow B\leftrightarrow A$)], there are several theoretical possibilities, $m_r^1\times V_1,m_r^2\times V_2,\cdots$, in the left-hand member (infinite without any other limitation).

That is, from Equation (16) we derive an element of uniqueness and an element of diversity, which, taken to the SM, correspond respectively to the identity of a class of particles and to the different generations of particles of that class, restricted in practice to the discrete number of known families. Different generations of particles that in the strict sense do not share a single normalization value b either, since this depends on the velocity field ${\stackrel{⌢}{\varphi }}_i$ itself, but that we can consider it as such, as it supposes a negligible correction in most cases (all except one), such as seen in [1] .

As the mass is expressed in Equation (16) we cannot say much more about it or, as we said, about its energetic representation, despite being well-defined and dimensionally homogeneous. A point that will be seen even more clearly if in Equation (16) we represent mass in its energetic form and, since in general $E=\hslash w=h\nu =h\times {\left[T\right]}^{-1}$, rewrite the constant h as an energetic magnitude $E_{\hslash }$ associated with the inverse of the time unit, according to its units $h=E_{\hslash }\times \left[T\right]=E_{\hslash }/{\nu }_{\hslash }$, that is,

$\begin{array}{c}\frac{E_r}{c^2}=m_r=\frac{h\times \left[T\right]\Delta ƛ}{V}=\frac{E_{\hslash }\times {\left[T\right]}^2\Delta ƛ}{{\pi }^2{a}^3}=\frac{E_{\hslash }\times {\left[T\right]}^2}{{\pi }^2{a}^2}\left(\frac{\Delta ƛ}{a}\right)\\ =\frac{E_{\hslash }}{{\pi }^2{\left(a/t\right)}^2}\left(\frac{\Delta ƛ}{a}\right)=\left(\frac{\Delta ƛ}{{\pi }^{2}a}\right)\frac{E_{\hslash }}{{\upsilon }_{\hslash }^2}=\left(\frac{\Delta \lambda }{2{\pi }^{3}a}\right)\frac{E_{\hslash }}{{\upsilon }_{\hslash }^2}=\Omega \frac{E_{\hslash }}{{\upsilon }_{\hslash }^2},\end{array}$ (19)

where it is indeed evident, given that Ω is dimensionless, that the first and last members of Equation (19), thanks to the additional time dimension within $\bcancel{\hslash }$ resulting from the integral solution reached in Equation (5), are dimensionally homogeneous, and that, consequently, so are all the members of the equation, and those of Equation (16) in the beginning. An additional time dimension that has allowed us to define a velocity ${\upsilon }_h$ (over two of the three a values), which we can correspondingly call our proper speed.

What has been said about mass $m_r$ is already enough for our purposes, that is, to achieve physical coherence, to put it in context, and to proceed further in the study of the terms. We can, however, say more on the basis of Equation (19), things which involve an exercise in prospection, an attempt to delineate the true nature of matter, and that of the space in which it arises as such and which is implicit in it, as well as of the true relation between it and radiation, and the true and necessary overlap, which we will analyse at the end, between the two factors seen, $m_r$ and $\text{sin}\left[\Phi \right]$, since it is the latter that allows the former to retain the apparently lost three-dimensionality in Equation (19) as a consequence of the conformation of the dimensionless factor Ω or absorption of a dimension in it.

Indeed, by means of Equation (19) it is not only evident not only what has already been referred to but also, taking into consideration the first and last members [which we write in Equation (20a) for clarity],

$\begin{array}{l}\stackrel{(\text{a1})}{\frac{E_r}{c^2}}=m_r=\stackrel{(\text{a2})}{\frac{E_{\hslash }}{{\upsilon }_{\hslash }^2}}\left(\Omega \right)=]\begin{array}{l}\Rightarrow E_{\hslash }=\left(\frac{1}{\Omega }\right)m_r{\upsilon }_{\hslash }^2\text{(b1)}\hfill \\ \Rightarrow E_r=E_{\hslash }{\left(\frac{c}{{\upsilon }_{\hslash }}\right)}^2\left(\Omega \right)\text{(b2)}\hfill \end{array}\\ \Downarrow \Downarrow \\ E_r=m_r{c}^2\text{(b3)}\Leftarrow E_r=\left(E_{\hslash }\frac{\Omega }{{\upsilon }_{\hslash }^2}\right)c^2\text{, (c)}\end{array}$ (20)

That the energy of the particle $E_r$ is proportional to the basic energy of wave $E_{\hslash }$. Being the proportionality factor, as we see in Equation (20b2), a combination of the one established by Ω itself and the square of the one established between the speed c associated to the (free) wave itself and the speed ${\upsilon }_{\hslash }$ (of which we did not know until now), which we could understand as a minimum or residual and internal speed of the particle (of the constrained wave associated to a), or, as we have already said, a proper or inherent speed. Besides being able to establish the usual form of proportionality in Equation (20b3), from Equation (20a1) or from the previous one, as a consequence of a different organisation and consideration, shown in Equation (20c), of the factors involved.

Consequently, and according to the unprecedented Equation (20b2), in the energy $E_r$of the particle is present a proportionality factor expressed through the speeds associated with the two states or forms of being of the particle, susceptible of being connected by a process (as it could be that of annihilation or its inverse) that can make us think that equally those speeds are the essence of that process. Or that these speeds are the true relationship, as opposed to the usual Equation (20b3) in which the (corpuscular) variable $m_r$ paradoxically computes the (corpuscular) energy value $E_r$ over the variable c (which is not corpuscular), as it follows from the equivalence of Equation (20a1), or as opposed to Equation (20b1), also unprecedented, which we can similarly obtain on Equation (20a2), in which the speed ${\upsilon }_{\hslash }$, which is part of this mass (and is corpuscular through the spatial dimension a), is related to an energy value, $E_{\hslash }$ associated to $\hslash$ which is not corpuscular or purely wave-like. That is to say, Equation (20b2) is the (only) expression that accounts, through both speeds, for the transformation or accommodation between some corpuscular and non-corpuscular energy, which homogenises the members, as opposed to the others, clearly heterogeneous without the presence of an adaptive element, since $m_r$ is not, and its equivalent in Equation (20c), although revealing, is not either.

A speed ${\upsilon }_{\hslash }$ that is a minimum not because there cannot be one less than it in the physical world but because for the particle, with $a=\Delta x^{\prime }$, any displacement of the variable x takes place over that minimum, which is an (internal) maximum in it, which implies its own size, so that a certain $dx\to 0$ at one extreme represents $\left(\Delta x^{\prime }+dx\right)=\Delta x^{\prime }$ at the other.

A speed ${\upsilon }_{\hslash }$ that can even make us think, according to its non-zero value, that speed c is as maximum because it is associated with a maximum limit (already defined in the field of velocities ${\stackrel{⌢}{\varphi }}_i=\left[0,c\right[$) as speed ${\upsilon }_{\hslash }$ is minimum because it is associated with a minimum limit (from which we could consider that in truth ${\stackrel{⌢}{\varphi }}_i=\left[{\upsilon }_h,c\right[$, that is, that any speed starts from that minimum ${\upsilon }_{\hslash }$). Which, in turn, can make us think that speed c (the same for all ${\stackrel{⌢}{\varphi }}_i$), responding to a mirror behaviour or the same way of developing, may obey a similar conceptualisation referring to a maximum size, that is, that of being defined through a spatial or topological constant, $\varsigma$, on which a dynamic functionality or additional condition is then implemented, as has occurred for ${\upsilon }_{\hslash }$ with respect to the two factors a affected and, therefore, in two of the three spatial coordinates of the toroid, without, however, affecting its toroidal shape or physiognomy. In other words, we can think that just as there is a minimum (proper) size for the wave packet, there is a maximum (proper) size $\varsigma$ when it is unfolded, intimately linked to c, as we will develop later [nothing extraordinary if we consider that we start from a SWP and the equivalence existing between a wave packet and a pulse (see C. 20 of [3] )].

Due to the additional dynamical functional ( $a\to {\upsilon }_{\hslash }$), which we have just quoted, and the equivalent mass dependencies on $E_{\hslash }$ in Equation (19) we can write the following mass energy expression:

$m_r=\left(\frac{\Delta ƛ}{{\pi }^{2}a}\right)\frac{E_{\hslash }}{{\upsilon }_{\hslash }^2}=E_{\hslash }\left(\frac{\Delta ƛ}{{\pi }^2{\upsilon }_{\hslash }^{2}a}\right)=E_{\hslash }\left(\frac{\Delta ƛ}{V/t^2}\right)=E_{\hslash }\left(\frac{\Delta ƛ}{\stackrel{⌢}{V}}\right),$ (21)

On which we can emphasize again that only two of the three spatial dimensions a of the volume of the toroid are affected (by time) and become speed, just as c is affected bidimensionally (by the two formants), and for this reason, it is c² in the energy expressions, and not c³ (as we would expect in some three-dimensional space), both in Equation (5a), we are working on, and in Equation (5b), characteristics of $m_r$, but not in Equation (5c), which, being purely wave-like, retains its one-dimensional character in the expression, i.e. one dimension less (as in the other cases) than its physical reality.

In the case of $E_k$, we pass through the symmetrisation process from a bi-one-dimensional system to a three-dimensional one characterised in the equations themselves by c³ (and a³), which finally decays to c² as a consequence of the integration process for the construction of the reference energy value (see Appendix A in [1] ), which is none other than that of the rest energy also associated, as can be seen in Equation (5b), to c². We could say that in the integration process, solved in Equation (8b), c² reflets the construction of the particle and the other variable c (through the speed $\upsilon$), of which (5b) does not specify, a proportionality factor with respect to rest, expressed by $\gamma$, which is derived from the dynamic process.

The system decays to c², as a consequence, as we said, of having combined two waves (and not three), and of having, therefore, only two of the dimensions (the original ${\upsilon }_{\hslash }$) that generating capacity or the construction. Being the third dimension (the remaining parameter a), consequently, something that arises or is created [which does not exist as such, in fact, in Equation (5b), or has yet another form], and creates the volume, the space proper, from two one-dimensional waves that do not have it in themselves. Two one-dimensional waves that form a two-dimensional pattern that, although it does not account for the volume, does account for the surface front or characteristic envelope that forms it, which we call the spherical wavefront, intimately related to it, as we shall see, in its state of expansion or dematerialization, in the same way as ${\upsilon }_{\hslash }^2$ is, although modelling the toroidal form, for the other state.

An analogous change to that made in Equation (20a1) to determine the value of energy $E_r$ as a function of rest mass and speed [Equation (20b3)], is that made in Equation (20a2) for $E_{\hslash }$, allowing us to establish, in a more formal way [Equation (20b1)], an alternative definition for mass (or understanding of what it is through its functionality). A definition that does not have to do with what it can energetically develop ( $E_r=m_r{c}^2$), but with what it is energetically in itself (in relation to the individual energy of a wave) when it is constrained in the particle, which is its true definition, the one we were trying to seek, and which in Equation (16) was still veiled.

This dual character of mass is represented clearly in the two starting Equations (20a1) and (20a2) in which the mass is the ratio of proportionality of two fractions (already referred to above for the speeds). The usual definition (extrinsic) of mass at rest, expressed in the first of these, is given by a ratio between the energy it is capable of delivering and the characteristic speed c² when this occurs, associated with the situation of maximum expansion or development of the velocity field ( ${\stackrel{⌢}{\varphi }}_c\equiv \left[0,c\right]$). The self-definition or intrinsic definition (expressed in the second one) does not speak of any process (at least not of expansion), is the internal relationship of a given energy pattern $E_{\hslash }$ (associated with the wave through h) and the characteristic speed ${\upsilon }_{\hslash }$, associated with the situation of maximum constriction or proper size a, which is that of the particle in question.

A self or state definition, and not a process definition (although it is also the end of a process) which, nevertheless, like the usual one, is hybrid or composed of elements of the two systems, states, or species, which is certainly paradoxical (as we have already mentioned). This will be better understood if, by cross-multiplying Equation (20a1) and (20a2), we obtain an equation free of these heterogeneities, since Ω is a simple dimensionless dividing factor, that is,

$E_{\hslash }{c}^2=E_r\left({\upsilon }_{\hslash }^2/\Omega \right)\equiv E_r{\varpi }_{\hslash }^2=cte,$ (22)

where it is emphasised that the energy $E_r$ associated with $m_r$, applied to the constrained kinetic surface ${\varpi }_{\hslash }^2$ that is natural to it (material wavefront effective) is equivalent to the unit energy $E_{\hslash }$ of the original wave applied to the surface that is natural to it (its two-dimensional expansion c²), that of the spherical wavefront. That is to say, these members are not only equal but are each composed of homogeneous elements of the same nature, which makes the equation a more essential or true-to-reality equation that differentiates states, a sort of invariant since it retains its value, which expresses the potential to be concretised in a certain energy value (in the form appropriate to their status). What will allow us, together with the previous analysis, to go deeper into the nature of the processes of creation and annihilation, and of the variables at play.

Indeed, the first member of Equation (22) highlights, in a first reading, that it is the energy $E_{\hslash }$ of a wave, applied to its whole spherical front c², which we can find (or serves as a reference) when a particle annihilates and passes to an immaterial state, and that is why the energy of a particle (when this occurs) comes as a function of c² (resolving the above paradox), and that, correspondingly (last member), it is the energy $E_r$ of a particle, applied to its effective material wavefront ${\varpi }_{\hslash }^2$, which represents (serves as a reference) when a particle is created.

In addition to finding in these equivalences that $E_r\gg E_{\hslash }$ because ${\varpi }_{\hslash }^2\ll c^2$, and that the smaller the wavefront ${\upsilon }_{\hslash }^2$ (or the effective one ${\varpi }_{\hslash }^2$) the greater the energy $E_r$, which is the only one we have any notion of in a process of creation, not being this value but the result of normalising the last member through ${\varpi }_{\hslash }^2$, given that c² (initial state) does not evolve (through ${\upsilon }_{\hslash }^2$) to ${\varpi }_{\hslash }^2=1^2$ (final state) but to an indeterminate ${\varpi }_{\hslash }^2$ value.

Normalisation which gives rise either to Equation (20c), in which it is emphasised (in a second reading, as a consequence of this normalisation) that it is not a unitary energy $E_{\hslash }$ which is applied on c² (or finally gives rise with $E_r$ by means of this application) but a density of this energy ( $E_h/{\varpi }_{\hslash }^2$) in this space, or to the equivalent Equation (20b2), where it is a unitary energy, although not applied to a continuous space but ascribed to a discrete number of elements ( $c^2/{\varpi }_{\hslash }^2$) in this space. Where it becomes clear that ${\varpi }_{\hslash }$ is also an intrinsic velocity which, because of its relation to c and $E_{\hslash }$, can be called unit velocity. A dual interpretation that, supported by the equations, is not at all strange if we take into account that which exists in itself between a pulse (finite single wave) and a wave packet, which implies the modulation of the former (with k wave-number).

Such a material system could have different values ${\varpi }_{\hslash }^2$ (different masses and different energies), for which the energy expansion of $E_h$ through c² is identical (third member), being able to change only the density of the (intrinsic) wave energy $E_h/{\varpi }_{\hslash }^2$ from which the same would take place, which is nothing but the density of wave energy to which the wave is forced in the process of material constriction. Since the constriction undergone does not depend on c² (which is common and universal, either as initial or final state), we will have to divide by this factor the value of the energy $E_r$ in Equation (22) or, better, in the different forms of Equation (20), to know the extent of that constriction in $E_h$:

$\begin{array}{l}{m}_r=\frac{E_r}{c^2}=\frac{E_{\hslash }{\left(\frac{c}{{\upsilon }_{\hslash }}\right)}^2\left(\Omega \right)}{c^2}=\frac{\left(\frac{E_{\hslash }}{{\upsilon }_{\hslash }^2}\Omega \right){\xcancel{c}}^2}{{\xcancel{c}}^2}=\left(\frac{E_{\hslash }}{{\upsilon }_{\hslash }^2}\Omega \right)=\frac{E_{\hslash }}{{\varpi }_{\hslash }^2}.\\ (23\text{b3)}(23\text{b2)}(23\text{c)}(23\text{a2)}\end{array}$ (23)

It follows that the same is constant and is the rest mass $m_r$, as we could imagine and appears in Equation (20a2), highlighting that, in accordance with the corresponding Equation (23) and what has been said above, the Equation (20c) reflects better the corpuscular energetic reality (in relation to the corpuscle) than Equation (20b2), which nevertheless expresses better what really happens in the particle (there is no density applicable to c² but an energy $E_h$ repeated a number of times $c^2/{\varpi }_{\hslash }^2$), which will be convenient to preserve the presentation of the starting unitary value $E_h$ and to differentiate it from its dynamics, and better, as we shall see in the following epigraph, to characterise the resulting or equivalent electromagnetic energy.

The formation of the particle requires the use (as formation energy) of the expanded energy, and its contraction and consequent finite spatial localisation. If the particle size were ${\varpi }_{\hslash }^2=1$ (for which we define an energy density $E_{\hslash }^o$), from Equation (22) we would have $E_r=E_{\hslash }^o{c}^2$, but this is not so, being, on the contrary, that generally ${\varpi }_{\hslash }^2<1$ and that only some ${\varpi }_{\hslash }^2$ values are possible, so that $E_h$ has to be expressed for ${\varpi }_{\hslash }^2<1$, that is, to increase its density value ( $E_h>E_{\hslash }^o$), until reaching the value ${\varpi }_{\hslash }^2$ of state, equivalent to increasing the ratio $c^2/{\varpi }_{\hslash }^2$ and the $E_h/{\varpi }_{\hslash }^2$ (which we call $m_r$) characteristic of the particle in question. The reason why a value $m_r$ is stable, and not another, is precisely what we have already explained in the theoretical body concerning generational changes. A $m_r$ corresponds to a value $E_k$, which in turn corresponds to the Lorentz factor $\gamma$ that reaches the condition (13) in the phase factor.

We have seen that the energy $E_r$ is determined with respect to the energy surface density [ $m_r$, expressed in Equation (20a2)], by the real dimension of the sample (that is, by c²) in the velocity field ${\stackrel{⌢}{\varphi }}_c\equiv \left[0,c\right]$, which finally takes the form given in Equation (20c), and that the same is equivalent to establishing how many units ${\upsilon }_{\hslash }^2$, affected by Ω, with energy $E_{\hslash }$, there are in that sample c², as expressed in Equation (20b2). That is, the number $c^2/{\varpi }_{\hslash }^2$ of units, according to Equation (23), which coincides, moreover, with the number of cycles C associated with the frequency, on which it is not difficult to compose or recover the frequency and put the value of the energy as a function of it, in accordance with the expression:

$\begin{array}{l}\left[E_r=m_r{c}^2\right]=E_{\hslash }\left(\frac{c^2}{{\varpi }_{\hslash }^2}\right)=\left[E_{\hslash }\left(\Omega \right)\left(\frac{c^2}{{\upsilon }_{\hslash }^2}\right)=h{\left[T\right]}^{-1}\left(\Omega \right)\left(\frac{{\varsigma }^2{\xcancel{\bcancel{t}}}^2}{a^2{\xcancel{\bcancel{t}}}^2}\right)\right]\\ \stackrel{\varsigma =na}{=}h{\left[T\right]}^{-1}\left(\Omega \right)\left(\frac{n^2{a}^2}{a^2}\right)=h\left(\frac{n^2\Omega }{t}\right)=h\left(\frac{C}{t}\right)=h\left(\frac{c^2/{\varpi }_{\hslash }^2}{t}\right)=\left[h\nu =E_r\right].\end{array}$ (24)

This shows that the number of cycles C is ultimately a ratio (affected by Ω) between the maximum size of the wave ${\varsigma }^2$ when it is expanded and the minimum size a² when it is constrained in corpuscular size.

We see, in short, that the real relation between the two usual ways of expressing $E_r$, explained in the initial and final brackets of Equation (24), can be seen in the central bracket of the same, where it is shown that the radiant energy of a mass $m_r$, as a function of the frequency and the energy unit $E_{\hslash }$ (associated with $\hslash$ or h), is related to the number of times that energy unit $E_{\hslash }$ is contained in that mass, which in turn is the squared function of the ratio between the speed c of the velocity field (maximum speed in a system) and the unit velocity ${\varpi }_{\hslash }$.

From there, we know the frequency $\nu$ for the mass of a given particle, and we are also able to know (although we do not develop it here) the values of Ω, which would allow us to calculate n and a, which would allow us to calculate (as already mentioned) $\varsigma$, that is, the spatial expansion or natural dimension of the wave associated with the speed c, and with it the relation between the two, and subsequently calculate ${\upsilon }_{\hslash }$.

It is worth pointing out several things in this respect. On the one hand, to reiterate that we have no guarantee of the existence of $\varsigma$, only that it is plausible under the finite character of the wave that characterises a pulse. On the other hand, to point out that, regardless of its existence, what has been said is correct, since $\nu$ is reached from the previous relationship between speeds, on which we can make the subsequent change (to space) as we wish, that is, for any value of time. Finally, although, indeed, we do not know of the existence of $\varsigma$ (as we do not know of ${\upsilon }_{\hslash }$, except for all that has been developed above), it is also true that we have $\nu$ as a constant of proportionality of two ratios (a frequency is always how many times something small occurs within something large), of which we have a known element in each of them, a and c, which allows us to know the other, in both, which justifies this frequency.

Although $E_r$ is only a total energy computation established from $E_h$, and there is, according to Equation (20c), no evolution between them, i.e. no dependence, we can conceptualise, according to Equation (22), such a relationship between the two, the particle maximum ( $E_r$) and the elementary wave minimum ( $E_{\hslash }$), insofar as we can establish an explicit connection or correspondence between these energies and their fields of application, that, as can be seen in Equation (25), lies in the ratio of the wavefronts and the proportion established for the same value $m_r$:

$\frac{E_r}{E_{\hslash }}=\frac{c^2}{{\varpi }_{\hslash }^2}=\frac{m_r{c}^2}{m_r{\varpi }_{\hslash }^2}=\frac{m_r\left[{\varpi }_{\hslash }^2\left(c^2/{\varpi }_{\hslash }^2\right)\right]}{m_r{\varpi }_{\hslash }^2}=\frac{\xcancel{h}\nu }{\xcancel{h}{\nu }_h}.$ (25)

That is to say, both energies are sustained in $m_r$ as an intrinsic quality, for a different wavefront or field of application, [as was already evident in Equation (20a1) and Equation (20a2), with which we could equally have reached the preceding equation]. It is further emphasised that for $E_r$ extended (electromagnetic) we are dealing with the same elementary wave $E_{\hslash }$ with that intrinsic quality, repeated a number of times in an expanded (serial) manner at c², which gives rise to multiple cycles and a frequency $\nu$, that repeating element being a single cycle, ${\nu }_h\equiv 1$, characterised by ${\varpi }_{\hslash }^2$ ( $\to {\upsilon }_{\hslash }^2\to a^2$). Whereas for $E_r$ constrained (corpuscular) we are dealing with the same wave repeated a number of times over itself (in parallel), which does not give rise to a travelling wave of a given frequency but to a stationary pattern of k waves (over the same cycle) or a wavenumber k (a wave packet), i.e. a different idea of repetition to the previous one, which we can express as follows:

$E_r=m_r{c}^2=h\nu =hC{\nu }_h=n^2\Omega h{\nu }_h\triangleq kh{\nu }_h=k{m}_r{\varpi }_{\hslash }^2=k{E}_{\hslash }=E_r,$ (26)

while on the other hand, it generates the additional dimension a, which is, in the end, what happens when the SWP is generated.

An additional dimension which gives volume (forms the toroid) but which may give the impression of being fictitious, since, in addition to it, a variable is incorporated in Equation (19), which absorbs or neutralises it (hence the dimensionlessness of Ω), as we have already mentioned, which brings us face to face with the requirement of having to combine both circumstances and to overcome the paradox and possible objection. An objection that may go unnoticed without an in-depth study of the equation or even (conversely) be overcome with what has already been presented, which has its own weight, without the need for additional arguments.

Indeed, before entering fully into the question, it must be observed that Equation (19) already establishes that the mass responds (dimensionally speaking) to the forms given in the two fractions [which we can see in Equation (20a) more clearly], so that, this not being a problem in Equation (20a1) for consigning the mass, it should not be a problem in Equation (20a2) simply because of the appearance of Ω as a converting element of the different nature of the two expressions, or the adaptation of the electromagnetic form to the corpuscular one, which is basically what it does and what we are basically dealing with in the whole development. Having said this, it is evident (complementing the above),

that the particle is not only $m_r$, but $M_r\equiv m_r\times \sin \left[\Phi \right]$, which by Equation (21) takes the form $M_r=\left(E_{\hslash }/\stackrel{⌢}{V}\right)\left(\Delta ƛ\sin \left[\Phi \right]\right)\equiv M_c\times M_{\omega }$. Consequently, what has to

be dimensionally homogeneous in Equation (19) is this expression, which is and remains as we have presented it in Equation (19) if we remove the dimensionless factor $\text{sin}\left[\Phi \right]$.

From there, as part of the aforementioned conversion, part $M_{\omega }$ of factor $M_c$ is required to act on $\stackrel{⌢}{V}$, which does not imply that $\stackrel{⌢}{V}$ of $M_c$ loses its volumetric character, given that although a of $\stackrel{⌢}{V}$ and $\Delta ƛ$ (which make up Ω) have the same dimensions, they have them in different phases or natures, which cannot compensate each other, neutralise each other, or bring their respective states together. What it does do, and hence the dimensional balance of the combination, when there is an energetic development in the form expressed by Equation (20a1).

To understand the basis of the aforementioned combination, which surely sheds light on what we are dealing with, we have to take into consideration firstly that for $\hat{\Omega }\equiv \Omega /2{\pi }^3$ in Equation (19), ${\hat{\Omega }}^{-1}=a\times \Delta \left(1/\lambda \right)=a\times \Delta \kappa$, which is how it appears in Equation (16). Secondly, that $a\times \Delta \kappa =\Delta x^{\prime }\times \Delta \kappa$, given that, although we have not said so explicitly, $a=x^{\prime }$ in Equation (4) determines a size thanks to the fact that it is referred to the point $x^{\prime }=0$ of an interval. Thirdly that ${\hat{\Omega }}^{-1}=\Delta x^{\prime }\times \Delta \kappa$ is an expression intimately linked to the construction of wave packets, which fulfils ${\hat{\Omega }}^{-1}\ge 1/4\pi$ (clearly dimensionless), i.e. an expression by means of which it is shown that there is a dependence between one variable and another in the construction of a wave packet. A dependence that can be calculated by the Fourier integral (see Sect. 3.4 of [4] ) and which is analogous, applying the Blogie-Einstein relations to the variables, to the principle of indeterminacy.

We see that the construction process itself of the SWP specifies this product and the value of inequality for that specific construction, i.e. that ${\hat{\Omega }}^{-1}$ is present in the construction of this SWP, which in some way represents an endorsement of the process or a signature attesting to its existence through the aforementioned indissoluble dependence of the variables of ${\hat{\Omega }}^{-1}$ in this construction. Regardless of the path or final affiliation of each of these variables: in this case, $\Delta x^{\prime }$ takes the path of being $\Delta x^{\prime }$ because it is already in itself a spatial dimension, and $\Delta \kappa$, of inverse dimension to the previous one, the one of maintaining, in addition to the mathematical condition, the dimensionality of the total system. A dependence represented by ${\hat{\Omega }}^{-1}$ which in its direct form, $\hat{\Omega }=\left(\Delta \lambda /a\right)$ [already present in Equation (19)], describes better than any other the nature of what we are dealing with, since it relates the width of the continuous spectrum of wavelengths used to form the particle (characteristic of the particle family) to the width of the particle, which gives an idea of the importance of the dimensionless parameter Ω, which ultimately becomes.

We can say that we have a two-dimensional system, the one that appears in Equation (20a1), and the system of Equation (20a2) with the same two-dimensionality, that of the two formants, which is provided in the process of symmetrisation with an additional dimension and its inverse dimension, of heterogeneous natures, in such a way that internally it remains two-dimensional [and hence the dimensional equivalence with the usual energy form given in Equation (20a1)] but that, on the other hand, externally it is three-dimensional with an extra one-dimensional undulatory component. An extra one-dimensional component which associated with the phase factor $\text{sin}\left[\Phi \right]$ is only a scale factor, which [as it appears in the last equivalence of Equation (27)] can be incorpored, with no dimensional impact, into the particle, which is thus strictly three-dimensional.

$\begin{array}{c}{M}_r=m_r\sin \left[\Phi \right]=M_c\times M_{\omega }=\frac{E_{\hslash }\Delta ƛ}{\stackrel{⌢}{V}}\sin \left[\Phi \right]=\frac{E_{\hslash }}{\stackrel{⌢}{V}}\frac{\sin \left[\Phi \right]}{\Delta k}\\ =\frac{E_{\hslash }}{\stackrel{⌢}{V}}\Delta ƛ\sin \left[\Phi \right]=\frac{E_{\hslash }\Vert \Delta ƛ\Vert }{\stackrel{⌢}{V}}\left(\frac{\Delta ƛ\sin \left[\Phi \right]}{\Vert \Delta ƛ\Vert }\right)\equiv m_c\times m_{\omega },\end{array}$ (27)

While dissociated (first equivalence) allows us to dispense with the phase factor he for mass $m_r$ (as we have been using it) and to say that it is both three-dimensional (because a³ physically is) and two-dimensional (because internally it is), this duality being a duality derived from the very essence of the process of formation, that is, of remaining internally what it was [and for this reason it is also a wave density ( $\Delta ƛ/\stackrel{⌢}{V}$)] despite being something else afterwards.

We have spoken of densification or change of density of the physical object, and we have also spoken of change of phase, which is a superior or broader concept than the previous one, which encompasses it but which implies something else, and which is necessary to take into account to understand, or at least interpret, the physical process that takes place, beyond what it mathematically represents through the evolution of $\Phi =\pi ,2\pi ,\cdots$ or the change from a certain interval of that evolution (phase) to another.

In line with this, when we have changed the variable x' to its own size a in Equation (4) by means of the Lorentz transformation, we have accepted this change of phase because we have not used Equation (4) to relate two relative systems but to relate all relative systems to a system that is not, which is obviously another phase or differentiated state. We have accepted it in the same way that we accept it in an ice cube, which in its solid phase retains its shape (regardless of the container it was originally given, and of all deformable liquid systems), and continues to retain its shape even if we throw it into the sea.

The question, therefore, is not whether or not we accept something by looking at what happens to the contents of a bucket of liquid water, but whether or not it is possible to fix its geometry or dimensional structure. Analyzing the equations of a group of waves without considering this, we can say that the group is dispersed and that it cannot represent matter, which is what obviously happens to a group of waves if you do not “freeze” it, and you do not fix its dimensions when throwing it into a sea of waves.

It is not only that by using a we have accepted the phase change, it is that we have used precisely the Lorentz transformation and the proper value a (unique and of convergence) to materialize (make physical) that phase change, to “freeze” the wave group and consider it to operate with it. To make physical that phase change is to establish internal cohesion forces stronger than those of dispersion, which forces them to maintain the form and travel as a whole, in the same way that a soap bubble maintains the form through surface tension, without which it would not be understood, the fact itself, which is so irrespective of our knowledge of this basis or how far off we are in our conjectures about it.

The fixation of the envelope is what nature has chosen to neutralise the forces of dispersion, to create matter and to give it form, regardless of whether we are able to notice its “surface tension” or to correctly imagine what it consists of. Further in this effort of imagination, we can think that nature does this by creating stationary wave packets, that is, by establishing boundaries at the ends of these packets, which in this case would be constructed by the process of symmetrisation whereby, forming a donut, one envelope would serve as a boundary for the other. If in addition, the distance of the extremes is that of the envelope itself in Equation (2), that is, the one that gives rise to size a, we would have that both waves would be authentically the same, in such a way that we can well consider that the wave overlaps reflected with the incident in each envelope or that each incident progresses (in a second identical cycle) and is added in the even envelope.

An assumption that we will have to model or develop, but which is based, like everything else, on the two formants, which make possible the construction of an initially static and self-sufficient physical system, a small universe from which it cannot escape or in which any attempt to do so (movement) does not represent a collapse, because it cannot occupy a different space and give rise to dispersion because it already occupies all of it: there is no difference between occupied and generated space but an internal advance that guarantees this and other dynamics that we will develop in future work.

We can see the phenomenon differently. When we are talking about wave functions we are talking about fields, that is, entities that support the development of the different variables, of which circumstantially we can know their value by applying certain operators, such as the Lagrangian of the system or the momentum operator. We can understand the creation of the particle, according to Equation (4), as a process of formation, which has a partial dissociation from the field associated with it, as a singularity therefore. In this case, the variables of the wave function undergo an evolution, and at some point the group of waves, which is governed by that function, becomes uncoupled from these variables by cancelling its wave phase with an inverse phase in the symmetrisation process. This is in essence the Lorentz transformation (4), as we have applied it, the conformation of a singularity of size a common to all x and all t, defined through x and t, and alien, however, to x and t. A transformation by which, in addition, another characteristic is acquired that is so closely related to the final singularisation of the particle as is the density of waves or wavenumber confined within it that we call mass. That singularisation, in this case, not only implies the definition of size and substance but the detachment of that space called size from the other generic space assigned to the spatial coordinate x, and this can be as much as creating the space, or said more specifically, a metric or possibility of dimensioning the space, that until that moment was only mathematical space and that from then on becomes physical space.

From there, there are two different spaces, two phases, two densities, inasmuch as the apparently local change implies the generalized creation of dense space, that is, of singularities with size, that enter into relations of size among them and substantiate the spatial dimension of phase Φ₁ on the basis of a phase Φ₀ in which there truly does not exist a real dimensionality of the dimension x. A phase Φ₀ that we could associate with the toroidal regions, which present themselves as “points” for phase Φ₁, created or configured on the external-discreet reference, but which, internally connected and undifferentiated, could be understood as an underlying grid of reference, another density, another universe. Phases Φ₀ and Φ₁ which, as I said and we see, have a differentiated space of development associated with them, a different field of speeds.

This also leads us to the idea that all the processes that we perceive are perceived in our phase, in material Φ₁, and that the luminous processes are the representation in our phase of characteristic processes of a different phase that are presented to us through the same as immaterial. There may be some phenomenology that is represented to us in our phase as it is and others that, to become evident, adapt to it. We have experience of what is represented in our phase, even if it does not belong to it, and equations for that experience, but we do not know of the other phases and neither what happens in the interface.

The absence of that knowledge is not problematic when we can reduce what we know or want to know to what happens in the context of a single phase (to the aforementioned equations), as is the case with electromagnetic interaction in our phase, but it is problematic when two phases are involved, and consequently the interface, in that it leads us, whether we are aware of it or not, to the erroneous interpretation of some phenomenologies, including the one concerning the predictable dispersive character of the wave group in our phase as opposed to the non-dispersive, or bound, character developed in the adimensional or dimensional interior of another species (phase) of the toroid.

Although it is true that knowledge limited to one phase is not problematic, it is also true that it involves a conceptual error when we treat different species undifferentiated, as we do with light when we include it by equations in our physical space (phase) without truly being. The reality is that material space is a singularity for the physical space of light, and for this reason, this drastic change regarding materiality, and the space of light is a singularity for material space, and for this reason, we perceive it to be an unattainable speed, singular in relation to the set of relative speeds with which we perceive ourselves in the material world. Attributes that are perceived in this way exclusively by virtue of our singularity, from our perception, since things are not something in themselves, but observed information, and neither does this materiality exist from the physical space of light nor a reciprocal “speed of light” from the same. In other words, the things of the universe are informational elements that some of us read in one way and others in another, physics being the set of those elements that we all read in the same way: one physics. Consequently, the materiality of the universe, that of wave groups as their functional elements, is a perception that is set in motion for us from a click to a range of vibration or densification because we as observers participate in that same range of densification, that is, we attend to that click, attending or not to others by virtue of the relation of inclusion or permeability. To say this without the equations presented may sound like an ancestral hunch. But with them, and in particular with Equation (5), which relates the different forms of materiality to the successive phases that have been overcome, the question changes, although it should be treated, and will be in the future (where we will talk of SWP and relativity), with more rigour.