The existence of two values of $Z$ (versus $b$) corresponding to the respective phases is not an anomaly or deficiency in the representation of the reality we are presenting (even if it is only in the case at hand) but a consequence of the physical system itself that we simply have to recognise and integrate, as we have done, through the $\alpha$; and an explicit consequence, according to Equation (23), Equation (24) and Equation (30) in [1] , of the existence of two different constants of normalisation ( $A_1$ y $A_2$), that is, one for each of the intervals or material phases, as detailed below:

$\begin{array}{l}{B}^2{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\left[\zeta \left(x\right)\right]\text{d}x}=A_0^2{\displaystyle \underset{0}{\overset{c}{\int }}\left[\zeta \left(\upsilon \right)\right]\text{d}\upsilon }\\ \stackrel{\leftarrow S}{=}{A}_1^2{\displaystyle \underset{0}{\overset{c}{\int }}\left[{\zeta }_{f+\omega }\left(\upsilon \right)\right]\text{d}\upsilon }+A_1^2{\displaystyle \underset{0}{\overset{{\upsilon }_1\left(c\right)}{\int }}\left[{\zeta }_k\left(\upsilon \right)\right]\text{d}\upsilon }\\ \stackrel{\leftarrow \pi }{=}{A}_2^2{\displaystyle \underset{0}{\overset{c}{\int }}\left[{\zeta }_{f+\omega }\left(\upsilon \right)\right]\text{d}\upsilon }+A_2^2{\displaystyle \underset{0}{\overset{{\upsilon }_2\left(c\right)}{\int }}\left[{\zeta }_k\left(\upsilon \right)\right]\text{d}\upsilon }\stackrel{\leftarrow 2\pi }{=}\left[\mathrm{...}\right]\end{array}$ (13)

A differentiation of intervals that is not motivated by the phases themselves, nor by the difference or relationship between limiting velocities in the phases (which are always presented as such), but by the individual limits of these velocities, or more explicitly by the size of the integration interval and the alteration of the normalisation constant in the successive $\sin \left[\Phi \right]=0$ [proportional to that expressed in Equation (10) for $Z$], as a function of this size, for values of $\upsilon$ significantly distant from $c$. As in our case with ${\gamma }_{\mu }$ for the muon, which moves away from the reference with respect to $c$, unlike what happens with ${\gamma }_e$ for the electron.

A difference that, far from being overlooked, leads us to make a study of the circumstances and a comparative analysis of the results of the Lorentz factors ${\gamma }_e$ and ${\gamma }_{\mu }$, from which explicitly results the maximum velocity that each type of particle can reach in its phase:

$\begin{array}{l}{\gamma }_e=206.764\Rightarrow {\upsilon }_1\left(c\right)=0.999988c\\ {\gamma }_{\mu }=16.817\Rightarrow {\upsilon }_2\left(c\right)=0.99823c,\end{array}$ (14)

as we have reflected in Equation (13), where we can see that each phase change $\left({\varphi }_1,{\varphi }_2,\cdots \right)$ corresponds to a previous velocity limit $\left({\upsilon }_1,{\upsilon }_2,\cdots \right)$ at which the particle develops and that defines an interval or field of velocities $\left({\stackrel{⌢}{\varphi }}_1,{\stackrel{⌢}{\varphi }}_2,\cdots \right)$, respect to the maximum velocity and the interval more general and complete ${\stackrel{⌢}{\varphi }}_0\equiv \left[0,c\right[$ of the original phase ${\varphi }_0$, and a measure, in terms of the velocities, of the relative size of the phase in progress [2] , which is, as we said, that defines the normalisation constant, or its relation to the previous one, seen through the different normalization constants in Equation (13), from which it also follows that we can no longer speak (except by default) of $\alpha$, $\chi$ and $\Re$, but of ${\alpha }_i$, ${\chi }_i$ and ${\Re }_i$, so that the results reached in Equations (10)-(12), corresponding to the first triad of masses, ascribed to the initial phase ${\Phi }_1$ and the starting mass $m_1=e$, would have to be called more specifically ${\alpha }_1$, ${\chi }_1$ and ${\Re }_1$, or directly $1\equiv m_1$, if such specificity were needed.

While the connection of the limits of integration with the normalisation constants is obvious, it is not clear how they operate with each other. The only connection in this sense is reached through the normalisation process carried out in Equations (10)-(13) of [1] in which, passing to the complex plane, we passed to a two-dimensional scenario in which the solution is a pole, which is only half a circumference ( $\pi$ value), which leads us to think that this connection is established by a variable (trigonometric), referred to the same, which defines precisely a portion of this value, of the $\pi /2$, given that we can circumscribe these variables to a single quadrant. That is, what we say is that in ${\upsilon }_i(c)={\alpha }_{i}c$, ${\alpha }_i$ is a sine, which represents an angle of a certain quadrant, that ${\alpha }_i=1$, which corresponds to $c$, represents the whole quadrant, and that consequently the relation between the partial velocity ( ${\upsilon }_i$) and the total velocity ( $c$), which is given by the sine, is the relation between the partial angle of the quadrant and the total angle of the quadrant. From there, that we can extrapolate these corresponding linear values ${\alpha }_i$ to angles, and more specifically to surfaces, is a consequence, as we said, of the extrapolation that exists through the complex calculus.

In our case, we find that ${\gamma }_e=206.767$, with limit ${\upsilon }_1\left(c\right)=0.999988c$, and ${\gamma }_{\mu }=16.817$, with limit ${\upsilon }_2\left(c\right)=0.99823c$, subtend with respect to $c$ angles of ${\theta }_1=89.72$ and ${\theta }_2=86.59$ degrees respectively and, consequently, certain relations with respect to $\pm \pi /2$:

$\begin{array}{l}{\theta }_0^1={\theta }_1/{\theta }_0={\theta }_1/{\theta }_{\pi /2}=\frac{89.72˚}{90˚}\cong 0.997\text{(a)}\\ \text{and}\\ {\theta }_0^2={\theta }_2/{\theta }_0={\theta }_2/{\theta }_{\pi /2}=\frac{86.59˚}{90˚}\cong 0.962(\text{b})\end{array}$ (15)

In which Equation (15b), in the absence of further analysis, seems to meet the requirements and the intention in Equation (10), i.e. to establish and justify proportionality factor between the normalisation integrals and their corresponding indices of phase $Z_i$ ( $\to b_i$), while the result reached in Equation (15a) confronts us with the idea of recurrence already presented and, in particular, with the possible existence of a proportionality factor ${\theta }_0^1\cong 1$ or transit over the previous phase ${\Phi }_0$, which would give rise to the specific values ${\alpha }_0$, ${\chi }_0$, and ${\Re }_0$ of the significant variables. A proportionality factor ${\theta }_0^i$ which, as we can see, express a greater deviation the smaller the angle and the lower the speed at which the transition takes place, what, in practice (as we have already mentioned), means that in all cases it is ${\alpha }_i\simeq 1$, and that the case ${\theta }_0^2$ for the electron, which we have just seen, can be considered a peculiarity and a simple subject of study.

It might be difficult to find another argumentation that could justify the result Equation (10), such as the one used on the proportionality factor ${\theta }_0^2$ of Equation (15b), a result that, as we see, is reached at the margin or independently of the one reached in ${\theta }_0^1$, which certainly may be contrary to some logic, since it is reached through the relation of a phase with the previous one, which is somehow represented by ${\theta }_0^1$. That is to say, we see that although the proportionality factor of each phase is referred through $Z_i$ to the preceding phase ( $Z_2/Z_1$) it is not referred, however, through ${\upsilon }_i$ to the preceding phase but to the original ${\upsilon }_0=c$, and that, consequently, the quantity ${\alpha }_1$ that applies to $Z_1$, as a coupling factor between phases ( $Z_2={\alpha }_1{Z}_1$), depends exclusively on ${\theta }_0^2$. In particular, we can appreciate that, according to the sequence established in Equation (13) and the value reached in Equation (10), we do not make ${\theta }_1^2=0.962\times 0.997\cong 0.960$ and that only Equation (15b) is effective.

This behavior makes it clear that it is not the kinetic term associated with ${\stackrel{⌢}{\varphi }}_1$ that serves as a reference but, as it appears in Equation (13), the whole phase member, and in particular the first term ( ${\zeta }_{f+w}$, which accounts for the changes between mass and energy), which always belongs to ${\stackrel{⌢}{\varphi }}_0$, that is, to the velocity field that rearranges the energy at the interface for $\sin \left[\Phi \right]=0$, as we characterise schematically below.

$A_0^2\left({\stackrel{⌢}{\varphi }}_0\right)\stackrel{\leftarrow S}{=}{A}_1^2\left({\stackrel{⌢}{\varphi }}_0+{\stackrel{⌢}{\varphi }}_1\right)\underset{\sin =0\to }{\overset{\leftarrow \pi }{=}}{A}_2^2\left({\stackrel{⌢}{\varphi }}_0\right)\underset{\leftarrow \sin =0}{\overset{2\pi \to }{=}}{A}_2^2\left({\stackrel{⌢}{\varphi }}_0+{\stackrel{⌢}{\varphi }}_2\right)$ (16)

It becomes clear that in all phases there is a possible or reference velocity field ${\stackrel{⌢}{\varphi }}_0\equiv \left[0,c\right[$, corresponding to the maximum velocity, which is unique and universal for all systems (as derived from special relativity), and that in all of them there is a velocity field ${\stackrel{⌢}{\varphi }}_i<{\stackrel{⌢}{\varphi }}_0$ defined or specified actually by the phases themselves. And that it is the relationship between each of them and the first one that determines the evolution of the system parameters, such as the normalisation constants, being ${\theta }_0^2$ ( ${\stackrel{⌢}{\varphi }}_2$ on ${\stackrel{⌢}{\varphi }}_0$) and not ${\theta }_1^2$ ( ${\stackrel{⌢}{\varphi }}_2$ on ${\stackrel{⌢}{\varphi }}_1$), which determines it at the first phase change ( $Z_1\to Z_2$). This does not mean that ${\stackrel{⌢}{\varphi }}_1$ does not affect the normalisation constant, it means that it does not have an effect on transit $Z_1\to Z_2$ or on $\delta Z$, but that has it on $Z_1$ or, to put it better, that has had it to be configured as such $Z_1$.

Deepening on the latter and the explanation of the apparent paradox (although we could assume any result ${\theta }_0^2\in \left[0.960,0.965\right]$), the reason for the non-intervention of Equation (15a) in Equation (10), from a more instrumental point of view of the transit processes ( $S,{\varphi }_1$), is that this equation is included in both transit processes (with a phase change ${\theta }_0^1$ and two phase changes ${\theta }_0^{1}x{\theta }_1^2\to {\theta }_0^2$, respectively), and that the second transit carries over (in the normalisation constant itself) the transformations experienced by the first one so that all the phases are finally referred to the initial phase ${\varphi }_0$ subtended by the initial angle ${\theta }_0=\pm \pi /2$. Consequently, the above does not mean that the influence of the previous velocity fields does not exist or disappears, but that it is integrated into the constant that we carry over to later phases. This can be reflected in Equation (13), written as follows:

${\alpha }_1=\frac{Z_2}{Z_1}=\frac{Z_0{\theta }_1^2}{Z_0{\theta }_0^1}=\frac{Z_0{\theta }_0^1{\theta }_0^2}{Z_0{\theta }_0^1}=\frac{Z_1{\theta }_0^2}{Z_1}={\theta }_0^2=\frac{{\theta }_2}{{\theta }_0},$ (17)

Where we see that the angle of the denominator corresponding to the first transition (referred to ${\varphi }_0$ by default) acts as a normalisation factor in such a way that the second one, although referred to ${\varphi }_1$, is effectively referred to ${\varphi }_0$, making it clear, on the other hand, that we are indeed starting from that original value $Z_0$, that is, from a standard or unique wave packet for this class of particles, which makes our instrumental requirement a necessary condition with physical meaning. From there, since the values $Z_1$ and $Z_2$ are already known, all that remains in this case is to find the initial value $Z_0$ (associated to ${\stackrel{⌢}{\varphi }}_0$) from $Z_1$ by Equation (15a), which we have already identified with the phase coupling factor ${\alpha }_0$, specific to that transition.

$Z_0=\frac{Z_1}{{\alpha }_0}=\frac{19511.456}{0.997}=19570.166$ (18)

Following a similar development to that initiated in (14) of [1] , we obtain, using the generalised wave function for $q=2/3$ and $\phi =0$, an expression equivalent to Equation (19) and finally to Equation (26) of [1] ,

${\zeta }^{\left(-2/3\right)}\left(\upsilon \right)=\left[\frac{\hslash \upsilon \sin {\left(\Phi \right)}^{-2/3}}{a^3{\gamma }^{-3}}-\left(\frac{-2}{3{\Phi }^{5/6}}\right)\frac{\hslash \upsilon \Delta k\cos {\left(\Phi \right)}^{-2/3}}{2a^2{\gamma }^{-2}}\right]+\left[-\frac{\hslash \omega \cos {\left(\Phi \right)}^{-2/3}}{a^2{\gamma }^{-2}}\right],$ (20)

except for the introduced exponent and the parenthesis or additional factor of the second term, consequence of the additional derivative of $d\left[f\left(\Phi \right)\right]$ with respect to $d\left(\Phi \right)$, in charge of that exponent This factor will undoubtedly affect how the energy of formation is reached for this type of particle, that is, it will introduce differences with respect to the energy transfer cycle seen for charged leptons, about which we cannot say much more at present. This is why we overlook it to focus on the change in behaviour of the term associated with kinetic energy, as a consequence of the exponent relative to charge $q$, since the term is comparable to the well-known corpuscular expression and can therefore provide us with more information about the underlying vibrational behaviour, performing an analysis similar to that developed with leptons, since it is fulfilled, similarly the Equations (30)-(32) of [1] , and according to the previous energy balance of Equation (23):

$E_k=A^2{\displaystyle \int {\zeta }_k\left(\upsilon \right)\text{d}\upsilon }=A^2\left(\frac{\hslash }{a^3}\right)\sin {\left[\Phi \right]}^{-2/3}{\displaystyle \int \frac{\upsilon }{{\gamma }^{-3}}\text{d}\upsilon },$ (21)

with:

$\sin {\left[\Phi \right]}^{-2/3}={\left[2b^{-2/3}\left({\left(\frac{\hslash }{\pi m}\right)}^{1/3}{\gamma }^{-1}\right)\right]}^{-2/3}={\left[2b^{-2/3}\delta {\gamma }^{-1}\right]}^{-2/3}.$ (22)

for which we can demand the following condition:

$\sin {\left[\Phi \right]}^{-2/3}=0,$ (23)

As we did (dispensing with this development) in Equation (38) of [1] , where we saw (and will now elaborate on) that it was satisfactory for:

${\left[\Phi \right]}^{-2/3}={\left[2b^{-2/3}\delta {\gamma }^{-1}\right]}^{-2/3}={\left[Z\delta {\gamma }^{-1}\right]}^{-2/3}=2\pi ,3\pi \left({\varphi }_1,{\varphi }_2\right)=\varphi ,$ (24)

where it is clearly shown, as we said in [2] , that it could be $\varphi =f\left(\Phi \right)$, as a more general case (associated with the exponent), and not necessarily $\varphi =\Phi$, i.e. that the effective phase $\varphi$ does not coincide with the phase $\Phi$ for the sequence.

A sequence that can be deduced from Equation (22), in an analogous way as we did for leptons with Equation (7) and Equation (9), both generalized in this case for any charge, and with the initial phase or particle of application discretionally defined by a number subscript or initial (if necessary) in the phase ratio $\chi$, and the derived factors $\alpha$ and $\Re$.

$\begin{array}{c}{\chi }_1=\frac{{\varphi }_2}{{\varphi }_1}=\frac{{\left[{\beta }_2{\left(\frac{1}{m_2}\right)}^{1/3}{\gamma }_2^{-1}\right]}^{-q}}{{\left[{\beta }_1{\left(\frac{1}{m_1}\right)}^{1/3}{\gamma }_1^{-1}\right]}^{-q}}=\frac{{\left[{\beta }_2{\left(\frac{m_2^2}{m_3^3}\right)}^{1/3}\right]}^{-q}}{{\left[{\beta }_1{\left(\frac{m_1^2}{m_2^3}\right)}^{1/3}\right]}^{-q}}\\ ={\left[{\alpha }_1{\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{-q}={\alpha }_1^{-q}\times {\Re }_1^{-q},\end{array}$ (25)

that for $q=2/3$ results:

${\chi }_1={\chi }_u={\alpha }^{-2/3}{\left(\frac{m_c^5}{m_u^2{m}_t^3}\right)}^{-2/9}\simeq 1.5\left(\propto \frac{3\pi }{2\pi }=\frac{3}{2}\right),$ (26)

for the estimated masses (and already presented in [1] ), and chosen with some freedom (which we will justify later) with respect to the known values, for an estimate of $\hat{\alpha }={\alpha }^{-2/3}=1.005\cong 1$, that is, of $\alpha =0.9949$, established through the angles subtended by the velocities, which we can obtain from these masses and from Equation (3).

Indeed, if we stick to the tabulated experimental data [4] :

$m_u={2.16}_{-0.26}^{+0.49}\text{MeV,}{m}_c=1.27\pm 0.02\text{GeV,}{m}_t=172.9\pm 0.4\text{GeV,}$ (27)

results in a range of values, $\left[\chi \right]\simeq {\Re }_{m1}^{-2/3}=\left(1.43,1.72\right)$, which contains the simple ratio of natural numbers reached in Equation (26) for ${\chi }_1$, and only this (which excludes others such as $\chi =1$ or $\chi =2$), from the values of Equation (24). From there, we can estimate the value of the masses, among the possible ones, which preserve this simple ratio Equation (26), a requirement that the values specifically meet:

$m_u=2.07\text{MeV,}{m}_c=1.27\text{GeV,}{m}_t=172.9\text{GeV,}$ (28)

and a spectrum over them, motivated by their margins of error, within the range of values of Equation (27).

In this case, only the measure of $m_t$, as well as the charged leptons, is obtained directly [and that is the coincident value in Equation (27) and Equation (28)], having from the other two only an estimate, which can be rectified and accommodated to Equation (26) by other means. As it is, in fact, by some of the other experimental measures if we take into consideration, since they are indirect, those that do not reach the highest probability (as presented in detail in Appendix B of [1] ).

The objective of this Section was not to calculate the masses of the quarks but to show that the masses already calculated are governed or must be governed by the different physical expressions derived from Equation (19) and thereby verify that Postulate II is fulfilled for particles of this type, or, in other words, verify that it is possible to satisfy Equation (26) with existing data.

It will be the objective, however, later on, when, by incorporating an additional requirement, we can rectify these masses and adjust them completely using Equation (25), highlighting that the values of Equation (27), supplied or calculated by other methods, are not fully reliable, nor, consequently, the methodology used to achieve them.

Although it will be later when we approach and base this theoretical possibility we will nevertheless advance its results here (obtained in [1] ) to show, now that we are talking about the masses of quarks, some correct data about them (para $\hat{\alpha }=1$), according to Equation (26), for the two transitions of Equation (24):

$m_u=2.368\pm 0.003\text{MeV,}{m}_c=1.361\pm 0.003\text{GeV,}{m}_t=172.9\pm 0.4\text{GeV}$ (29)

which we present in Table 2 , together with the other additional parameters for such transitions:

In which we see that now (as a more general case) we can indeed establish a relation or sequence on a single value $Z=Z_1=Z_2$ (versus $\beta$) and an initial mass for each row, which is nothing more than that corresponding to each of the phases in Equation (25), which take this form:

${\varphi }_i={\left[{\beta }_i{\left(m_i^2/m_{i+1}^3\right)}^{1/3}\right]}^{-q}={\left({\beta }_i{\bar{\gamma }}_i\right)}^{-q}$, (30)

by which $m_{i+1}$ evolves from its initial value $m_i$ to that other one in which the effective phase corresponding to the inertial mass is reached, which defines the initial mass of the next cycle, constituting the practical expression or verification of our postulates. A sole expression or sequence that accounts for the value of the particles of a class, and which is even continuous in sections if we make $\varphi ={\varphi }_i+{\varphi }_{i-1}$, that is, if (taking $i=2$) we establish it over the previous phase ( ${\varphi }_{i-1}={\varphi }_1$), as represented graphically in Figure 2 :

A previous phase ( ${\varphi }_{i-1}={\varphi }_1$) introduced for the purpose we have just indicated, which for ${\varphi }_2$ is none other than the initial phase $\phi$ incorporated in Equation (19), which will take the values of that phase $\phi ={\varphi }_1={\Phi }_1^{-q}$, while this of the previous phase $\phi ={\varphi }_0={\Phi }_0^{-q}$, corresponding to ${\varphi }_1$, will generally be zero ( ${\phi }_u=0$), as for electrons ( ${\phi }_e=0$). Being ${\phi }_c=2\pi$ the advance of the previous phase for the following transit (which for the corresponding lepton would be ${\phi }_{\mu }=\pi$). It is understood that this is an interpretation or modelling of what is happening, being other than ${\phi }_c=0$ and ${\phi }_{\mu }=0$, which would make each phase, once the new mass is configured, start from the same situation, from the same phase (since ${\stackrel{⌢}{\varphi }}_0\equiv \left[0,c\right[$ is not exceeded).

The choice of one option or the other will depend, as discussed in [1] , on whether we grant a different degree of materiality to each transit or whether, on the contrary, we restrict this division to what is already presented to us as material (corpuscle) and non-material (phase factor), which in itself depends on what nature itself has chosen as a path. Although the second (and third) generation particles that we detect (in our phase) respond to the second case, it may be that there is an undetectable universe of particles associated with the first one.

Similarly, for $q=-1/3$ we obtain:

${\zeta }^{\left(-1/3\right)}\left(\upsilon \right)=\left[\frac{\hslash \upsilon \sin {\left(\Phi \right)}^{1/3}}{a^3{\gamma }^{-3}}-\left(\frac{1}{3{\Phi }^{2/3}}\right)\frac{\hslash \upsilon \Delta k\cos {\left(\Phi \right)}^{1/3}}{2a^2{\gamma }^{-2}}\right]+\left[-\frac{\hslash \omega \cos {\left(\Phi \right)}^{1/3}}{a^2{\gamma }^{-2}}\right]$ (31)

Restricting our analysis to kinetic energy, with

$E_k=A^2{\displaystyle \int {\zeta }_k\left(\upsilon \right)\text{d}\upsilon }=A^2\left(\frac{\hslash }{a^3}\right)\sin {\left[\Phi \right]}^{1/3}{\displaystyle \int \frac{\upsilon }{{\gamma }^{-3}}\text{d}\upsilon }$ (32)

and

$\sin {\left[\Phi \right]}^{1/3}={\left[2b^{-2/3}\left({\left(\frac{\hslash }{\pi m}\right)}^{1/3}{\gamma }^{-1}\right)\right]}^{1/3}={\left[2b^{-2/3}\delta {\gamma }^{-1}\right]}^{1/3}.$ (33)

Fulfilling that:

$\sin {\left[\Phi \right]}^{1/3}=0,$ (34)

which, in this case, is given for:

${\left[\Phi \right]}^{1/3}={\left[2b^{-2/3}\delta {\gamma }^{-1}\right]}^{1/3}={\left[Z\delta {\gamma }^{-1}\right]}^{1/3}=2\pi ,\pi \left({\varphi }_1,{\varphi }_2\right)=\varphi$ (35)

A sequence that we will put in the equivalent form:

$\varphi ={\left[2b^{-2/3}\delta {\gamma }^{-1}\right]}^{1/3}={\left[Z\delta {\gamma }^{-1}\right]}^{1/3}=4\pi ,2\pi ,\pi \left({\varphi }_1,{\varphi }_2,{\varphi }_3\right),$ (36)

because it is a form that would allow the incorporation of ${\varphi }_3=p\pi$ $\left(\text{:}p\in \mathbb{N}\right)$, that is, a third transition for the assumption of a fourth generation, which the other classes of particles by this particular (the increasing evolution of the effective phases $\varphi$) allow, as well as being backed up. In addition to being obliged or having no other choice, as we will see in Section 5, by the additional and unavoidable requirement, already mentioned above. We must also remember that, although there is no transition to the fourth generation, there is in all cases the phase $\left[{\varphi }_3\right]$ interval in which the third generation takes place.

A sequence achieved for $\varphi$, which is demonstrated once again by simply replacing the approximate values of the masses in Equation (25), adapted in this case to the new exponent (1/3):

${\chi }_1={\chi }_d=\frac{{\varphi }_2}{{\varphi }_1}={\alpha }^{1/3}{\left(\frac{m_s^5}{m_d^2{m}_b^3}\right)}^{1/9}\simeq 0.5\left(\propto \frac{1\pi }{2\pi }\right),$ (37)

in which, with $\alpha =1.01295$ and $\hat{\alpha }={\alpha }^{1/3}=1.0042$, the ratio ${\chi }_1$ is left entirely to the masses, which for the listed values of the masses:

$m_d={4.67}_{-0.17}^{+0.48}\text{MeV},m_s={93}_{-5}^{+11}\text{MeV},m_b={4.18}_{-0.02}^{+0.03}\text{GeV},$ (38)

provides a range of values $\left[\chi \right]\simeq {\Re }_{m1}^{1/3}=\left(0.518,0.588\right)$.

A result deviating from that required in Equation (37), from which it follows that these masses must also be redefined with the additional condition, and, again, that the indirect calculation parameters used to obtain Equation (38) are not sufficiently balanced or achieved, here, with ${\chi }_1\notin \left[\chi \right]$, with even more justification.

As in the previous case and to present correct data of the masses, here where we are talking about them, we will make an advanced use, together with the application of Equation (36) and Equation (37), of these definition criteria, which results:

$m_d={5.395}_{-13}^{+19}\text{MeV,}{m}_s={83.88}_{-0.31}^{+0.49}\text{MeV,}{m}_b={4.18}_{-0.02}^{+0.03}\text{GeV}\text{.}$ (39)

Values that also correspond to the measures listed experimentally [4] , if we take into consideration those recorded that do not reach the highest probability (as presented in detail in Appendix B of [1] ), which we present in Table 3 , as in the previous cases, along with the others additional parameters for these transitions:

Values that we can also achieve and represent by Equation (30), for ${\phi }_d=0$ and ${\phi }_s=4\pi$ ( Figure 3 ):

And which, in a similar way to the previous one, are not only those corresponding to the exact values $\varphi$ defined in Equation (36) but to the exact value $\chi$, defined in Equation (37), that, together with the results of Equation (12) for Table 1 and Equation (26) for Table 2 , can be presented in a new table ( Table 4 ).

Continued

Emphasizing that although the triad gives us a result $\chi$ that defines it, the true relationship of the particles is between them each two, that is, the one derived from the phase, or phasic situation between one and the next in the sequence, given by Equation (30).

On Equation (25), the change of sign will mean that the ratio between phases is inverted, which in the case of leptons will be:

$\bar{\chi }=\frac{{\bar{\varphi }}_2}{{\bar{\varphi }}_1}={\left[\alpha {\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{-1}={\alpha }^{-1}\times {\Re }^{-1}=\hat{\alpha }{\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{-1/3}=\frac{{\varphi }_1}{{\varphi }_2}={\chi }^{-1}$ (40)

Therefore, the following is fulfilled:

$\chi \bar{\chi }=1$ (41)

This gives a value $\bar{\chi }\approx 0.5$ which implies that the sequence applicable to the phase is:

$\bar{\varphi }={\left(2{\bar{b}}^{-2/3}\delta {\gamma }^{-1}\right)}^{-1}={\left(\bar{Z}\delta {\gamma }^{-1}\right)}^{-1}=2\pi ,\pi \left({\bar{\varphi }}_1,{\bar{\varphi }}_2\right).$ (42)

That is, this or a proportional one and, in any case, inverse to the one given for leptons in Equation (6). It is also possible to verify that:

${\varphi }_1{\bar{\varphi }}_1={\varphi }_2{\bar{\varphi }}_2$ (43)

From the sequences in Equation (6) and Equation (42) the corresponding value $\bar{Z}$ is quickly reached:

$\frac{Z\delta }{\pi }=\gamma =2\pi \bar{Z}\delta \Rightarrow \frac{Z}{\bar{Z}}=\left(2\pi \right)\pi =2{\pi }^2\Rightarrow \bar{Z}={19.7392}^{-1}Z$ (44)

with which, applied to Equation (5a), we can obtain the associated value $\bar{b}$. Operative that is valid for both ${\bar{Z}}_1$ and ${\bar{Z}}_2$.

With the phase sequence in Equation (42) and the $\bar{Z}$ obtained by Equation (44) or manually (as we did with the particles through their masses), values for antiparticles are achieved that are completely in agreement with the associated particles, since neither the initial mass nor $\bar{\delta }=\delta$ changes through Equation (5b), whereas $\bar{\gamma }=\gamma$ as evidenced in Equation (44). Results that can be put into Table 5 :

From here:

$\frac{{\bar{Z}}_2}{{\bar{Z}}_1}=\alpha =\frac{950.778}{988.4619}\cong 0.962$ (45)

Value $\alpha$ identical to that already found, and which also corresponds to the ratio of the angles subtended by the speeds associated with ${\gamma }_e$ and ${\gamma }_{\mu }$, with which it also complies:

$\begin{array}{c}\bar{\chi }={\left[\alpha {\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{-1}={\alpha }^{-1}\times {\Re }^{-1}={0.962}^{-1}\times {2.079}^{-1}\\ =1.039\times 0.481=0.5=2^{-1}={\alpha }^{-1}{\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{-1/3}\end{array}$ (46)

That is, a product of inverse factors to those reached for the particles, with which an inverse value is reached for $\bar{\chi }$ to that reached with them, as we had already advanced, composed with the same factors $\alpha$ and $\Re$ (which do not change), fulfilling, on the other hand, the expectations regarding antileptons in the developed framework.

For these, in even more abbreviated form, we have:

$\bar{\chi }=\frac{{\bar{\varphi }}_2}{{\bar{\varphi }}_1}={\left[\alpha {\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{2/3}={\alpha }^{2/3}\times {\Re }^{2/3}=\hat{\alpha }{\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{2/9}=\frac{{\varphi }_1}{{\varphi }_2}={\chi }^{-1}$ (47)

This gives a value $\bar{\chi }\simeq 2/3$, for $\hat{\alpha }\simeq 1$, which implies that the sequence applicable to the phase is:

${\left[\Phi \right]}^{2/3}={\left[2{\bar{b}}^{-2/3}\delta {\gamma }^{-1}\right]}^{2/3}={\left[\bar{Z}\delta {\gamma }^{-1}\right]}^{2/3}=3\pi ,2\pi \left({\bar{\varphi }}_1,{\bar{\varphi }}_2\right)=\bar{\varphi }$ (48)

that is, this one or a proportional one and, in any case, inverse to the one given for particles in Equation (24), which also fulfils Equation (43).

From the sequences in Equation (24) and Equation (48), the value $\bar{Z}$ associated with it is quickly reached:

$\frac{Z\delta }{{\left(3\pi \right)}^{-3/2}}=\gamma ={\left(2\pi \right)}^{-3/2}\bar{Z}\delta \Rightarrow \frac{Z}{\bar{Z}}={\left(6{\pi }^2\right)}^{-3/2}\Rightarrow \bar{Z}=455.697Z$ (49)

With the phase sequence of Equation (48) and the $\bar{Z}$ obtained by Equation (49) or manually (as we did with the particles through their masses), values for antiparticles are achieved that are fully in agreement with the associated particles, which we may reflect in Table 6 :

Similarly, we have:

$\bar{\chi }=\frac{{\bar{\varphi }}_2}{{\bar{\varphi }}_1}={\left[\alpha {\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{-1/3}={\alpha }^{-1/3}\times {\Re }^{-1/3}=\hat{\alpha }{\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{-1/9}=\frac{{\varphi }_1}{{\varphi }_2}={\chi }^{-1}$ (50)

This gives a value $\bar{\chi }\simeq 2$, for $\hat{\alpha }\simeq 1$, which implies that the sequence applicable to the phase is:

${\left[\Phi \right]}^{-1/3}={\left[2{\bar{b}}^{-2/3}\delta {\gamma }^{-1}\right]}^{-1/3}={\left[\bar{Z}\delta {\gamma }^{-1}\right]}^{-1/3}=\pi ,2\pi ,4\pi \left({\bar{\varphi }}_1,{\bar{\varphi }}_2,{\bar{\varphi }}_3\right)=\bar{\varphi }$ (51)

which is inverse to the one given for particles in Equation (36) and also fulfils Equation (43).

From the sequences in Equation (36) and Equation (51), the value $\bar{Z}$ associated with it is quickly reached:

$\frac{Z\delta }{{\left(4\pi \right)}^3}=\gamma ={\pi }^3\bar{Z}\delta \Rightarrow \frac{Z}{\bar{Z}}={\left(4{\pi }^2\right)}^3\Rightarrow \bar{Z}={\left(61528.908\right)}^{-1}Z$ (52)

With the phase sequence in Equation (51) and the $\bar{Z}$ obtained by Equation (52) or manually (as we did with the particles through their masses), values for antiparticles are achieved that are completely in agreement with the associated particles, which we may put in Table 7 :

We have to draw attention to the fact that the evolution of phases with a negative exponent (positive charge) is increasing and those with a positive exponent (negative charge) is decreasing, as shown, respectively, in Figure 2 and Figure 3 , i.e. the phase value ${\varphi }_i$ is reached, according to condition $\text{sin}{\left[\Phi \right]}^{-q}=0$, by an evolution of the phase from $\varphi =0$ or from $\varphi \approx \infty$, as the case may be, giving rise to development intervals of the type $\left[{\bar{\varphi }}_1\right]=\left[0,2\pi \right]$ y $\left[{\varphi }_1\right]=\left]\infty ,\pi \right]$.

The treatment of this issue, in addition to being a very complex one and deserving a separate detailed study (as we will do in another paper), is practically irrelevant to our objectives now, so it is not necessary to dwell on it. Just to say in this respect, as an advance, that trying to explain why some phases are one way or another without the help of the particle that supports them makes no sense, that is to say, a phase without that support runs, without further ado, from one end of the universe to the other (from right to left, we could say) and without the possibility of establishing a point of reference on it. It is when we establish that reference (the particle) that we can say what happens up to it or what happens from it, regarding the phase, that is to say, when two differentiated movements of phase are established from the point of view of the particle.

Two distinct movements that we can differentiate graphically to the right ( $|\leftarrow$) and left ( $\leftarrow |$), respectively (as if they were two parts of the same continuous phase). The first starts from $\infty$ with the intention of carrying out the path $\left[\varphi \right]=\left]\infty ,0\right]$ that finally only progresses to the values of phase change, that is, to the closest to the particle, and concretely to the first $\left[{\varphi }_1\right]=\left]\infty ,\pi \right]$. The second, starts from 0, with the intention of carrying out the path $\left[\bar{\varphi }\right]=\left[0,\infty \right[$ that finally only progresses to the values of phase change and concretely to the first $\left[{\bar{\varphi }}_1\right]=\left[0,2\pi \right]$, as closest to the particle. It is these positions close to the particle that are reached or completed, either on one side or the other, because it is the proximity to the particle that is likely to constitute a state, regardless of whether the traveling wave has a plus or minus sign.