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\title{\textbf{\huge{ Transition Radius Method }
\author{\textbf{M.D.~Holte}}}}
\date{23 September 2006}

%\title{\textbf{\large{ Transition Radius Method Extended}
%date{Extended version  September 23,  2006
%\\ Previous version  November 12,  2003

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\begin{abstract} Solving the $R_{22}$ Ricci expression iteratively yields a way to define both a transition radius and a new expression for determining fundamental boson energies.  This transition radius can be defined as a radius value where the three spatial dimension f(r) metric component is equal to the same higher dimensional metric component, and the benefit of this transition radius is that it can be used to determine the energy of  fundamental boson types and fundamental particle types.  This method fits a total of fifteen boson types including %associated with twelve fundamental particles plus t
the W, Z, and X bosons.  Also, letting a higher dimensional radius go to zero for an infinite number of spatial dimensions predicts an energy for a new boson type.  The transition radius method seems to be a preferable way to resolve a part of the hierarchy problem because it fits fundamental particle energies,  it yields a simple expression for boson energy, and it is derived from general relativity. 
%This extended version examines the effect of changing the units of 2M from centimeters to Planck lengths on particle masses and fit parameters.

%[© 2006  M. D. Holte]
\end{abstract}


\section{Introduction}

Previously, Schwarzschild solutions have been modified for spatial dimensions higher than three by raising the power of r in (1 – 2M/r) as described in \cite{myers1986}. For example, r would be changed to $r^2$ for four spatial dimensions, to $r^3$ for five spatial dimensions.  As an alternative, at large distance from a massive object, the  higher dimensional Schwarzschild radius could change in a way that is similar to the way a flat space radius changes with higher dimensions.

%Previously, Schwarzschild solutions have been modified for spatial dimensions higher than three by raising the power of r in (1 – 2M/r) as described in \cite{myers1986}. For example, r would be changed to $r^2$ for four spatial dimensions, to $r^3$ for five spatial dimensions.  There may be a better way to connect a metric to flat space as the value of the radius r becomes large.  It seems that at a large distance from a massive object a higher dimensional Schwarzschild radius should change in a way that is somewhat similar to the way a flat space radius changes with higher dimensions.
    
To explore how general relativity connects to flat space in higher dimensions we %should
examine flat space radius expressions to get an idea of how the r term should be changed for higher spatial dimensions.  For Euclidean space,  where a three dimensional Cartesian coordinate system applies, the radius squared can be found from the expression $r^2 = (x1 – x1^*)^2 + (x2 – x2^*)^2 + (x3 – x3^*)^2$  for the two end points $(x1,  x2, x3)$ and $(x1^*, x2^*, x3^*)$.  Extending flat space to five  spatial dimensions this radius squared becomes $r^2 = (x1 – x1^*)^2 + (x2 – x2^*)^2 + (x3 – x3^*)^2 + (x4 – x4^*)^2 + (x5 – x5^*)^2$  by extending the distance formula by two additional terms.  We see that this five dimensional radius squared in flat space is corrected for extra dimensions by the addition of two more  nonlinear terms.  This  suggests that the radius term in (1 – 2M/r) should be corrected for higher dimensions by adding nonlinear terms to r squared, and not by raising the power of  r.  However, such a correction can not come from an expression which describes flat space because the Schwarzschild solution describes curved space.
    
General relativity offers guidance about 
%The best place to look for the 
nonlinear terms to be added to the radius term in (1 – 2M/r) to correct it for higher spatial dimensions.
%is to look within general relativity.  
Going back to the derivation of the Schwarzschild solution, as shown in Robert Wald’s book \textit{General Relativity} on page 123, we see that the Ricci expressions $R_{22}$ and $R_{33}$ were not used to derive the Schwarzschild solution \cite{wald1984the}. In this work, we explore these two Ricci expressions which were set aside to determine if they can be used to find the nonlinear terms that need to be added to the radius term in (1 – 2M/r) to correct it for higher spatial dimensions.  Since $R_{22}$ = $R_{33}$ we only need to look at $R_{22}$ more closely. Integration of $R_{22}$ gives a dimensionally corrected radius expression which matches our expectations because its form is a sum of nonlinear terms. %\cite{holte2003transition} 
    
\indent These nonlinear terms, which correct the radius used in (1 – 2M/r) when there are more than three spatial dimensions, are of the form A(ln(r))$^n$ where A is a factor that changes for each integer value of n, r is the radius of a thin-shell bubble as in Israel \cite{Israel:1966rt}, n is the number of spatial dimensions minus three, and $\ln{(r)}$ is the natural logarithm of the radius.  These dimensionally corrected radii are derived from $f_{i} (r)$ used in an expression which has the form $(SLOPE)_{i} [ f_{i} (r) ]^{-1} + 1$ where best values of SLOPE  are  used  to smoothly  connect  this  new expression to (1 – 2M/r) over a radius range of highest energy.  Functions $f_{i}(r)$ are changed to dimensionally corrected radii by $R_{i} (r) = –3 f_{i} (r)$ so that $R_{i} (r)$ has a positive value where i is the number of spatial dimensions.  It is important to note that $f_{i} (r)$ is the dimensionally corrected radius expression $R_{i} (r)$ divided by – 3 , and $f(r)$ is the factor multiplying $dt^2$ in the metric; so these two functions are completely different.

\indent To find the value of SLOPE for any spatial dimension requires making a starting guess for the value of (1 – 2M/r).  We need to make the best possible starting guess for (1 – 2M/r) which will anchor the lowest possible value of M consistent for Schwarzschild black holes.  A suitable value is the approximate maximum neutron-star mass of 2.2 solar masses \cite[p.205]{thorne1994black}.
%is  I have selected evaluating 2.2 solar masses, after reading page 205 of Kip Thorne’s book on black holes where he pointed out an approximate maximum neutron-star mass of two solar masses.\cite{thorne1994black}  
%The expression (1 – 2M/r) had 2M in centimeter units in the previous paper titled Transition Radius Method; however, 
When 2M is in units of Planck lengths it is (1 – $4.076 \times 10^{38}$/r) at 2.20 solar masses when M is “nongeometrized”.  
As will be discussed in section 12, the results will apply from 2.2 solar masses up past 3 million solar masses and to the mass of the universe.
%As I point out in section 12, I  found that this method, I call new extended TRM, applies from 2.2 solar masses up to 3 million solar masses and up to the mass of our universe. 

\indent Determining the value of SLOPE for each spatial dimension involves averaging a slope estimating the following expression SX (short for slope-estimating expression) for two to five SX values.  This SX is given by SX = (1 – 2M/r) $(f_{i} (r))$, and in all but one case these SX values were only determined for values of radius which changed by an order of magnitude from one radius value to the next.  By averaging different numbers of SX values, the range of radius values over which (1 – 2M/r) and (SLOPE) [ $f_{i} (r) ]^{-1} + 1$  are coupled is varied depending on the number of spatial dimensions involved. 
%Expecting to be able to average the same number of SX values for all numbers of spatial dimensions would have been more bizarre than assuming there is variation in the number of SX values to be averaged for different numbers of spatial dimensions.

\indent Once a SLOPE value is determined for a given number of spatial dimensions, the transition radius values are determined by  finding radius values  for  2.20  solar  masses  which  satisfy      $(1 – 4.076 \times 10^{38}/r)$  =  (SLOPE)  [$f_{i} (r) ]^{-1} + 1$  for each number of spatial dimensions.  For five spatial dimensions and for seven through twenty-four spatial dimensions I find two transition radii for each; however, four and six spatial dimensions both have an infinite number of transition radii.  The significance of these differences will be discussed later.

\indent I have found that the energy of a boson type is proportional to the reciprocal of its transition radius cubed where a boson type energy is the energy of a specific particle/antiparticle annihilation at rest mass. These boson type energies match masses for fundamental particle pairs or energies for fundamental bosons. A total of fifteen different boson types are fit by this transition radius method.  For all of these bosons their TRM fit energies can match their expected values exactly.  In addition to these bosons, a  boson for an infinite number of spatial dimensions is defined which may represent a boson unification energy.

%\indent For the previous version of this transition radius method it was assumed that a rescaling of the radius and M in (1 – 2M/r) was unnecessary because the SLOPE parameter would compensate for any rescaling.  I have had doubts about this, and decided to test this assumption by changing the units of 2M so its values goes from 652,100 for the previous paper to 4.076 x $10^{38}$ Planck lengths when M is 2.2 solar masses for this extended version The data have been updated in this paper everywhere this change of M has shifted values, and comparisons are made in section 7.  Slight shifts in the predicted energies of fundamental particles are seen after changing the of units for M, and new findings at the LHC might be used to determine if  2M = 4.076 x $10^{38}$ when M is 2.2 solar masses, for this  extended version, gives the best results or if the previous version gives a better fit of particle energies. Much of the text is unchanged from the previous paper since major concepts remain the same.

\indent This transition radius method also gives a transition radius value which relates to the size of the universe at time equals zero. One of the two transition radius values for seven spatial dimension is 1130 Planck lengths which is very close to an estimated bounce radius of 1000 Planck lengths, based on an expected energy density at time equals zero, which was derived by Carmen Molina-París and Matt Visser.\cite{molinaparis1999}  Also, the transition radius method implies that our universe was initiated by a ‘presingularity’ inside a black hole in a parent universe or our universe resulted from a bounce.  I use the word ‘presingularity’ to describe an object having a radius  between one Planck length and 30,000 Planck lengths where this object at some time was capable of gravitational collapse.

\indent Finally, I suggest ways that the transition radius method might be used to advance other theories. Throughout this paper I will be using five abbreviations.  Ds will be short for spatial dimensions.  For example, 3 - Ds is shorthand for three spatial dimensions.  Next, Ns is short for the number of spatial dimensions. The shorthand symbol hdf stands for higher dimensional $f(r)$ which means an $f(r)$ for more than 3 - Ds.  Another abbreviation is SX which is short for a  SLOPE estimating expression. Finally, the overall transition radius method will be abbreviated TRM.
%, and this paper presents the extended version of TRM – also called TRM extended  or  extended TRM.

     
\section{Higher demensional metric component}

The transition radius method (TRM) 
%began in February of 2001 when I solved the Ricci 
begins with solving the Ricci expression $R_{22}$ for $f$ in a way that gives a sum of nonlinear terms, this $R_{22}$ expression is defined in \textit{General Relativity} by Robert Wald on page 123 of his book.\cite{wald1984the} This $R_{22}$  is the Ricci expression which was not used to derive the Schwarzschild solution as shown on pages 123 - 124 of the same book. 

\subsection{\textit{Another look at the Ricci expressions}}


\indent Solving expression $R_{22}$ for $f$ is achieved by first simplifying the expression by substituting both $h = \frac{1}{f}$ and $h^’ = – f^{-2}$ into $R_{22}$. The $R_{22}$ expression follows:
\begin{equation}
    R_{22} = -(\frac{1}{2})(rfh)^{-1}f^{'} + (\frac{1}{2})(rh^2)^{-1}h^{'}+r^{-2}(1-h^{-1})
\end{equation}

    Expression 2.1 is identical to expression 6.1.37 from reference 5. Substituting $\frac{1}{f}$ for h in 2.1 gives:
\begin{equation}
    0 = – (\frac{1}{2}) (r^{-1}) f^{’} + (\frac{1}{2}) (r^{-1}) f^{2} h^{’}  + (r^{-1}) \frac{(1 – f)}{r}        
\end{equation}
    Substituting $(–f -2)$ for $h^{’}$ in expression 2.2 and multiplying both sides by 2r simplifies it to:

\begin{equation}
    0 = – f^{’} + (– f^{-2}) f^{2} + 2\frac{(1 – f)}{r}
\end{equation}

This expression reduces to:
\begin{equation}
    0 = – f ’ – 1  + 2 \frac{(1 – f)}{r}                    
\end{equation}

    Expression 2.4 does not match expression 6.1.40 on page 123 of Wald’s book because it is derived from expression $R_{22}$ only. Let’s rearrange expression 2.4 then integrate both sides:
\begin{equation}
    f^{’} = 2 \frac{(1 – f)}{r} – 1  
\end{equation}

\begin{equation}
    ∫ df = ∫ [\;(2\;\frac{(1 – f )}{r}) – 1 \;]\; dr
\end{equation}
\begin{equation}
    f + C1 = 2 ∫ [\;\frac{(1 – f)}{r}\;]\; dr – ∫ \;dr
\end{equation}
    Two additional constants of integration will result from integrations on the right hand side of 2.7, but to simplify the expressions I will assume that all  constants of integration are zero.   Integrating expression 2.7 gives: 
\begin{equation}
    f = 2 ∫ (\frac{1}{r})\:dr \;–2 ∫ (\frac{f}{r})\;dr – r
\end{equation}
\begin{equation}
    f = − r + 2\, \ln{(r)} − 2 \;∫ (\frac{f}{r})\;dr
\end{equation}

I have found that expression 2.9 can generate a series of functions by guessing the function f and substituting it into the right hand side of expression 2.9.  The series of functions that f can generate will be called $f_{i}$ expressions.The first guess of f is:

\begin{equation}
    f = 2 \ln{(r)} + (\frac{–r}{3}) – 2\;(\ln{(r)})^{2}
\end{equation}

This first guess is substituted into the right hand side of expression 2.9 giving an expression for $f_{i}$:

\begin{equation}
    f_{i} = – r + 2 \ln{(r)}\, –2 ∫ [\; (2 \ln{(r)} –  \frac{r}{3} +\frac{(− 2) (\ln{(r)})^{2})}{r} \;]\; dr
\end{equation}  

\begin{equation}
    f_{i} = − r + 2 \ln{(r)} − 2 \;[\; 2\; (\frac{1}{2}) (\ln{(r)})^2 − \frac{r}{3} + (−\frac{2}{3}) (\ln{(r)})^3 \;] 
\end{equation}

\begin{equation}
    f_{i} = (\frac{− r}{3}) + (2 \ln{(r)}) + (− 2) (\ln{(r)})^{2} + (\frac{4}{3}) (\ln{(r)})^{3}                                              
\end{equation}

Successively higher terms of $f_{i}$ can be generated by using the fourth term of expression 2.13 as a guess for f to be substituted into the third term of expression 2.9.  Let’s separate out this third term of expression 2.9 so that it stands on its own as a term  generating function:

\begin{equation}
    \text{Term Generator} = − 2 ∫ (\frac{f}{r})\; dr                    
\end{equation}

\begin{equation}
    \text{Term 5} = − 2 ∫ [\;(\frac{4}{3}) (\frac{\ln{(r)})^{3}}{r} ]\; dr  = − 2 [(\frac{1}{3}) (\ln{(r)})^{4}\;  ] 
    =  (\frac{−2}{3}) (\ln{(r)})^{4}                
\end{equation}
    The process is repeated by substituting Term 5 into expression 2.14 for f to generate Term 6.   In general, Term (J + 1) is generated by substituting Term J into expression 2.14 for f  and solving.  This process generates successively higher terms for $f_{i}$   shown below:
    
    \begin{equation}
 \text{Term}\; 6 = − 2 ∫ [\;(\frac{− 2}{3}) \frac{(\ln{(r)})^{4}}{r} \;] \;dr \text{ so... }\rightarrow \text{Term}\; 6 = (\frac{4}{15}) (\ln{(r)})^5
 \end{equation}


\begin{equation}
    \text{Term}\; 7 = (\frac{− 4}{45}) (\ln{(r)})^6                                   
\end{equation}

\begin{equation}
    \text{Term}\; 8 = (\frac{8}{315}) (\ln{(r)})^{7}                                    
\end{equation}

\begin{equation}
    \text{Term}\; 9 = (\frac{− 16}{2520}) (\ln{(r)})^8                            
\end{equation}
This process can go on to generate an infinite number of terms, but let’s derive a general expression for $f_i$ by inspecting the way successive terms above change relative to each other.  A repetitive trend starts in expression 2.13 with the second term 2 ln r.  By comparing the second term to the higher terms listed above I found that the following expression generates successive terms:
\begin{equation}
   \text{Term}\; (n + 1) = [\;\frac{2\; (− 2)^{n – 1}} {n!}\;] (\ln{(r)})^{n}          
\end{equation}
    A general expression for $f_{i}$ can be formulated using expression 2.20 as shown below:
\begin{equation}
    f_{i} (r) = (\frac{− r}{3}) + \sum_{n=1}^{k} [\; ( 2 (− 2)^{n} – \frac{1} {n!} ) (\ln{(r)})^{n}\; ]                                                           
\end{equation}
    
In expression 2.21 the left hand side $f_{i}$ (r) is a function where (r) means $f_{i}$is a function of the radius r.   The subscript i is not yet defined in relation to n.  When n = 0 expression 2.21 reduces to −1 so having n = 0 does not generate the next term after ($\frac{− r}{3}$) in expression 2.13.  However, when n = 1 for expression 2.21 the term 2 ln r is generated.  I noticed that the ($\frac{− r}{3}$) term corresponds to a vector for 3 - Ds (three spatial dimensions) so adding 2 ln r should correspond to 4 - Ds.  If we let the subscript i designate Ns (the number of spatial dimensions), then when i = 4 the value of n is n = 1 so n = i – 3 for expression 2.21 above.  Therefore, if the summation in expression 2.21 starts with n = 1 this first term of the summation generates $f_{4}$ (r).  It follows that summing from n = 1 to n = 2 will generate $f_5$ (r) or in general summing 2.21 from    n = 1 to n = k gives $f_{3+k}$ (r) so that i = 3 + k when the summation goes from n = 1 to n = k.  This fully defines the relationship between the subscript  i  and the summation parameter n.
    
\indent Expression 2.21 combined with the paragraph following it defines a series of higher dimensional expressions that have some relationship to the (3 + 1) dimensional Schwarzschild solution.  Notice that the first term on the right hand side of expression 2.21 is equal to r after both sides of expression 2.21 are multiplied by -3. If we let $R_{i}$(r) = -3 $f_{i}$ (r) we see that $R_{i}$ (r) appears to define a series of higher dimensional radius values.  This suggests that the higher dimensional radius $R_i$ (r) should replace the radius in (1-2M/r) to give the modified form  (1-KM/$R_{i}$)  (r) ) where K is some constant.  If this comparison to  (1-2M/r)  of the (3 + 1) dimensional metric is correct, then the best way to relate $f_{i}$ (r) to  (1-2M/r)  would be to compare  (1-2M/r)  to ($\frac{1 + (SLOPE)}{f_{i} (r)}$) which contains a plus sign because $f_{i}$ (r) is less than zero.  In my original notes I wrote this expression as   (SLOPE) [$f_{i} (r) ]^{-1}$ + 1    to  put  it  in  the   form mx + b so I will use this second form of the expression below. 
    
\indent This term  (1-2M/r)  comes from the (3 + 1) dimensional Schwarzschild solution which is:
\begin{equation}
    s^{2} = − (\frac{1 – 2M}{r}) dt^{2} + (\frac{1 – 2M}{r})^{-1} dr^{2} + r^2 dΩ^{2}
\end{equation}
Here $s^{2}$ is the spacetime metric, M is the mass of the static spherically symmetric object, t is time,  r is the radius, and $dΩ^2$  =  ($dθ^{2}  + sin^{2} θ dϕ^{2}$) as presented by Wald.  This metric is referred to as an exterior metric because it describes spacetime exterior to the massive object.   Later, I will show that TRM can be extended to an interior metric. 
    
\indent After attempts at graphically matching hdf = ((SLOPE) [$f_{i} (r)]^{-1} + 1)$ to (1 – $4.076 \times 10^{38}$/r), it became apparent that a SLOPE estimating expression abbreviated SX helps to speed up the process of comparing these two expressions.  This SX parameter is:


\begin{equation}
  SX = (\frac{− 2M}{r}) (f_{i} (r))                                           
\end{equation}

    The variables M and r are the same as defined previously.   SX will be averaged over a range of radius values for each Ns to determine the most likely value for SLOPE.
    
\indent Please examine the two tables below numbered 1 and 2 to see how comparison of hdf to (1 – $4.076 \times 10^{38}$/r) provides insights into processes occurring over a wide range of radius values. The abbreviation hdf means higher dimensional   f (r);  I am jumping ahead with my definitions by setting  hdf = (SLOPE) [$f_{i} (r)]^{-1} + 1$ , but using an abbreviation reduces the amount of typing so I might as well use an abbreviation that is going to apply later.  The tables which follow will compare hdf to (1 – $4.076 \times 10^{38}$/r) for two numbers of spatial dimensions 9 - Ds and 10 - Ds where M takes on a value corresponding to 2.20 solar masses.

\indent A striking phenomenon seen in these tables is the way  hdf  changes from a large negative value to a large positive value for radius values below 4 Planck lengths.  These changes are labeled inflation triggers, but their effects may not trigger inflation until after the metric flips to its mirror image solution which happens because the spacetime metric is $s^{2}$.

%  Rik copies a table thing from another paper as an example.
\begin{table}[H]
    \centering
    \begin{tabular}{|c|c|c|c|}
        \hline Radius (r) & [$f_9(r)^-1$] & (2.08x10$^{39}$[$f_9(r)^-1$]+1 & 1-4.076x10$^{38}$/r \\ \hline
        10$^{30}$ & − 3.00 (−30) &– 6.240 (9)& – 4.076 (8)\\
        10$^{10}$ &− 2.99 (− 10) & – 6.219 (29) & – 6.219 (29)\\
        10$^{9}$ &− 2.946 (− 9) &– 6.128 (30) & – 4.076 (29)\\ \hline 
        10$^{8}$ & − 2.754 (−8) & – 5.728 (31) & – 4.076 (30) \\ \ 
        10$^{7}$& − 2.154 (−7) & – 4.480 (32) & – 4.076 (31) \\ \
        10$^{6}$ &− 1.193 (− 6)&– 2.481 (33) & – 4.076 (32)\\ \hline 
        10$^{5}$& − 5.094 (− 6) &– 1.060 (34) & – 4.076 (33)\\ 
        10$^{4}$\textbackslash{} & − 2.282 (− 5)&  – 4.747 (34) \tiny{\textbackslash{}(transition radius is} &– 4.076 (34) \\ 
        10$^{3}$/ &− 1.437 (− 4) & – 2.989 (35) \tiny{5776 Planck lengths)} & – 4.076  (35)\\ \hline 
        10$^{2}$\textbackslash{} & − 1.881 (− 3) &–  3.912 (36) \tiny{\textbackslash{}(transition radius is)} &– 4.076 (36)\\ 
        10/ & − 0.1285& –  2.673 (38) \tiny{/ 90 planck lengths} &–         4.076 (37) \\ 
        5  & − 0.8277 & –  1.722 (39) &– 8.152  (37) \\ \hline
        4 & -1.724& -3.586(39) & – 1.019 (38)\\
        3 &  − 6.694 &– 1.392 (40) \tiny{\textbackslash{}Inflation} &  – 1.359 (38)\\
        2 &  + 12.24 & + 2.546 (40) \tiny{/ Trigger} & – 2.038 (38) \\ \hline 
        1.6 &  + 13.17 &+ 2.739 (40)  & – 2.548 (38)\\
        1.4 & + 43.26 & + 8.998 (40)   \tiny {\textbackslash{} Inflation} &– 2.911 (38) \\ 
        1.2 & − 10.59 & – 2.203 (40)   \tiny{/ Trigger} &  – 3.397 (38) \\ \hline
        
    \end{tabular}
    \caption{Radius$(r)$  versus (SLOPE) $[f_9 (r)]^{-1}$ + 1  and (1 – $4.076 \times 10^{38}$/r) at 2.20 solar masses for 9 spatial*  dimensions where $(r)$ is in Planck lengths and $f_9(r)$ is: 
$f_9 (r) = (\frac{− r}{3}) + 2 \ln (r) + (− 2\ln (r^2)) + (\frac{4}{3} \ln (r^3)) + (\frac{− 2}{3} \ln (r^4)) + (\frac{4}{15} \ln (r^5)) + (\frac{− 4}{45} \ln (r^6))$} %\newline

\indent [at 2.20 solar masses the nongeometrized (1 – $(\frac{2M}{r})$) = (1 − $4.076 \times 10^{38}$/r) ] **
    %\label{tab:eventrate}
\end{table}


* SLOPE is an average of the 3.5 largest SX values (3 largest SX averaged then averaged with the average of  
   the  4 largest SX) so Nsx = 3.5.

** Columns 3 and 4 are in nongeometrized units per table F.1 on page 471 of General Relativity by Robert Wald \cite{wald1984the}.
%The value of 2M has been changed from 652,100 to 4.076 x $10^{38}$  for this extended version.

‡ Note that an abbreviated form of scientific notation is used above  where − 3.00 (−30) = − 3.00 x $10^{−30}$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[ht!]
    \centering
    \begin{tabular}{|c|c|c|c|}
        \hline Radius (r) & [$f_{10}(r)^-1$] & (-3.786x10$^{39}$)[$f_{10}(r)^{-1}$]+1 & 1-4.076x10$^{38}$/r \\ \hline
        10$^{30}$ & − 3.00 (−30) &+ 1.136 (10)& – 4.076 (8)\\
        10$^{10}$ &− 3.069 (− 10) &+ 1.162 (30) & – 4.076 (28) \\
        10$^{9}$ &− 3.358 (− 9) &+ 1.271  (31) &– 4.067 (29)\\ \hline 
        
        10$^{8}$ & − 5.546 (− 8) & + 2.100 (32)& – 4.076 (30) \\ \ 
        10$^{7}$& + 3.944 (− 7) & – 1.492  (33) & – 4.076 (31) \\ \
        10$^{6}$ &6.244 (− 7)&– 2.362  (33)  & – 4.076 (32)\\ \hline 
       
        10$^{5}$\textbackslash{}& 2.064 (− 6) &– 7.814 (33)     \tiny{\textbackslash{} [ transition  radius  is 1.32(4)]} & – 4.076 (33)\\ 
        /10$^{4}$/ & 1.011 (− 5)&  – 3.828 (34)  / & – 4.076 (34) \\ 
        \textbackslash{}10$^{3}$ &8. 262 (-5) & – 3.127  (35)    \tiny{\textbackslash{}[ transition  radius  is /260 Planck lengths ]}   & – 4.076  (35)\\ \hline 
       
        10$^{2}$ & 1.704 (− 3)&– 6.451 (36) &– 4.076 (36)\\ 
        10 &+  1.071 & – 4.055 (39) \tiny{\textbackslash{}  Inflation} &– 4.076 (37) \\ 
        5  & − 2.009 &+ 7.606 (39) \tiny{/  Trigger} &– 8.152  (37) \\ \hline
        
        4 & − 3.030& + 1.147 (40) & – 1.019 (38)\\
        3 & − 9.967 &+ 3.774 (40) \tiny{\textbackslash{}Inflation} &  – 1.359 (38)\\
        2 &   + 11.96 & – 4.528 (40) \tiny{/ Trigger} & – 2.038 (38) \\ \hline 
       
        1.6 &  + 13.15 &– 4.978 (40)  & – 2.548 (38)\\
        1.4 & + 43.23 & – 1.637 (41)   \tiny {\textbackslash{} Inflation} &– 2.911 (38) \\ 
        1.2 & − 10.59 & + 4.009 (40)   \tiny{/ Trigger} &  – 3.397 (38) \\ \hline
        
    \end{tabular}
  \caption{Radius r versus (SLOPE) [$f_{10} (r)]^{-1}$ + 1 and (1 – $4.076 \times 10^{38}$/r) at 2.20 solar masses for 10 spatial*dimensions where r is in Planck lengths and f10 (r) is:\\ $ f_{10} (r) = (\frac{− r}{3}) + 2 \ln (r) + (− 2) (\ln (r))^{2} +
(\frac{4}{3}) (\ln{r})^{3} + (\frac{− 2}{3}) (\ln(r))^{4} + (\frac{4}{15}) (\ln (r)^{5} + (\frac{− 4}{45}) (\ln(r))^{6}  + (\frac{8}{315}) (\ln(r))^{7} $ 
[at 2.20 solar masses the nongeometrized (1 – 2M/r)  =  (1 – $4.076 \times 10^{38}$/r)  ]}


\end{table}
   
    
  






* SLOPE is an average of the 3 largest SX values so Nsx = 3.






% Please add the following required packages to your document preamble:
% \usepackage{booktabs}


\subsection{ Coupling hdf to 1-2M/r }

\indent The point of making an effort to couple hdf  to (1 – 4.076 x 10\textsuperscript{38}/r) is to get the higher dimensional versions of the metric component     f (r) to merge with the value of the 3 - Ds metric component over a range of radius values.  For Tables 1 and 2 above the first question that might come to mind is how the SLOPE values were determined for each of these tables.  A SLOPE estimating parameter SX is defined to determine SLOPE for hdf.

\indent To determine what form this SX should have the hdf and (1-2M/r) expressions are closely examined to see where they are linked to energy level.  For the geometrized (1 – 2M/r) the second term – 2M/r is proportional to energy divided by radius because mass is equal to E/c\textsuperscript{2}.  Also, (SLOPE) [f\textsubscript{i} (r)]\textsuperscript{$-$1 } is the part of hdf which corresponds directly to  $-$2M/r  so it is also proportional to energy divided by radius or E/r.  The ratio of these two E/r expressions gives:

\begin{equation}
    \frac{E_{1}/r_{1}}{{E_{2}/r_{2}}} = \frac{−2M/r} { (SLOPE) [f_{i} (r)]^{− 1}}
\end{equation}

\indent From previous examination of f\textsubscript{i} (r) it was shown that R\textsubscript{i} (r) $=$ $-$ 3 f\textsubscript{i} (r) can be considered a higher dimensional radius.  This means that the expression [f\textsubscript{i }(r) ]\textsuperscript{$-$1} is proportional to the reciprocal of a higher dimensional radius so that SLOPE should be proportional to energy.  To estimate the value for SLOPE the left side of expression 2.24 above can be set equal to one and after rearrangement this gives:
\begin{equation}
    SLOPE = \frac{−2M/r} { [f_{i} (r)]^{-1}}                            
\end{equation}       

\indent This shows that when the energy/radius ratio is equal to one the expression 2.25 gives the SLOPE, but this expression generates a large number of SLOPE values because it gives a different SLOPE for each value of radius.  To estimate the best value of SLOPE the SLOPE parameter in expression 2.25 can be replaced by a parameter called SX which will be averaged over a range of radius values to estimate SLOPE.  The expression for SX is:
    
\begin{equation}
    SX = \frac{−2M/r}{ [f_{i}(r)^{−1}]}  = {− (2M/r)}[f_{i}(r)]
\end{equation}
\indent Please notice that this SX expression can be quickly calculated using the values in columns two and four in each table above by using the nongeometrized form of the expression:

\begin{equation}
    SX = \frac{[(1 – 4.076 \times 10^{38}/r) − 1 ]}{[f_{i} (r)]^{−1}}
\end{equation}

\indent Expression 2.27  is  calculated  for  each  value  of radius in the previous tables and the largest SX values are averaged together to estimate SLOPE.  Assuming that the largest SX values are the best estimators for the value of SLOPE is the same as assuming that 3 - Ds and higher spatial dimensions will prefer to couple at the highest shared energy levels where the left side of expression 2.24 is one.  However, it was found that the single highest SX value does not give reasonable boson energy predictions for 5 - Ds.  This may be due to the need to couple the 3 - Ds metric component to hdf over a range of radius so that the coupling is not limited to too narrow a range of radius.  The best number of largest successive SX values to average together (for SX determined for each radius value in the tables above) ranges from 1.1 to five for the spatial dimensions from 5 - Ds to 25 - Ds.  The SX values included in this averaging in all but one case are calculated for successive radius values where each radius is ten times larger than the previous radius.   The one exception to this rule occurred for 5 - Ds because some of its largest SX values occur below 10 Planck lengths.

\indent I will abbreviate the best number of SX values to average together to estimate SLOPE as Nsx.   The table for 4 - Ds is not presented above, but it shows that for 4 - Ds the value of SX remains constant over radius values ranging from 10\textsuperscript{6} to 10\textsuperscript{10} Planck lengths.  This means that 4 or 5 values of SX can be averaged together over this radius range without changing SLOPE.  It was found that for 4 - Ds there are actually an infinite number of transition radius values for which hdf equals (1 – $4.076 \times 10^{38}$/r), but it is not necessary to average more than four largest SX values corresponding to successive radius values in column one to determine the best value for SLOPE for 4 - Ds. A table of Ns (the number of spatial dimensions) versus Nsx will be presented later to show how Nsx varies with Ns; this Nsx data is fit assuming Nsx varies smoothly versus odd Ns and versus even Ns separately which was a trend that reinforced itself as more data was fit.

\indent This answers the question concerning how SLOPE values are determined for tables 1 and 2. The noted areas in column three of these tables will be explained in the next section.

\section{ Transition and inflation trigger radii}

\indent Transition radii and inflation trigger radii are two of the three main characteristics common to most of the 5 - Ds through 25 - Ds data tables showing the coupling hdf to (1 – $4.076 \times 10^{38}$/r).  The third characteristic is whether the SLOPE value is positive or negative.   After  defining  what I mean by a transition radius and an inflation trigger radius, I will present all three characteristics for all spatial dimensions from 5 - Ds to 25 - Ds in tables 3 and 4 below for 2.20 solar masses; one table is for spatial dimensions with positive SLOPE values and the other table is for spatial dimensions with negative SLOPE values. 


\subsection{Transition Radius Defined}

\indent A transition radius is any specific radius value where hdf $=$ (1 – 2M/r) for a given value of nongeometrized M.  In most cases there are two transition radii associated with each Ds.  However, for 4 - Ds and 6 - Ds there are an infinite number of transition radii because hdf $=$ (1 – $4.076 \times 10^{38}$/r) over a wide range of radius values.  This phenomenon seems to occur naturally for 4 and 6 spatial dimensions.  For 4 - Ds and 6 - Ds a specific transition radius value can not be defined because infinitely many are possible.

\indent I have set Nsx to four for 25 – Ds.  The reason I stopped at 25 spatial dimesions is because string theory predicts that it is the highest number of  dimensions possible for bosons is (25 + 1) as shown by Yoichiro Nambu. \cite[p.93]{kaku1995}  This limitation to 26 spacetime dimensions means that the maximum number of spatial dimensions should be limited to 25.  
%This is the only place within TRM where another theory has altered the TRM model.  In the previous paper the value of Nsx was set to one for 25 – Ds , but in this extended version 

\indent I will be referring to transition radii as either r\textsubscript{LT }or r\textsubscript{ST} where the former is the symbol for larger transition radius and the latter is for the smaller transition radius.  Both of these are equivalent to the transition radius  r\textsubscript{T} ; I  have  altered these subscripts merely to keep track of two transition radii for each spatial dimension.

\subsection{Inflation Transition Radius Defined}

\indent An inflation trigger radius is a radius value where the value of hdf  goes from a large negative value to a large positive value with a relatively small decrease in the  radius value.  Inflation triggers usually occur for radius values less than 4 Planck lengths.  However, exceptions to this rule are seen for 4 - Ds and 6 - Ds, as well as, for the even numbered spatial dimensions which are greater than 8 - Ds.

\indent I have not determined if the smaller inflation triggers cause the value of the spacetime metric to change sign or if the bubble shrinks to a radius of one Planck length before the spacetime metric changes sign as the bubble grows.  Other theories suggest than the singularity will go below a radius of one Planck length, but TRM seems to be suggesting that the radius should go no lower than one Planck length.

\indent Resolving at what bubble radius collapse should reverse to expansion is beyond the scope of this paper so I will merely comment on the number of inflation trigger radius values predicted by TRM and speculate on the number of stages of inflation this implies


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]

\centering
\begin{tabular}{|c|c|c|c|c|c|} 

\hline %\textbf{Ns}&  \textbf{rLT}& \textbf{rST}  &   \textbf{SLOPE}%
&&&& \textbf{Larger} &\textbf{Smaller}\\ &&&& \textbf{Inflation}& \textbf{Inflation}\\\textbf{Ns}&  \textbf{rLT}& \textbf{rST}  &   \textbf{SLOPE}& \textbf{Trigger} & \textbf{Trigger}\\ &&&& \textbf{Radius}& \textbf{Radius} \\&&&& \textbf{Range} & \textbf{Range}\\ \hline

     4 &-    &-    &  1.359 (38)   & 10 – 100   & -\\
     5 &68.5 &5.06 &   2.979 (38)  &  -          &- \\
     6 & -& -&   1.359 (38)        & 103 − 104   & - \\ \hline
     
   
    
    
   7& 1075& 13.3& 5.87 (38)& 2 – 3 &1.2 – 1.4\\
    9&5780 &90.1& 1.943 (39) &2 – 3 & 1.2 – 1.4 \\
    11  &    5.50(4) &500 &6.999 (39)  &2 – 3 &1.2 – 1.4 \\ \hline 
     
    
    13 &  5.26 (5) &2.73 (3)& 2.6762 (40)& 2 – 3& 1.2 – 1.4\\
    15& 2.12 (6) &3.15 (4)& 1.0924 (38) & 2 – 3& 1.2 – 1.4\\
    17 &6.20 (6)& 4.97 (5) &4.352 (41) & 2 – 3& 1.2 – 1.4\\ \hline
   
    19 &1.78 (8) &1.132 (6) &1.444 (42) & 2 – 3& 1.2 – 1.4\\
    21 &1.42 (9) &7.65 (6) &5.508 (42)& 2 – 3& 1.2 – 1.4\\ 
    23 &5.00 (9) &1.078 (8) &2.112 (43)  & 2 – 3& 1.2 – 1.4\\ \hline 
   
     25& 5.00 (9)	&−	&8.27 (43)&	2 – 37& 	1.2 – 1.4 \\ \hline 
     
 
          



\end{tabular}
\caption{Ns for Negative SLOPE versus Transition Radii, SLOPE, and Inflation Trigger Radius Ranges*}
\end{table}
* The large and small transition radii, as well as, the inflation trigger radii are in units of Planck lengths.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|} 

\hline
&&&& \textbf{Largest}& \textbf{Medium} &\textbf{Smaller}\\ &&&& \textbf{Inflation}& \textbf{Inflation}& \textbf{Inflation}\\\textbf{Ns}&  \textbf{rLT}& \textbf{rST}  &   \textbf{SLOPE}& \textbf{Trigger}& \textbf{Trigger} & \textbf{Trigger}\\ &&&& \textbf{Radius}&\textbf{Radius}& \textbf{Radius} \\&&&& \textbf{Range} & \textbf{Range} & \textbf{Range}\\ \hline


        8 &2260& 33.5 &– 8.49 (38)&  −&-&1.2 – 1.4\\
       10 &1.31 (4)& 259&– 3.673 (39)&5 – 10& 2 – 3&1.2 – 1.4\\
       12 &8.49 (4)&  1980&– 1.486 (40)& 10 – 100&2 – 3&1.2 – 1.4 \\ \hline 
 
       14 &4.24 (5) &1.93 (4) &– 5.923 (40) & 10 – 100 & 2 – 3 &1.2 – 1.4 \\
       16 & 3.40 (6)&1.30 (5) & – 2.396 (41) & 10 – 100 & 2 – 3 &1.2 – 1.4 \\
       18 &3.30 (7) &7.42 (5) &– 8.70 (41) & 10 – 100 & 2 – 3 &1.2 – 1.4 \\ \hline
     
       20 &4.85 (8) &3.03 (6) &– 2.94 (42) & 100 − 103 &2 – 3 &1.2 – 1.4  \\
       22 &3.30 (9) &2.29 (7) &– 1.10 (43) &  100 − 103&2 – 3 &1.2 – 1.4  \\
       24 &7.00 (9) &5.00 (8) &– 3.932 (43) & 100 − 103 &2 – 3 &1.2 – 1.4  \\ \hline
       
     


\end{tabular}
\caption{Ns for Negative SLOPE versus Transition Radii, SLOPE, and Inflation Trigger Radius Ranges*}
\end{table}
* The large and small transition radii, as well as, the inflation trigger radii are in units of Planck lengths.  For
    both tables these results are for a static spherically symmetric object of 2.20 solar masses.   




\subsection{Initial Observations}

\indent Tables 3 and 4 contain all of the key characteristics of the tables for the coupling of   hdf  to (1 – $4.076 \times 10^{38}$/r).  Showing all of these coupling tables would have been too tedious so I’ve summarized their features in the two tables above.  It is advantageous to split these tables so all positive SLOPE values are in one table and negative SLOPE values are in the other.  This split presentation emphasizes that all the spatial dimensions having positive SLOPE values have no more than two inflation trigger radius ranges, and almost all spatial dimensions with negative SLOPE have three inflation trigger ranges.  This strongly suggests that there is a significant different between spatial dimensions having positive SLOPE values versus those with negative SLOPE values.  In section 2.2 it made sense to think of SLOPE as proportional to energy.  If this idea is carried over to tables 3 and 4 it implies that spatial dimensions having a positive SLOPE have positive energy and spatial dimensions having a negative SLOPE have negative energy.  This assumption will affect the energy values predicted for bosons later.

\indent Skimming through the two columns of transition radius values in the above two tables, I came across one transition radius which stood out from the rest.  In table 3 the larger transition radius for 7 - Ds is  1130 Planck lengths.  A         bubble having a radius of   1130      Planck lengths may interact in such a way with this  transition radius that it is momentarily restricted to this size.   As Carmen Molina-París and Matt Visser pointed out in their paper, the energy density at a bounce suggests that the bubble should rebound from a radius of about 1000 Planck lengths \cite{molinaparis1999}.  This finding suggests the larger  transition radius for  7 - Ds  has some   influence  over  the size  of  the   bubble  which  inflates  to  become  our   universe.  If this speculation is correct, it may be that the initial formation of large-scale structure is influenced  by  the  larger  transition  radius  for  7 - Ds;  this influence may be due to this transition radius causing a momentary slowing of the collapse so the bubble pauses near a radius of 1130 Planck lengths.  

%\indent Variation within the previous TRM model indicates that the true value for the larger transition radius at 7 - Ds most likely falls within a range of 1130   $\pm$ 75  Planck lengths ($\pm$ one sigma).  This 1130 Planck length transition radius value is 5.1 percent larger than the 1075 value found in the previous paper.  For this range I would favor the lower half since that half of the range would be closer to the radius estimated by Molina-París and Visser.  This assumption favors having the true larger transition radius for 7 - Ds fall between 1000 and 1130 Planck lengths.  If I were asked to make my best guess at the range of the bubble radius when large-scale structure was first influenced I would pick a radius ranging from 1065 to 1130 Planck lengths. For now, I will assume that the 1130 Planck length bubble radius applies for extended TRM, and I will present 1130 Planck lengths in the data tables that follow.

\indent Inflation trigger radii are the last bit of characteristic data on which to elaborate.  I will not determine how small a collapsing object gets when it obeys the restrictions imposed by this TRM model because such an analysis is beyond the scope of this paper.  However, I will speculate that it may be preferable to have the bubble collapse to the 1.2 – 1.4 Planck length radius range so that the smallest radius inflation triggers associated with the negative energy spatial dimensions (in table 4) get a chance to participate in the initiation of inflation.  This  may be  preferred  because  a very high negative energy density  could  be  generated inside the bubble if the most energetic spatial dimension reached during the collapse is a negative energy spatial dimension, and an energy density dominated by a negative energy density is needed to get the bubble to begin expanding as Alan Guth pointed out in \textit{The Inflationary Universe}. \cite[p.266]{guth1997the}  Analysis of this with other models  would be needed to be more confident about this speculation.  However, if the bubble were to begin expanding from a radius range of 1.2 – 1.4 Planck lengths it would follow that there should be at least two dominant stages of inflation.  One stage of inflation would be dominated by the inflation trigger event at a 1.2 – 1.4 Planck length radius range, and the second stage of inflation would be dominated by  the  set  of  inflation  triggers  at a 2 – 3 Planck length radius range. 

\indent A third inflation event which occurs over a much wider range of radius values (associated with the largest inflation trigger radius range in table 4) may also contribute to initiating inflation.  Depending on how small the bubble gets before inflation is initiated, there may be as many as three dominant inflation stages, but one or two stages seems most likely.  The inflation trigger for 4 - Ds occurs in the same radius range as that for the third inflation event above so its minor effect should merge with the third inflation event.  TRM can predict the precise transition radius value where these inflation triggers initiate by looking for a sign change in hdf.

\indent Only the inflation trigger associated with the 6 - Ds stands alone at a radius range which does not match any of the others.   This  6 - Ds inflation trigger may momentarily slow the collapse of the bubble long enough for the 1130 Planck length transition radius associated with the 7 - Ds to influence large-scale structure.

\section{Nsx distribution for SX}

\indent Previously, in section 2.2, it was shown that the best SLOPE value should be determined by averaging several of the largest SX values.  The number of SX values that need to be averaged to get the best estimate for SLOPE is called Nsx.  A value of Nsx $=$ 4 worked best for 4 - Ds, 5 - Ds, and 6 - Ds.  However, the most reasonable way for TRM to get close to predicting a specific boson energy makes it necessary for Nsx to ramp down from Nsx $=$ 4  to Nsx $=$ 1.13 at 16 - Ds.   Furthermore, it is advantageous for Nsx values to climb  up  toward Nsx $=$ 4 again after 16 - Ds  to fit a specific boson energy at 22 - Ds.

\indent These restrictions on Nsx are \textit{ad hoc}, but there may be a theoretical explanation for the Nsx distribution below.  The following Nsx distribution in table 5 for the spatial dimensions from 4 - Ds up to 25 - Ds is assumed for all initial SLOPE values determined by TRM.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\centering

\begin{tabular}{|c|c|c|c|}
\hline
  \textbf{Even} &\textbf{Nsx for} & \textbf{Odd } & \textbf{Nsx for}  \\ \textbf{ Ns} & \textbf{Even Ns} & \textbf{Ns}	& \textbf{ Odd Ns}\\ \hline
      4 & 4 &5 &4 \\
      6 & 4 &7 & 4\\
      8	 & 3.5&9 &3.875 \\ \hline            
    
     10 &3.25 &11 & 3.75 \\
     12 &3 &13 &3.668 \\
     14	&1.969 &15 &2.758  \\ \hline 
     
     16 & 1.13& 17 & 2.27 \\
     18 & 1.781& 19 & 4.024 \\
     20 & 3.49 &21  & 4.032 \\ \hline
   
     
     22 & 4&23 &4 \\
     24 &4 &25 &4 \\
     26 &4 & 27& 4\\ \hline 
     
  
             
          

  


\end{tabular}
\caption{Even Ns and Odd Ns versus Nsx – when 2M = $4.076 \times 10^{38}$}% for extended TRM}
\end{table}
*  See Appendix A for more details


\indent Seeing Nsx = 3.75 in table 5 may be a bit puzzling so I will describe what such a value means when averaging SX values for 7 - Ds.  When we average values to determine a mean value we always think of averaging a number of values which is a whole number.  When we are asked to average 3.75 values this does not make sense unless we regard it as a shorthand for a more complex averaging technique.  In this case, to average 3.75 SX values means to combine averages for whole numbers of SX values.  First, find the Nsx = 3 and Nsx = 4 averaged SX values which are SLOPE$_3$ = (SX1 + SX2 + SX3)/3 and SLOPE$_{4}$ = (SX1 + SX2 + SX3 + SX4)/4, respectively.   Next, to find the SLOPE3.5 , which is a 3.5 SX average, we just average the two previous slopes so that SLOPE${_3.5}$ = (SLOPE$_3$ + SLOPE$_4$)/2.  Now averaging 3.75 SX values can be defined as averaging SLOPE$_{3.5}$ with SLOPE$_4$ or SLOPE$_{3.75}$ = (SLOPE$_3.5$ + SLOPE$_4$)/2 which completes the definition for an average of 3.75 SX values.  Those who have a problem with this fractional averaging technique may prefer to think of Nsx = 3.75 as a shorthand way of indicating the complex averaging technique defined above.

\indent The values of Nsx in table 5  are likely to be very close to their true values.  If some other theory is developed to explain the smooth shape of this Nsx distribution, fitting the  Nsx values listed in table 5 would be an indication that such a theory is on the right track.


\section{Boson energy from transition radius}

\indent To resolve the part of the hierarchy problem involving fundamental boson energies a model must give accurate boson energy $E_B$ predictions over a wide range of boson energies.  Immediately after finding the transition radius values listed in tables 3 and 4, I wondered if the volumes associated with these transition radius values are connected to the energy of some types of particles.  I started to think about these transition radius volumes which are (4/3) $\pi$ (r\textsubscript{T}) where r\textsubscript{T} is the transition radius defined above which can be either r\textsubscript{LT} or r\textsubscript{ST} as defined previously.  While thinking of these transition radius volumes and Planck energy I found a simple equation for boson energy. 

% Editors note See for example \cite{Chung2016,Olivi-Tran20}.  
% These papers are what come up in a name-brand search engine with keywords like dimensions boson mass.   They were all published after this work, except for Chung's early effort on arXiv.  They are authors and works similar to Holte in many ways.  This note is hiding in the LaTeX, not part of the finished pdf.
% @article{Chung2016,
%    title = "{The Accurate Mass Formulas for Leptons, Quarks, Gauge Bosons, the Higgs Boson, and Cosmic Rays}",
%    year = "2016",
%    journal = "J. Mod. Phys",
%    volume = "7",
%    number = "12",
%    doi = "10.4236/jmp.2016.712144",
%    eprint = "arXiv:hep-ph/0003237"
%}   

%@incollection{Olivi-Tran20,
%    author = {Nathalie Olivi-Tran},
%    title = {Theoretical Calculations of the Masses of the Elementary Fermions},
%    booktitle = {Accelerators and Colliders},
%    publisher = {IntechOpen},
%    address = {Rijeka},
%    year = {2020},
%    editor = {Ozan Artun},
%    chapter = {5},
%    doi = {10.5772/intechopen.91846},
%    url = {https://doi.org/10.5772/intechopen.91846}
%}
%  Also Dao Vong Duc, Nguyen Mong Giao, Vietnam
% doi 10.4236/jmp.2013.47133

\indent  As the radius of a boson increases from 1.0 Planck lengths to some transition radius value the energy of larger bosons should decrease proportional to the volume ratio given by the expression VR $=$ (4/3) $\pi$ (1.0)\textsuperscript{3} /[(4/3) $\pi$ (r\textsubscript{T})\textsuperscript{3}] or simplifying VR $=$ 1 / (r\textsubscript{T}).  The maximum energy possible in the smallest region of space is the Planck energy which is Planck energy $=$ 1.22 x 10 GeV. \textsuperscript{ }  The energy that a boson volume can hold should decline from Planck energy where a boson with a one Planck length radius is at Planck energy, and this boson energy should decline in a way such that boson energy is proportional to VR. I assume this rule holds from one transition radius boson type to the next.   Therefore, the energy that a gauge boson volume can hold decreases as the radius increases where the boson’s energy can be found by multiplying the Planck energy by VR, giving:

\begin{equation*}
  E_{T} = ± (1.22 x 10^{19} GeV)VR \rightarrow E_{B} = E_{T}  = ± \frac{1.22 x 10^{19}  \text{GeV}} {(r_{T})}^{3}
\end{equation*}


\begin{equation}
  E_{B} = \frac{E_{P}}{((r_t + M)^3 - A) }
\end{equation}
% \raggedleft (5.28')

for $A \neq 0, A \neq 1, r_{t} \ge 1$ for $r_{t} > 20$ and $A = -1$ only

%\text{for} r_{t} \gt 20 \text{and} A = -1 \text{only})$

\begin{equation*}
    \text{or an alternate form}\;    E_{B} = \frac{E_{P}}{(r_{T})^{3}} 
\end{equation*}                                        
%\doublespacing(5.28'') 

% remove pm sign, per authors notes  E_{B} = \pm \frac{E_{P}}{(r_{T})^{3}}   

\indent where $E_{B}$  is energy of a boson at this transition radius, $E_{T}$  is the energy corresponding to transition radius $r_{T}$ (where $r_{T}$ is in Planck lengths),  and $E_{P}$  is the Planck energy.                                    

\indent When a particle/anti-particle pair annihilates it may produce more than one boson; the E\textsubscript{B} value above is the total energy of the rest mass particle/anti-particle pair. Expressions 5.28 which defines transition energy will be used to predict the energy of bosons which have radius values equal to some transition radius.  I have assumed that these  boson energy values (E\textsubscript{B}) can be found from E\textsubscript{B} $=$ E\textsubscript{T} so expression 5.28 gives a boson energy when a transition radius is entered for r\textsubscript{T}.  I will assume that all of the particle/anti-particle annihilation energy goes into a single boson of energy E\textsubscript{B}.   This links a set of fundamental boson energies to transition radii derived from the transition radius method, and TRM is derived from general relativity so it follows that a set of specific annihilation boson energies is linked to general relativity. 

\indent Previous methods for finding particle energy have been based on wavelength.   One of these methods, called ``Special-Scale Relativity" by L. Nottale uses an expression based on wavelength and the Planck scale plus several other factors to predict particle energy.  \cite{nottale1997} Such an expression is a very good way to make a quick estimate of the energy of a particle or boson, but such methods  were not derived directly from general relativity.

\indent Alternatively, TRM uses expression 5.28 to predict annihilation boson energy from transition radius values derived from general relativity.  This makes TRM a more basic model because it predicts annihilation boson energies without using particle mass or wavelength.

\indent Expression 5.28 will be used to predict boson energies for the transition radius values  for both versions of TRM. Tables 6 and 7 below show the predicted boson energies for
%extended
TRM.  These predicted boson energies are labeled E\textsubscript{LT} and E\textsubscript{ST} which correspond to the transition radii r\textsubscript{LT} and r\textsubscript{ST}, respectively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering

\arrayrulecolor{black}
\begin{tabular}{|c|c|c|c|c|}

     \hline Ns&   rLT&	ELT ‡& rST &EST \\
    &&(Boson Energies)&&(Boson Energies)   \\ \hline         

       4 &− & −&− &  − \\ 
       5 &69 &+ (-)  3.71 (13) &5.05 & + (-)  9.47 (16) \\
       6 &− &− & − &− \\\hline
       
           
    	               
           
       7&1128  * &+ (-)  8.50 (9) &   13.0&+ (-)  5.55 (15) \\
       8&2265 &– (+)  1.05 (9) &33.4 &− (+)  3.27 (14) \\
       9&7456 & + (-)  2.94 (7) &76.8 &+ (-)  2.69 (13) \\\hline	      
       
      10&1.494 (4) &− (+)  3.66 (6) &237.6 &− (+)  9.10 (11) \\
      11&5.53 (4) & + (-)  7.21 (4)&499.5 &+ (-)  9.79 (10) \\
      12&8.54 (4) &− (+)  1.96 (4) &1969 &− (+)  1.60 (9)  \\\hline
      
      
      13&3.699 (5) &+ (-)  241  &3667 & + (-)  2.47 (8)\\
      14&4.237 (5) &− (+)  160.4  &1.927 (4) &− (+)  1.705 (6) \\
      15&1.696 (6) &+ (-)  2.50 & 3.76 (4) & + (-)  2.295 (5) \\\hline
      
       
      
      16&1.21 (6) &− (+)  6.89 &3.264 (5) &  − (+)  351\\
      17&7.77 (6) &+ (-)  0.026  & 4.059 (5) & + (-)  182.4 \\
      18& 1.48 (7)&− (+)  3.76 (-3) &  1.502 (6) &− (+)  3.60 \\\hline
      
      
      19&1.781 (8) &+ (-)  2.39 (-6) &1.132 (6) &  + (-)  8.41\\
      20&3.63 (8) &− (+)  2.55 (-7) & 3.861 (6) & − (+)  0.212 \\
      21& 1.49 (9)&+ (-)  4.26 (-9)  &7.66 (6) &+ (-)  0.0271 \\\hline
      
      
     
      22&3.39 (9)  &− (+)  3.13 (-10) &2.287 (7) &− (+)  0.00102  \\
      23&1.113 (10) &+ (-)  8.85 (-12) &5.26 (7) &  + (-)  8.38 (-5)\\
      24& 4.72 (10) &− (+)  1.16 (-13) &9.75 (7) &− (+)  1.32 (-5) \\\hline
      
      25 & 7.462 (10)	&+ (-)  2.9363 (-14)& 4.137 (8)     &+ (-)  1.723 (-7)\\\hline               



\end{tabular}
\caption{Ns versus $r_{LT}, ELT, r_{ST}$, and EST for 2.20 solar masses where $r_{LT}$  and  $r_{ST}$  are in Planck lengths and ELT  and EST  are in GeV energy units (for TRM using table 5 when 2M = $4.076 \times 10^{38}$).}
\end{table}
‡  The possibility of negative energies are due to $E^2 = p^2 c^2 + m^2 c^4$  and to sign changes in SLOPE values.
* The radius at Ns = 7 per TRM when Nsx = 4  for 2.2 solar masses  is 1128 Planck lengths.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering
\caption{Ns versus Boson Type, TRM Predicted Boson Energy, and Measured or Estimated Boson Energy  where Boson Energies are in GeV}
\begin{tabular}{|c|c|c|c|} 
\hline &&TRM & Measured \\
Ns & Boson Type & Predicted Boson & or Estimated \\ 
&& Energy (GeV) & Boson Energy (GeV)\\\hline 

 
          5 & X like boson  &+ (-)  3.71 (13) ,  9.47 (16) &$10^{14}$ to $10^{15}$   * \\ 
          7 & X like boson & + (-)  5.55 (15) & $10^{14}$ to $10^{15}$   * \\ 
         13	& W boson &241  (triple) &80.2        *\\ \hline

         
         14 &W boson &– (+)  160.4    (double) &80.2        * \\
         15 &Charm/anti-charm & + (-)   2.50 & 2.50\\
         17 &Z boson &+ (-)   182.4    (double) &91.2          * \\
         18  &Up/anti-up quarks &– (+)  3.76 (-3) &2 (-3)  to  10 (-3) \\ \hline 

         
         
         & e-neutrino/anti-νe& & \\
        21 &annihilation boso &+ (-)  4.26 (-9) &<  6.0 (−9) \\
         &e− e+ annihilation & & \\
        22	&negative energy photon  &– (+)  0.00102 &+ 0.00102 \\ \hline
        
  


\end{tabular}
\end{table}
* From a paper titled \textit{Elementary Particles and Cosmology} by I L Rozental’, Russian Academy of Sciences, Uspekhi Fizicheskikh Nauk, 1997.\cite{ILrozental'1997} 

\indent
The boson energies in table 7 show that TRM gives a reasonable approximation for at least seven specific boson types; very little SLOPE adjustment is needed to fit boson energies. %because the previous version of TRM made it possible to select better intial Nsx estimates for extended TRM.  With little adjustment extended TRM  predicts the boson energies associated with all six types of quark/anti-quark annihilation bosons. 


Getting reasonably close to predicting boson energies for so many different boson types for bosons over a wide range of energies is a good sign that 
%extendedTRM and previous TRM are both 
TRM is able to fit fundamental particle energies so that the pattern of particle energies is explained by the smooth change of the single parameter Nsx over even numbered spatial dimensions  and by a smooth change of Nsx over odd numbered spatial dimensions.

Table 8 below shows that most New SLOPE values are 6.25 x 10\textsuperscript{32}  times larger than the Previous SLOPE values, and this ratio comes from the reciprocal of 1.6 x 10\textsuperscript{-33} centimeters per Planck length which was expected.

\indent To make it easier to duplicate this work I have included an Appendix B where I have listed a BASIC language computer program used to calculate the f\textsubscript{i} (r) values.   Computerizing the entire TRM process will speed E\textsubscript{B} estimation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|} 

\hline 
Ns & New *& New SLOPE & &&& \\ 
& SLOPE & Divided by & Ext *** & Ext *** & Ext ***& Ext *** \\
&&     Previous& $r_{LT}$ & $E_{ST}$ & $r_{ST}$ & $E_{ST}$ \\
&&SLOPE ** && (Gev) & & (Gev) \\ \hline 


      7 &5.87 (38) &6.10 (32)  &1128  ‡ &8.50 (9) &13.0  &5.55 (15)  \\
     13 & 2.6762 (40)&NA    &3.699 (5) & 241  &3776 &2.47 (8) \\
     15 &1.0924 (41) &6.25 (32) & 1.696 (6) &2.50 &3.76  (4) &2.295 (5) \\ \hline 
	


     16&– 2.396 (41) &6.25 (32)  &1.21 (6) &6.89 &3.264 (5) & 351 \\
     17&4.352 (41) &NA &7.77 (6) &0.0260 &4.059 (5) &182.4 \\
     18&–  8.70 (41)  & 6.25 (32) &1.48 (7) & 3.76 (-3) &1.502 (6) & 3.60\\ \hline
	

    

     19&1.444 (42) &6.25 (32)  &1.781 (8) &2.39 (-6) &1.132 (6) & 8.41 \\
    20   &– 2.94 (42)  &6.25 (32)  &3.63 (8) &2.55 (-7) &3.861 (6) &0.212 \\ \hline  






\end{tabular}
\caption{.  Ns versus New SLOPE, New SLOPE/Previous  SLOPE, $r_{LT}$, $E_{LT}$, $r_{ST}$, and $E_{ST}$}
\end{table}
*  New SLOPE means the result obtained when the value of 2M used is $4.076 \times 10^{38}$  Planck lengths.

**  Previous SLOPE means the result obtained when the value of 2M used was 652,100  centimeters.

%***  Ext  refers to extended TRM for which 2M = 4.076 x $10^{38}$  Planck lengths. 

‡  This transition radius is for Nsx = 4   for Ns = 7  at 2.2 solar masses when 2M = $4.076 \times 10^{38}$.

NA = not applicable due to change from single particle energy to multiple particles.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering

\begin{tabular}{|c|c|c|c|} 


\hline 
         Ns	& Boson & Extended  TRM & Measured or \\
         & Type &  Predicted & Estimated  \\
         &&   Boson Energy& Boson Energy \\    
         && (GeV) & (GeV) \\ \hline 
         
         
          
       5 and 7 &X like boson &  3.71 (13) , 9.47 (16) , 5.55 (15) & $10^{14}$ to $10^{15}$   * \\
       13 and 14 &W boson & + (-)  \textbf{241} (triple)  and \textbf{160.4} &80.2        * \\
        15	&Charm/anti-charm boson &+ (-)  2.50 &  2.50         ** \\ \hline 

                  

         16 &top/anti-top quark boson &– (+)  351 &+ 349          ** \\
         17 &Z boson &+ (-)  \textbf{182.4}     (double) &182.4         * \\
         17 &tau-neutrino/anti-t-neutr. &0.026 &<  0.036       ** \\
         18	& Up/anti-up quark &– (+) 3.76 (−3) & 2 (−3)  to 10 (−3)  **\\
         &negative energy boson & & \\ \hline 
         

    
         18 &tau/anti-tau negative &– (+) 3.60 &+ 3.60              ** \\
            &energy boson & & \\
         19 &μ-neutrino/anti-μ- &+ (-)  3.76 (– 3)  &<  3.8 (−3)        ** \\
            & neutrino boson & & \\
         19	&bottom/anti-bottom &+ (-)   8.41  &8.40               ** \\
            &quark  annihilation boson & & \\ \hline 
            
         20 &    muon/anti-muon &– (+)  0.212  &+ 0.212              ** \\
            &negative energy boson & & \\ 
            &or strange/anti-strange & & \\
            &negative energy boson & & \\ 
         21 &e-neutrino/anti-νe & + (-)  4.26 (– 9) &<  6.0 (−9)        ** \\
            &annihilation boson & & \\
         22 &e− e+ annihilation &– (+)  0.00102  &+ 0.00102 \\
         &negative energy  photon & & \\ \hline 



\end{tabular}

\caption{Ns versus Boson Type, TRM Predicted Boson Energy, and Measured or Estimated Boson Energy  where Boson Energies are in GeV} 
\end{table}

* From a paper titled \textit{Elementary Particles and Cosmology} by I L Rozental’, Russian Academy of Sciences, 
   Uspekhi Fizicheskikh Nauk, 1997. \cite{ILrozental'1997} 
** Modified from a paper titled Report of The Working Group on The Future of Accelerator-Based Particle
     Physics in Europe, ECFA/01/213, 13 September 2001.\cite{L.foa2001}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 10 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\centering

\begin{tabular}{|c|c|c|c|} 

\hline 
     Even Ns &	New SLOPE for Even Ns &	   Odd Ns &  New SLOPE for Odd Ns \\ \hline 
     
          4 &1.359 (38) &5 &2.979 (38) \\
          6&1.359 (38) &7 & 5.87 (38)\\
          8  &–  8.490 (38) &9  &1.943 (39)  \\  \hline 
  
                 
         10 &– 3.673 (39)  &11 & 6.999 (39) \\
         12 &– 1.486 (40) &13 &2.6762 (40) \\
         14	&– 5.923 (40) &15 &1.0924 (41)  \\            
         
                 
         16 &– 2.369 (41) &17 & 4.352 (41) \\
         18 &– 8.70 (41) &19 &1.444 (42) \\
         20	&– 2.94 (42) &21 &5.508 (42)  \\ \hline             
         
     
       
         22 &– 1.10 (43) &23 &2.112 (43) \\
         24 & – 3.932 (43) &25 &8.270 (43)  \\      \hline    	

             

\end{tabular}

\caption{Even Ns vs. New SLOPE and Odd Ns vs. New SLOPE   (2M = $4.076 \times 10^{38}$ at 2.2 solar masses) }
\end{table}

\indent Tables 10 and 11 show the best values of SLOPE, r\textsubscript{LT }, and r\textsubscript{ST} for TRM when it is solved assuming a static spherically symmetric object of 2.20 solar masses.  
%I will use the data in these two tables later (section 7) to complete development of extended TRM.

\indent Table 9 shows the best TRM predicted boson energies in comparison to measured or estimated boson energies.  The data in this table is for TRM when solved for an object of 2.20 solar masses.  It is puzzling that R\textsubscript{22} which is associated with an exterior metric should give a model which fits boson energies so well; I will explain why in  section 9.

\indent TRM can fit exact boson energies for at least thirteen boson types (as shown in table 9).  These include the three bosons X, W, and Z boson. Also included are the quark/anti-quark annihilation positive energy bosons for quark types charm, bottom, and top, and the quark/anti-quark annihilation boson energies are nearly identical for the up and down quarks so both are predicted from one value.  Two additional bosons which both have negative energy results for TRM are the muon/anti-muon annihilation boson and the strange/anti-strange annihilation boson. Finally, the electron/positron annihilation boson, and the tau/anti-tau annihilation boson can be fit with no SLOPE value adjustment.   The  boson  energy  predicted  by TRM for the $\mu$-neutrino/anti-$\mu$-neutrino annihilation boson is close to  its  upper  limit.

\indent Thus, TRM can fit the boson energies of all six quark/anti-quark annihilation bosons plus boson energies associated with all six lepton/anti-lepton bosons, plus the W, Z , and X.

\indent There may be other bosons which can be fit by TRM. For example, the electron/positron annihilation energy which fits at Ns $=$ 22 when Nsx $=$ 4 can also be fit at Ns $=$ 17 for Nsx $=$ 3.978.  Many of the TRM predicted boson energies above remain unmatched to known particles; these new particles may be found in the future. Additional boson matches may be possible, but I will move on to other concepts.



\section{A Boson of infinite dimensions}

\indent There is another boson not listed in the above tables which can be predicted by expanding expression 2.21 out to an infinity number of spatial dimensions.  Expression 2.21 contains a summation expression which can be summed over an infinite number of terms.  Since each term added to this summation describes the next spatial dimension, expanding the summation to an   infinite   number   of   terms   is  the   same  as examining an infinite number of spatial dimensions.

\indent To find this boson energy for an infinite number of dimensions we start with expression 2.21 from section 2.1 which is:
\begin{equation*}
f_{i} (r) = (-\frac{r}{3}) + \sum_{n=1}^{k} [ (\frac{ 2 (− 2)^{n – 1}} {n!} ) (\ln( r))^n]   
\end{equation*}
  
\text{Focusing on the summation portion only we have:}

\begin{equation}   
\text{Summation} = \sum_{n=1}^{k} [ \frac{( 2 (− 2)^{n – 1}} {n!} ) (\ln(r))^{n} ]
\end{equation}
Letting $x=\ln(r)$ and simplifying gives:
\begin{equation}
 \text{Summation} = -(\frac{1}{2}) (2) \sum [(−1)^{n} (2)^{n} (\frac{x^{n}}{n!})]    
\end{equation}
or 
\begin{equation*}
    \text{Summation} = −\sum_{n=1}^{k} [(−1)^{n} (2)^{n} \frac{x^{n}}{n!}]
\end{equation*}
when k goes to infinity this summation goes to:
\begin{equation}
    \text{Summation} = − (e^{−2x − 1}) = (1 − e^{−2x}) \; \; \;  (\mathrm{for} k=\infty)         
\end{equation}
%($k=\infty$)

If x is replaced by ln(r) this becomes:

\begin{equation}
    \text{Summation (k = $\infty$) }= (1 − r^{−2}) 
\end{equation}

Substituting this into expression 2.21 gives: 

\begin{equation}
    f_{i} (r) = (\frac{− r}{3}) + (1 – r^{−2})  \; \text{for i = $\infty$}     
\end{equation}


    As I pointed out on in the second paragraph after expression 2.21 in section 2.1 when f\textsubscript{i} (r) is multiplied by a factor of $-$3 the result is an expression for the higher dimensional radius which is R\textsubscript{i} (r) $=$ $-$3 f\textsubscript{i }(r)  or:                                         
\begin{equation}
    R_{i} (r) = r − 3 \sum_{n=1}^{k} [ (\frac{ 2 (− 2)^{n – 1} } {n!} ) (\ln(r))^{n} ]    
\end{equation}
An expression for the metric component f (r), which will be derived in a later section, has the form:
\begin{equation}
    f (r) =  (1 – \frac{KM}{R_{i}(r)})                                       
\end{equation}
where K is a constant dependent on i.  Expression 6.35 becomes singular when R\textsubscript{i} (r) $=$ 0 which occurs when f\textsubscript{i} (r) $=$ 0.  This shows that finding a value for r where expression 6.33  goes to zero will represent a minimum higher dimensional radius for TRM.  Solving f\textsubscript{i} (r) $=$ 0 gives: 
\begin{equation}
    r = 3 (1 − r^{−2})
\end{equation}

Solving 6.36 for r gives r = 2.532089 Planck lengths.  When this value of r is substituted into expression 5.28 to solve for energy we get:
\begin{equation}
    E_{T} = ± 1.22 x \frac{10^{19} GeV}{(2.532089)^{3}} \rightarrow  E_{B} = E_{T} = ± 7.515 x 10^{17} \text{GeV}             
\end{equation}


\indent This 7.515 x 10$^{17}$ GeV energy represents the energy \cite{rees2000just} for a higher dimensional radius R\textsubscript{i} (r) equal to zero for an infinite number of spatial dimensions.  Whether this energy  corresponds to a unification has yet to be determined, but this value is at an energy which makes it look like a reasonable unification candidate. 

%\\ \indent his new boson with a transition radius of 2.532089 Planck lengths and a transition energy of 7.515 x 10\textsuperscript{17} GeV should have a name which describes its nature.  My first thought for a name is ‘infiniton’, but this name sounds too much like inflaton.  A variation on this name which may be preferred is ‘infinitron’.  The first part of this name suggests infinite dimensions, and adding a ‘tron’ ending prevents confusion with inflaton.      This ‘tron’ ending is usually reserved for fermions so using it breaks this rule, but we may never be certain whether an ‘infinitron’ is a boson or a fermion anyway.   I do not know if ‘infinitron’ has been used for some other particle already; if not, it seems appropriate





\section{Determining f (r) expressions}


\indent Expression 6.35 has only been assumed based on R\textsubscript{i} (r) being like a higher dimensional radius. To derive an expression for f(r) from an hdf expression which incorporates the variable M will require very close inspection of the way variables in these expressions change as M is varied.  I will show how I am able to incorporate M into an hdf expression.  %The following analysis differs from that done for the previous TRM paper only in that the factor dividing M is adjusted for changing the units of M from centimeters to Planck lengths after expression 8.45.

\indent Before the form of the hdf expression was known all there was to work with was [f\textsubscript{i}(r) ]\textsuperscript{$-$1  }and the expression (1 – 2M/r).  When I made my first attempts to couple these two expressions a SLOPE value was multiplied by [f\textsubscript{i}(r) ]\textsuperscript{$-$1}  to achieve  a fit to the (1 – 2M/r ) expression.  To incorporate a geometrized M into this hdf expression a parameter M/(nongeometrized M) can be compared to hdf  values as M is varied  while the radius is fixed at 1000 centimeters.

\indent Using this comparative method, a table of data was generated, making the initial 2.0 solar mass assumption.  The values in table 16 below shows how the comparative method is used with radius fixed at 1000 centimeters.

\indent From table 16 the hdf expression can be multiplied by M/296,400 without changing its magnitude where the hdf expression initially was (SLOPE) [f\textsubscript{i}(r) ]\textsuperscript{$-$1} before it was known that the value one should be added to it.  The third column of this table can be thought of as equal to the first guess at hdf  for the range of radius where the hdf  couples to (1 – 2M/r) in a way that the two expressions are nearly equal.  From the Schwarzschild solution it is known that the first metric component is f(r) $=$ (1 – 2M/r) so we can also say that f(r) $=$ hdf expression where the two expressions are nearly equal near transition radii.

       This results in the following expression for the hdf:
\begin{equation}
    f (r) = (\frac{M}{296400}) (SLOPE) [f_{i} (r) ]^{−1}      
\end{equation} 

From table 11  it is obvious that there is a difference of one between columns two and three.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 16 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering
\caption{M versus (1 – 2M/r), − 2M/r, and (M/296,400) where r = 1000 centimeters for 2.0 solar masses.%$ per the previous TRM version
}
\begin{tabular}{|c|c|c|c|} 
\hline 
\textbf{M} & \textbf{(1 – 2M/1000)} & \textbf{− M/500} \textbf{(M/296,400)}\\ \hline
10,000 &− 19 &− 20  &0.0337 \\
25,000 &− 49 &− 50 &0.0843 \\
50,000   &− 99 &− 100 &~0.169 \\

\hline
100,000 &− 199 &− 399 &− 592 \\
200,000 &− 200 &− 400 &− 593 \\
296,400&0.337 &0.675 &1.00 \\

\hline 
\end{tabular}
\end{table}

The effect of this difference on expression 7.38 is that a value of one must be added to the right hand side of expression 7.38 to get a final f (r) which matches (1 – 2M/r) as closely as possible.  

\indent After adding one to the right hand side of 7.38 the result after rearranging and changing units is:
\begin{equation}
    f (r) =  1 + [ \frac{(SLOPE^*) \frac{M}{1.853 x 10^{38}} }{{f_i}(r)}]
\end{equation}

\indent SLOPE$^\ast$ has an asterisk because it is corrected for a change of length scales.  The value 1.853 x 10\textsuperscript{38} corresponds to an object of two solar masses so its value will increase in proportion to the mass of the object in solar masses divided by two.  Expression 7.39 is an intermediate form of f (r) which incorporates M into an hdf expression, but an unseen factor in expression 7.39 multiplies 1.853 x 10\textsuperscript{38} and this unseen factor is the ratio of the mass of the object in solar masses divided by two.

\indent A preferred form for expression 7.39 substitutes R\textsubscript{i} (r) into the expression, and this can be done because we know from a previous definition that f\textsubscript{i} (r) $=$ R\textsubscript{i} (r)/($-$3) .  When this substitution is made the final form for the expression for f (r) is:
\begin{equation}
    f (r) =  1  −  [ \frac{(SLOPE^*) \frac{M}{6.177 x 10^{37}} }{R_{i} (r)}]
\end{equation}

\indent A set of higher dimensional metric components can be generated for a higher dimensional Schwarzschild solution for a 2.20 solar mass object by using the SLOPE values from table 10 from section 5.  For these SLOPE values the value 6.177 x 10\textsuperscript{37} in expression 7.40 must be increased by 2.2/2.0  which  gives  6.784 x 10\textsuperscript{37 } so the expression for f (r) for table 10 data is:


\begin{equation}
     f (r) =  1  −  [ \frac{(SLOPE*) \frac{M}{6.784 x 10^{37}} }{R_{i} (r)}]
\end{equation}

\indent Again SLOPE is changed to SLOPE$^\ast$ to remind us of a change of units for M. Using expression 7.41, the SLOPE  data   from   table 10 is used to generate the set of seven higher dimensional Schwarzschild f (r) metric components for all of the spatial dimensions satisfying 4$ ≤$ Ns$ ≤ $10 to show the trend.  % See table 17  on the next page for 7.41  where f (r) is a function of M and the higher dimensional radius $R_{i}(r)$.
     
\indent The form of the f (r) expressions in table 17  are nearly identical to (1 – 2M/r), except that  r  is replaced by Ri (r) (see expression 6.34). The constants multiplying M increase in magnitude as Ns increases, and the constants multiplying M for 8 - Ds and 10 - Ds are both negative.  I should also point out that for 2.20 solar masses the constants multiplying M for 4 - Ds and 6 - Ds are equal to two so the  f (r)  expressions   for 4 - Ds  and for 6 - Ds  match  the form of (1 – 2M/r).
       
\indent Such a matching of  the form of (1 – 2M/r) suggests that the extra dimensions associated with 4 - Ds  and 6 - Ds  are both of infinite extent and are hidden from observation in such a way that Newton’s law of gravitation is  not changed by having more than three spatial dimensions.  There are several ways to hide extra dimensions of infinite extent.  The way to hide these dimensions
%, that occurred to me in the previous TRM paper,  was 
is to imagine that extra dimensions of infinite extent can align with our (3 + 1 ) dimensions at every point in spacetime in such a way that the overlapping extra dimensions merge into a set of  dimensions that look like three spatial dimensions at every point in spacetime.   Such a dimensional alignment would require extremely high energies to separate since they have remained undetected.  Also, the reason that these aligned dimensions do not change Newton’s law of gravitation may be because what we observe every day is a set of three spatial dimensions which is a merged overlay at every point in spacetime. 

\indent This explanation for how extra dimensions of infinite extent might go undetected would extend to the other extra dimenions in TRM such as        5 - Ds  and  spatial dimensions  7  through 25. 

\indent Another possibility is that all of these extra dimensions are virtual dimensions which have no influence on Newton’s law  in our  (3 + 1) dimensions.    

\indent As was pointed out previously, whenever a negative constant multiplies M it suggests that the associated spatial dimensions prefer negative energy or a negative energy density.    Therefore, both 8 - Ds and 10 - Ds should prefer negative energy  while spatial dimensions  4 - Ds,  5 - Ds,  6 - Ds, 7 - Ds, and 9 - Ds prefer positive energy.   How these negative versus positive energy preferences manifest themselves within the bubble is beyond the scope of this paper.

\indent Each f (r) expression listed in table 17 can be substituted into the Schwarzschild solution to  create a new higher dimensional version of the Schwarzschild solution for each value of Ns (where h (r) = 1/f (r) ), and the result of making a set of substitutions of this type is to create a set of higher dimensional Schwarzschild solutions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%% Table 17 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering

\begin{tabular}{|c|c|} 
\hline 
\textbf{Ns} & \textbf{f (r) = 1 − ~[ (SLOPE*) M/6.784 x $10^{37}$]/Ri (r)}  \\\hline
4           & f (r) = 1 − 2.00 M/$R_4$(r)  \\ \hline
5           & f (r) = 1 − 4.39 M/$R_5$(r) \\ \hline
6           & f (r) = 1 − 2.00 M/$R_6$(r)  \\ \hline
7           & f (r) = 1 − 8.65 M/$R_7$ (r)  \\ \hline
8           & f (r) = 1 + 12.51 M/$R_8$ (r) \\ \hline
9           & f (r) = 1 − 28.64 M/$R_9$ (r) \\ \hline
10          &  f (r) = 1 + 54.14 M/$R_{10}$ (r)\\
\hline 
\end{tabular}
\caption{Ns versus f (r) using expression 8.48 for table 11 data (SLOPE* is adjusted for 2M = $4.076 \times 10^{38}$ at 2.2 solar masses)
}
\end{table}




%Question: Continued Section 7 ?%



\section{Why does an exterior metric work?}

\indent It seems quite strange that basing TRM on an R\textsubscript{22} equation which appears when deriving an exterior solution like the Schwarzschild should result in a model which fits so many boson energies so well, especially since we are located in a universe which should be  described by an interior solution.   I will examine the equations used to derive an interior solution, which can be found in \textit{General Relativity} by Robert Wald, to see if there is a way to explain why an R\textsubscript{22} expression yields a TRM model which fits boson energies so closely.\textsubscript{ }

\indent For an interior solution the expressions 6.2.3 and 6.2.4 from \textit{General Relativity} apply \cite{wald1984the}. If we add these two expressions the result is:
\begin{equation}
    8π(ρ + P) = (r h^{2})^{−1} h’ + (r f  h)^{−1} f '
\end{equation}
where h and f are the metric components of a static spherically symmetric spacetime,  r is the interior radius of a spacetime bubble, $\rho$ is density, and P is pressure.  This equation can be simplified and rearranged as:
\begin{equation}
    8π(r h) (ρ + P) = (\frac{h’}{h}) + (\frac{f ’}{f} )                     
\end{equation}

Let us see what happens to expression 8.43 as the radius r becomes very large.  It can be argued that for regions of space exterior to stars and planets   when  r   is  sufficiently   large   the   term ($\rho$ + P)  in  this expression 8.43 will approach a value of zero.        


From the right hand side of expression 8.43 the term  f ’/f  is approximately equal to 2M/r\textsuperscript{2} since f  is approximately equal to one and f ’ is approximately equal to 2M/r\textsuperscript{2}.  The left hand side of 8.43 must be smaller than 2M/r\textsuperscript{2} for expression 8.43to be valid so let’s make the left hand side smaller than M/(25 r\textsuperscript{2}) which gives:
\begin{equation}
    8π (r h)(ρ + P) < M/(25 r^2) 
\end{equation}
Since the left side of expression 8.43, which is 8π (r h) (ρ + P), goes to zero as r gets large, where expression 8.44 is satisfied, it follows that expression 8.43 becomes:
\begin{equation}
    (h’/h) + (f ’/f) = 0                                               
\end{equation}
Expression 8.45 is identical to the intermediate expression 6.1.38 (see page 123 of Wald’s General Relativity \cite{wald1984the}) that gives the Schwarzschild exterior solution. From this result it follows that h = 1/f and h’ = − f$^{−2}$ as shown in the same reference.
    
\indent If we now subtract expression 6.2.4 from expression 6.2.3 (as shown on page 125 in \cite{wald1984the}) this gives:
\begin{equation}
    8π(ρ − P) = (r h^{2})^{−1} h’ + 2 r^{−2} (1 – h^{−1}) − (rfh)^{−1} f ’
\end{equation}
Rearranging 8.45 gives:

\begin{equation}
    8π (ρ − P) = − (r f h)−1 f ’ + (r h2)−1 h’ +
                        + 2 r −2 (1 − h−1)                           
\end{equation}

When 8π (r h)(ρ + P) < M/(25 r2) we showed above that h = 1/f and h’ = − f$^{−2}$ .  Substituting h and h’ into expression 8.46 gives:
\begin{equation}
    8π (ρ − P) = \frac{− f ’}{r} + (\frac{f^{2}}{r})(−f^{−2}) + 2 r^{−2} (1 –f)       
\end{equation}

\begin{equation}
    8π r (ρ – P) = (– f ’ – 1 ) + \frac{2 (1 – f )}{r}
\end{equation}

For a universe as large as ours P = 0 is very likely, but ρ = 0 needs more inspection.  I will briefly assume that both P and ρ go to zero for a universe with a radius as large as the radius of our universe.  When ρ and P are both zero the left  hand side of 8.49 goes to zero giving:

\begin{equation}
    (− f ’− 1) + \frac{2 (1 – f)}{r} = 0                                  
\end{equation}
Expression 8.50 is identical to expression 2.4 of section 2.1, and this is the expression on which TRM is based. 

\indent However, there is a problem with the assumptions made above.  If we rearrange expression 8.44 it becomes:
\begin{equation}
    r^{3} < \frac{M}{(200 \pi h)} (ρ + P)^{–1}                           
\end{equation}

We can substitute the values h = 1, M = 326,000 and P = 0 in the above equation to give:
\begin{equation}
    r^{3} <  \frac{518.8}{ρ}       
\end{equation}

\indent To see why there is a problem with expression 8.52 we have to substitute an appropriate value for the density in the denominator of the right hand side.

\indent The average density of our universe if Ω = 1 is five atoms per cubic meter, and this density may be much lower than a five atoms per cubic meter value as Martin Rees pointed out in Just Six Numbers \cite[p.73]{rees2000just}.  If the value of ρ is assumed to correspond to five atoms per cubic meter, the resulting density is about 8.35 x 10$^{−30}$ g/cm3  in units called nongeometrized by Robert Wald.  To geometrize this density of  8.35 x 10$^{−30}$ g/cm3 the value must be multiplied by G/c$^{2}$ which is 7.41 x 10$^{−29}$ cm/g per table F.1 page 471 of Robert Wald’s General Relativity. \cite{wald1984the} This gives a geometrized density of 6.19 x 10$^{−58}$ cm$^{−2}$ for our universe if Ω = 1.  Substituting this geometrized density  value into expression 8.52 for ρ  gives:
\begin{equation}
    r^{3} < 8.38 x 10^{59} cm^{3}                                           
\end{equation}
This is the same as requiring the radius to be less than 9.43 x 10$^{19}$ cm or less than about 100 light-years. 

\indent This requirement that r is less than 100 light-years means that the density must be much greater than was assumed.  Radiation density can not be ignored for universe radius values less than 100,000 light-years, but even when our universe had a radius of 100,000 light-years its density would be too high for the low density assumption made to reduce expression 8.49 to expression 8.50. This 100 light-year radius requirement of expression 8.53 completely contradicts the low density assumption made for expression 8.50, and we must conclude that the exterior form of TRM does not apply in the most simple way to the interior of the bubble for an early universe.  Only when a universe is many billions of years old, where M is extremely large and the density is very low, may the interior form of TRM per expression 8.49 reduce to 8.50.
       This leaves two possible ways for TRM to influence the energies of annihilation bosons for our early universe:
\begin{enumerate}
    \item  Expressions  8.53   and  8.59  must  be  solved
       directly,  making   the  expressions  for  higher
       dimensional radius more complicated.
    \item  Before  inflation  there  was  one,  a few,  or
        many black holes in our universe. Later, at         
        the  time  of  quark  formation,  there were 
        many   black  holes   interior  to   the  bubble,   
        where these black holes have nearly identical   
        masses, so  quark  formation  was  influenced
        by   being   exterior   to   these    black holes where the   exterior  form   of  TRM  applies.
\end{enumerate}       

\indent If the second possibility is correct this means that TRM based on the exterior Schwarzschild solution  can be applied directly for our early universe without having to go back to expressions 8.53 and 8.59 .   

\indent Therefore, the exterior form of TRM only  applies for the interior of a bubble if a pre-inflation bubble was filled with one, a few, or many  black  holes, and the post-inflation universe was filled with one, a few, or many primordial black holes. For our universe to be as homogeneous as it is one would expect that many black holes existed before and after  inflation.



\section{Where is the graviton?}

\indent The short answer is that TRM does not directly predict a boson energy which is an obvious candidate for the graviton.  A boson energy associated with the 25 - Ds is about 9.8 x 10\textsuperscript{-11} GeV which is far too high for the graviton. 

\indent To directly predict a boson energy which is orders of magnitude lower than a photon’s energy requires a transition radius which is greater than 10\textsuperscript{15} Planck lengths.  However, I have limited TRM to a maximum of 25 spatial dimensions so it will agree with string theory, but a transition radius value above 10\textsuperscript{15} Planck lengths would require more than 25 - Ds.   Predicting a specific graviton energy requires specifying a number of spatial dimensions far greater than 25, but there are so many values of Ns > 25 that it is impossible to guess which Ns is the correct one to pick for a graviton.  The two-step cancellation mechanism described on page 20 is a way around this problem of  fitting the graviton within 25 spatial dimensions.

\indent However, there is another approach to this problem which may make it possible to find the effect of a graviton within the limits of a 25 spatial dimension spacetime.   
%From the Previous TRM paper,  I noticed
Notice that the larger transition radii for 24 - Ds is 7.00x10$^9$ Planck lengths and the single transition radius for 25 - Ds is  5.00 x 10\textsuperscript{9} Planck lengths.  The energy of a boson corresponding to the larger transition radius for 24 - Ds is -3.6x10$^{-11}$ GeV versus 9.8x10$^{-11}$ GeV for the 25 - Ds boson.   If the value of Nsx for 24 - Ds were adjusted slightly the absolute values of these two energy values could be much closer together.  In that case, the graviton may be the effect of a sum of 24 - Ds and 25 - Ds boson energies where the former has a negative energy boson and the latter has a positive energy boson.  If the absolute value of the energies for these two bosons were very close their sum would almost cancel, leaving a very small net energy.  These boson energies are extremely sensitive to the value of Nsx used for TRM so a second model would be needed to determine the best Nsx values for 24 - Ds to find a graviton effect.  This shows that there is a potential explanation for the graviton effect within a 25 spatial dimension TRM, but it requires additional theory to determine the correct net energy value.     

%\indent Even though TRM does not directly predict graviton energy it still does an excellent job of fitting a very wide range of particle/antiparticle energies (see section 7 pages 16 - 22).

\section{Wavelength versus Transition Radius}

\indent A relationship between quantum theory wavelength and TRM transition radius which applies at annihilation energy can be derived by equating the expression E $=$ h c/$\lambda$ with a variation of expression 5.28ff.

\indent In general, expression 5.28ff can be modified to account for the fact that boson energy is usually carried by two bosons of equal energy when annihilation occurs.  This modification gives the following expression for boson energy:
\begin{equation}
    E_{BK} =\frac{ E_{P}} {(K r_{T}^3 }                                           
\end{equation}
where K = 1  for an unsplit boson or  K = 2  for two bosons of equal energy like electron/positron annihilation photons.

\indent Equating the energy term in E = h c/λ with EBK  in expression 10.54 gives an expression which can be rearranged to give an expression for the wavelength λ:
$E_P$ /($K r_{T}^{3}$) = hc/λ      which rearranges to give

\begin{equation}
    λ = (\frac{Khc}{E_{P}}) (r_{T})^{3}                                          
\end{equation}
where h is Planck’s constant, c is the speed of light, and EP is Planck energy.  To get the wavelength λ in expression 10.55 in units of meters the following values are substituted:h = 6.625 x 10–34   J-sec, c = 2.998 x 108 m/sec, and $E_{P}$ = 1.954 x 10$^{9}$ Joules.   After making these substitutions, expression 10.55 simplifies to:

\begin{equation}
    λ =  (1.0165  x 10^{– 34} ) K  (r_{T})^{3}                        
\end{equation}

where λ is in meters and $r_T$  is in Planck lengths.
This expression was checked for the electron which has an energy of  510,000 eV, which is equivalent to 8.17 x 10$^{–34}$ Joules, by using the  expression λ =  h c /(8.17 x 10$^{–34}$ J) which gives a wavelength of 2.431  x 10$^{–12}$  meters.  The same value  of   λ  resulted  for  expression   10.56 when K = 2 and when $r_T$ = 2.287 x 10$^{7}$  Planck lengths.  Thus, expression 10.56  gives the connection between wavelength and transition radius provided the correct integer value of K  is used.

\section{Discussion and conclusions}

\indent This transition radius method looks at the definition of higher dimensions in a new way, and this leads to several new findings. First, integrating the Ricci expression R\textsubscript{22} generates expression 2.21 which defines f\textsubscript{i}(r), and the reciprocal of this f\textsubscript{i}(r) function can be coupled to (1 – 2M/r) in a way which yields transition radii that can be used to predict many boson energies.   When boson energies are predicted from transition radius values, TRM uses the expression E\textsubscript{B} $=$ E\textsubscript{T} $=$ $\pm$ 1.22 x 10 GeV/(r\textsubscript{T})  where E\textsubscript{B} is boson energy, E\textsubscript{T} is transition energy, and r\textsubscript{T} is the transition radius.  TRM fit boson energies match the energies of at least fourteen known boson types plus the X bosons.  A second finding which results from TRM is a set of inflation trigger radius ranges.  These inflation triggers are clustered around three distinctly different radius ranges, but additional work is needed to determine how many of these inflation triggers initiate   different   stages   of   inflation.    A  third finding is that the larger transition radius (r\textsubscript{LT}) for 7 - Ds predicts a bubble radius of 1130 Planck lengths which is compared with a previous bubble radius estimate giving a most likely bubble radius from 1065 to 1130 Planck lengths which may be the radius when large-scale structure formation was initially influenced. Finally, TRM fits the energies and masses of known fundamental particles, and TRM predicts the existence of particles at energies above 300 GeV.  These are the main findings made possible by the development of TRM.

%\indent Many of the findings in the previous  TRM paper are still valid so previous discussion from that paper will be presented again, containing modifications consistent with extended TRM.   The comparison of the extended TRM to the previous TRM will be addressed near the end of this section.  If you prefer to review the  comparison of extended versus previous TRM first, skip ahead to page 32  columns 1 and 2  or go back to section 7.  \cite{holte2003transition}

\indent In addition to the above findings, the energy of a boson with an infinite number of dimensions is found in  section 6  by  deriving  expressions  6.29 and  6.33, then solving expression 6.36 to find its transition radius.  From this transition radius the boson energy for a boson with an infinite number of dimensions is E\textsubscript{B} $=$ $\pm$ 7.515 x 10\textsuperscript{17} GeV.  This boson energy may represent one of the unification energies for bosons.  As I suggested previously, an appropriate name for such a boson might be an ‘infinitron’.

\indent The next highest boson energy fit by TRM is that for the X-like boson associated with 5 spatial dimensions which has a predicted E\textsubscript{B} for its smaller transition radius of 9.47 x 10$^{-16}$ GeV. 

\indent To appreciate the wide range of boson energies that TRM fits one should note that TRM gives a reasonable estimate for the energy of the e-neutrino/anti-e-neutrino annihilation boson.  From table 9 the TRM predicted energy for this boson is 4.26 x 10\textsuperscript{$-$9} GeV so the energy of a single e-neutrino is predicted by TRM to be 2.13 x 10\textsuperscript{$-$9} GeV. Thus, TRM predicts boson energies spanning 26 orders of magnitude when the ‘infinitron’ is included or it predicts boson energies spanning 25 orders of magnitude when the X-like boson for 5 - Ds is considered the highest boson energy.

\indent In addition to boson energies, inflation trigger radius ranges are predicted by TRM.  These events occur in radius ranges shown in tables 3 and 4.  This model suggests that a ‘presingularity’ may never be allowed to collapse to a radius smaller than 1.0 Planck length because these inflation triggers all occur at radius values above a 1.0 Planck length radius.  Additional work is needed to determine how these inflation triggers affect inflation, especially any influence these triggers may have on the number of stages of inflation.  TRM provides evidence which supports the idea that inflation may have occurred in one, two, or three stages.

\indent Key concepts that I developed which made TRM possible are:
\begin{enumerate}
    \item Deriving higher dimensional $f_{i}$(r) expressions
     from two of the four Ricci expressions which     
     had previously been set aside when the   
     Schwarzschild solution was derived.

    \item Defining transition radii as the radii where
     higher dimensional $f(r)$ expressions are equal 
     to the (3 + 1) dimensional $f(r)$ expression 
     (1 – 2M/r).

    \item Defining an averaging parameter SX and a
     distribution for the number of largest SX
     values (Nsx) to be averaged to determine the
     first approximations for SLOPE values. 

    \item Finding an equation which predicts boson   
     energy from a transition radius.  This 
     equation is $E_B$ = ± 1.22 x 10$^{19}$ GeV/$(r_T)^{3}$  where
      $r_T$ is the transition radius.  This also means
     that (4 π/3  $(r_T)^3$ ) $E_B$ =  ± 5.11 x 10$^19$ GeV.  Both
     equations apply for (rest mass) particle/anti-
     particle annihilation bosons.
\end{enumerate}       
       
    These key concepts make it possible for TRM to fit a large number of annihilation boson energies  and fundamental particle masses.

\indent A higher dimensional radius R\textsubscript{i} (r) defined by expression 6.34 is used in the higher dimensional f (r) expressions.  The f (r) expressions for 4 – Ds  through 10 - Ds are listed in table 14, and they show that even numbered spatial dimensions starting with 8 - Ds have negative SLOPE values which imply that negative energies occur for these dimensions.

\indent A complete  list  of  these  SLOPE  values for   4 $\leq$ Ns $\leq$ 25 is found in table 10.   These SLOPE values affect the degree to which the (1 – 2M/r) metric component for 3 - Ds is coupled to a specific higher dimensional f (r) expression.   If this SLOPE parameter were renamed in a way which reflects its function, it might be called an ‘interdimensional’ intersection factor  or  IIF.   This naming emphasizes the intersection of the higher dimensional metric components to the    (3 + 1) dimensional metric component (1 – 2M/r) of the Schwarzschild solution at  transition radii.

\indent From the higher dimensional f (r) expressions listed in table 14 a set of higher dimensional Schwarzschild solutions can be derived by substitution.

\indent For 7 - Ds a unique transition radius is obtained which nearly matches a prediction of the size of a bubble at a bounce made by Carmen Molina – París and Matt Visser.\cite{molinaparis1999} I have estimated a bubble radius range of 1065 to 1130 Planck length radius from TRM.  At this bubble radius the large-scale structure of our universe may have been first influenced.    

%\indent One shortcoming of TRM is that it may not directly predict the energy of a graviton.  There may be ways to use other theories in conjunction with TRM to find graviton energy, but this is still speculation. This shortcoming does not detract from the ability of extended and previous TRM to fit the annihilation energies of fundamental particle pairs over a wide range of energies.

%\indent A concern that I had about TRM initially was that it implied that a black hole having a small mass may have generated everything in our universe.  I could not see how   a ‘presingularity’ inside such a black hole could  produce a universe filled with as much mass-energy as is contained in our universe.  Fortunately, I came across two ways that mass-energy can be dramatically increased.  First, in a book by John Gribbin I read that Werner Israel, Eric Poisson, and A. E. Sikkema found that a spinning black hole does not rotate smoothly and ``quivering of the outer horizon" produces blue-shifted gravitational waves some of which fall into the black hole.\cite{Poisson:1989zz, gribbin1998in} As Gribbin points out this ``inflow of blue-shifted gravitational radiation will carry energy and produce an extraordinary increase in the mass inside the black hole." \cite{ibid}  TRM is not based on a spinning black hole, but it may be possible to extend TRM to the Kerr metric. The second way that mass-energy is increased within a bubble is by the presence of the Higgs field allowing the energy density of the false vacuum to remain constant during inflation which Alan Guth explains more clearly on page 170 of his book \textit{The Inflationary Universe}.\cite{guth1997the}  These two mechanisms for increasing mass-energy within the bubble convinced me that a ‘presingularity’ inside a black hole of relatively small mass could generate a universe having as much mass-energy as our universe.

%\indent The recent work of Roberto Emparan and Harvey S. Reall suggests that pre-inflation black holes may have a ring shape.\cite{emparan2002} It might also be assumed that ring black holes could spin around more than one axis so that the volume they sweep out while spinning would resemble a coreless spheroid.  If the number of these pre-inflation black holes were known, it might be  that their spheroid spin volume would pack a bubble of 1065 to 1130 Planck length radius in a way  that maximum packing density might be used to estimate the size of each pre-inflation black hole.

\indent Assuming that TRM must apply indirectly through black holes for the interior of the bubble, as pointed out in section 9,  also changes the way we have to think about the 1065 to 1130 Planck length bubble radius.  TRM requires that this bubble radius be influenced by being external to at least one black hole, but the only place that could happen is when the bubble is in the parent universe.  It follows  that the parent universe which generated our universe should have had a set of black holes.   Therefore, for a 1065 to 1130 Planck length bubble radius to be controlled by TRM the parent  universe must interact with the child universe and the parent universe must contain many black holes in its interior.        

\indent For the previous TRM version, a possible resonance is seen for the top/anti-top annihilation boson.  This annihilation energy can be fit as sum of two bosons of the same energy for the  smaller  transition  radius  at   18 – Ds   when Nsx $=$ 4.0.  If these three boson types having Nsx resonances are excluded, the Nsx  values for the remaining boson types  have  a  range  which tightens up to 1.78 $\leq$ Nsx $\leq$ 4.  It seems likely that most boson types will be described by a resonance of two different Nsx values where one value is for a single boson type and the other is for the sum of two boson type energies.  This means that many boson types will be fit by a resonance between two different Nsx distributions.  Based on the findings for the three boson types above, the Nsx values will be very close to a 4.0 value for the Nsx distribution where two boson type energies are summed.  This trend suggests that there are two intersecting Nsx distributions (see Appendix A). 

\indent  For a given boson type which resonants between two Nsx values it seems that the difference between its Nsx values is related to the massiveness of that boson type.  For example, for the previous version of  TRM the  Nsx  differences are  0.023, 1.73,  and 2.90 for the e\textsuperscript{+}e\textsuperscript{- }boson, Z boson, and top quark/anti-top quark annihilation boson, respectively.   

\indent If the total number of positive energy boson types for TRM is the sum of  the number of single transition radius positive energy boson types plus the number of possible paired positive energy boson types,  then the maximum total  number  of  positive energy gauge boson types (Nb) should be given by Nb $=$ 39 + [(39)(38)/2 ] + 1 $=$ 781 when negative energy bosons do not interact with positive energy bosons.  If negative energy boson types fully combine with positive energy boson types, this  Nb  value  could  be  as  large  as  Nb $=$ [(780)(780) – 780] + 780 + 1; however, this Nb value seems way too large. In either case, the number of observable gauge boson types (Nobs) should be much less than Nb because one boson type of  each pairing may have an energy which is orders of magnitude less than the energy of the other boson type so the paired energy would be indistinguishable from the energy of one of the unpaired boson types.  The first case above gives Nobs < 781, assuming that a positive energy boson only combines  with a negative energy boson having  the  same  absolute value of  energy (see Appendix C for more details).

%\indent Another way to look at number of particle types implied by TRM is to count only the new fundamental particle pairs, excluding particle combinations.   This approach gives either 23 new fundamental particle types or 51 new fundamental  particle  types.    For  the  first  case, only  the  Ns versus Nsx distributions in Tables 12 and 13 are considered so there are 40 total types minus 19 known types (in bold case) plus one ‘infinitron’ plus one graviton precursor giving 23  new types.  The second case,  when the Ns versus  Nsx distributions in Tables 14  and 15 are also considered, gives 28 more new types (bold case values not counted) for a total of 51 new fundamental particle types for TRM.   This assumes that we consider the  five bold cased values near X boson energies as known particle types (even though they are still theoretical); this X boson count could also be six. These new particle type counts assume that we truncate counting new particle types for Ns values greater than 25.          

\indent The transition radius method shows considerable promise based on its ability to fit as  many as fifteen particle types over a wide range of energies.  Most transition radii occur in pairs for each Ns, and fitting one of these transition radii for a given pair to a known particle energy necessarily anchors a second predicted particle energy.  

\indent It is possible that many of the unmatched negative energy bosons could be  part  of  the  dark  energy   which   is  being investigated today; although, it is likely that these negative energy bosons have  much lower energies today.

\indent For TRM the extra dimenions may hide in our three spatial dimensions as a perfectly aligned overlay at every point in spacetime, and these extra dimensions may hide by limiting access to quantum entities provided an equilibrium has been established (to satisfy conservation of energy).  For TRM the extra dimensions do not have to curl up at small dimensions to be undetectable.  Thus, TRM suggests that our three spatial dimensions may be each a composite of eight infinite dimensions, and one or more infinite dimensions may be free to  move between  them at  a  rate  comparable  to or equal to light speed.  A better interpretation of TRM may be to assume that each of our three spatial dimensions is a composite of an infinite number of extra dimensions of infinite extent.   Another way that extra dimensions of infinite extent might avoid influencing Newton’s Law is if extra dimensions of  infinite extent are all virtual.

\indent Either expressions 8.43  and 8.49 must be solved to determine what happens interior to the bubble or we can use the exterior form of TRM (expressions 2.21 through 5.28) if we assume that many black holes of nearly identical masses were present inside a bubble during quark formation.

\indent For the first case,  the effect of equation 8.49  may cause boson type energies to be different than they are today, and this implies a  contradiction of the uniformity of the atomic mass  of   hydrogen   throughout   the   observable universe. This apparent contradiction can be resolved if annihilation boson energies matched the energies determined by TRM using expressions 8.43 and 8.49.  TRM suggests that particle masses or energies all shifted from early universe values to current values once our universe exceeded a radius of 100,000 light-years.

\indent In the second case, where many black holes are present inside the bubble at the  time  of   quark   formation,  expressions  2.21 through 5.28 can be used directly to determine annihilation boson energies and inflation triggers.  Throughout the period of quark formation these early universe black holes do not have to maintain constant mass for quark masses to remain constant throughout this period due to findings shown below. 

\indent The structure of a quark may also restrict a quark to a more probable energy so that the boson type energy restrictions of TRM provide boson type energies close to what is needed for formation of each quark flavor, then quark structure dictates the precise average energy of quark formation from these bosons.  Determining how much control TRM has over the precise average energy of a quark will  likely require other theories, but it can be said that TRM restricts the allowed energy of each quark flavor (and other particle types) to a narrow range of energy by controlling the energy of the annihilation boson types from which they form.

%\indent I had a concern about a possible shift in  boson energies due to a supermassive black hole at the center of a galaxy.  Using the extended version of TRM, I reevaluated the boson energies for a galaxy center black hole of 3 million solar masses for all fifteen observable annihilation  bosons  fit  by  TRM  (where  the $\tau$-neutrino/anti-$\tau$-neutrino boson was included in this reevaluation).  The energies of these observable annihilation boson types for the 3 million solar mass black hole were compared to those found for a 2.20 solar mass black hole.  Analysis of the metric component (1 – 4.076 x 10\textsuperscript{38}/r) and the way SLOPE is determined shows that the  extended version of TRM will give the same particle energies for any mass above 2.2 solar masses so this version will apply at least up to the mass of our universe. 

\indent Finding that TRM allows a wide range of black hole masses to give nearly identical sets of observable boson energies also means that early universe black holes do not need to have nearly identical masses to produce identical patterns of particle masses.  Nearly identical black hole masses are not necessary, but they could be nearly  identical due to other processes.

\indent The severe spacetime distortions produced by pre-inflation black holes when the radius of their collapsing bodies is equal to an inflation trigger radius are likely intense enough to produce an inflation triggering effect interior to their event horizons.  The ‘presingularities’ within these pre-inflation black holes  are likely  so  tightly  packed together before inflation that they are within each other’s event horizon when inflation initiates.

%\indent  Previous work showed that minimum bubble radius might be based on the anticipated effect of bubble rotation on bubble contents.  Since both versions of TRM are based on the Schwarzschild solution for a static black hole the anticipated effect of bubble rotation can only be pasted into the TRM model.  Several assumptions are made about a rotating bubble to conclude that it should be filled with many ‘presingularities’.  First, the asymptotic freedom of quark/gluon plasma at very high energy density should minimize the viscosity to the point that the positive energy contents of the bubble may continue to behave like a superfluid during collapse.  Second, uniformly rotating superfluids generate multiple vortices within the fluid.  Third,  multiple vortices may each produce  ‘presingularities’ within the  rotating bubble.  Fourth, the findings of Hod and Piran show that infalling blue-shifted radiation should prevent ring singularity formation within a rotating black hole so the bubble may contain   a pair of  ‘presingularites’.  \cite{hod1998}It is possible that each of these two   ‘presingularities’ may contain many ‘presingularities’.  Fifth, the work of Hochberg and Kephart shows that squeezed vacuum should generate a negative energy density which would allow inflation to intiate, and this negative energy may infall into pre-inflation ‘presingularities’ to accelerate their evaporation preventing them from completely dominating large-scale structure.  \cite{hochberg1991} Hawking  radiation should also speed evaporation. The influence of ‘presingularities’ on large-scale structure would also be limited by the energy density smoothing effect of spacetime distortions created by inflation triggers and the density smoothing effect of the beginning of inflation. Combining these speculations and models suggests that it is likely that the positive energy contents of a spinning ‘presingularity’ (bubble) within one black hole will contain  many ‘presingularities’ within it.   The reason for discussing the anticipated effects of bubble rotation is to show that basing a TRM model on the Kerr solution and the Emparan-Reall solution are reasonable next steps. 

%\indent After I finished the above paragraph, I searched the Internet and came across a website, science.nasa.gov, which summarized the results  seen for BEC spinning by W. Ketterle et al done at MIT.  \cite{phillips2002my}  This summary shows a set of uniformly sized vortices throughout the volume of a spinning Bose-Einstein  condensate, and this pattern of vortices is what I anticipate could be present inside of spinning ‘presingularities’ inside of black holes.
%@article{phillips2002my,
%title = {My Pet Neutron Star, science.nasa.gov},
%journal = {Science@ NASA},
%year = {2002},
%author= {Phillips T }}


%\indent I would like to revisit boson energies fit by  the previous TRM to discuss a few energies which I have neglected to mention up to this point.  A few days before completing the previous TRM paper, I used TRM to fit the energies of a glueball and a gluon. 

%\indent First, the glueball energy is fit by a 1.80 GeV energy which occurs for 18 - Ds when the tau/anti-tau boson energy is split in half.  This 1.80 GeV energy is only 0.1 GeV off from the 1.7 GeV energy  mentioned  in the text \textit{Q is for Quantum} by John Gibbin.  \cite{gribbin1998}This   glueball energy can also approach 1.5 GeV if an energy associated with 20 - Ds is subtracted from 3.60 GeV or (3.60 – 0.212) , then the resulting energy is divided by two.

%\indent Also, I found three ways that one-third of the rest mass energy of the proton, which is about 0.313 GeV, can be fit by TRM.  First, the energy of 2.50 GeV associated with 15 - Ds may split in half three times or 2.50/8 $=$ 0.313 GeV. Second, a minor contributor to this energy may be fit by splitting the energy of a composite of two boson types.  This composite energy  occurs   when  the  0.273  GeV  energy  for 21 - Ds having Nsx $=$ 4.0 for the smaller transition radius is added to the 0.379 GeV energy for 15 - Ds having Nsx $=$ 4.0 for the larger transition radius.  The composite energy of 0.652 GeV is split in half to give 0.326 GeV.  Third,  this energy in the proton is closely fit by the  energy associated with smaller  transition  radius  at 16 - Ds when Nsx $=$ 2.5  which gives a boson energy of 0.3104 GeV; however, this third way will coexist with the top/anti-top fit at Nsx $=$ 1.1.  Some of these three alternative energy states may be in resonance, providing some of the proton’s mass. 

%\indent A comparison of extended TRM where 2M $=$ 4.076 x 10\textsuperscript{38}  to the previous TRM was explored in section 7  in tables 12 – 15.  These tables revealed many similarities and several significant differences between the extended TRM and the previous TRM model.   The most striking similarity is that the Nsx versus odd Ns distribution and Nsx versus even Ns distribution retain the same shape for extended TRM versus previous TRM.  Also, the range of Nsx values for both extended and previous cases are similar.

%\indent  With a few exceptions, due to changing the units of M in (1 – 2M/r) from centimeters to Planck lengths,  most of the particle masses fit   by TRM  shift   by   less   than   fifteen  percent  when Nsx $=$ 4 (see table 14).  Particle masses can be fit exactly by shifting Nsx values slightly (see table 12).  Also, changing the Nsx value at Ns $=$ 17  so that the Z boson is fit at E\textsubscript{B} $=$ 182.4 GeV  allows  the  tau neutrino  energy  to  be  fit at Ns $=$ 17  also  which is an improvement on previous TRM results (see table 12).

%\indent  Improving the smoothness of the Nsx versus Ns distribution for TRM reveals  a  particle energy at 160.4 GeV (see Ns $=$ 14 , Nsx $=$ 1.969  in table 13)  which is double 80.2 GeV. On the other hand,  if Nsx $=$ 2 at Ns $=$ 14 , then  E\textsubscript{B} $=$ 147 GeV which may be a Higgs particle.

%A plot of Nsx versus Even Ns is made smoother by letting Nsx $=$ 2.6 for Ns $=$ 12, and a plot of Nsx versus Odd Ns is made smoother by letting Nsx $=$ 3.4 for Ns $=$ 13  (see Appendix A), giving  E\textsubscript{B} $=$ 430 GeV which may be Higgs boson.

%\indent The most likely asymmetric decay for an E\textsubscript{B} value near 430 GeV is  2 (80.3) + 3 (91.2) $=$ 434 GeV which is two W bosons plus three Z bosons.  Symmetric decay of an annihilation boson energy  E\textsubscript{B} $=$ 434 GeV  will give particle energies of         217 GeV and/or 145 GeV.  Whether both of these are detected will depend on how much they interact with matter. 

%\indent Assuming that  Nsx $=$ 4  or Nsx is very close to four  for all  Ns  values produces a different Nsx versus Ns distribution which is used to predict a most likely pattern of particle masses (or energies) above 300 GeV which may be detected at the LHC.  When  2M $=$ $4.076 \times 10^{38}$, the most likely pattern of  symmetric decay particles is:  346, 387, 461, 516,  692,  773,  922,  1031,  1383,  and 1547 GeV.  When  2M $=$ 652,100 the most likely pattern of symmetric decay particles is:  350, 373, 470, 497, 705,  745,  940,  993,  1410,  and 1490 GeV.  By symmetric decay I mean the products of a decay have identical particle masses.    Only one of these two patterns is likely to be detected at the LHC,  which one depends on which value of 2M matches how nature works, and I anticipate that the former pattern is more likely to match empirical data.   Some of these symmetric decay particles may not be detectable if they do not interact with matter.

%\indent Asymmetric particle decay is another possibility, and it is where the resulting decay products have different masses.  Such asymmetric particle decay is more difficult to predict (see suggested rules in section 7 between tables 13 and 14).   Should asymmetric particle decay occur it will add to the complexity of the pattern of particle masses, having masses above  300 GeV,  which are detected at the LHC.  If none of these predicted particle energies in the previous paragraph are detected, either they are all noninteractive like dark matter, the Nsx $=$4 distribution for all Ns values is not valid, or neither version of TRM  fully describes how nature works with regard to particle masses and energies.  At this time,  the first two of these reasons seem the most likely explanation for these predictions not matching any LHC  results.

\indent  TRM suggests that the black hole metric influences rest masses and energies of fundamental particles might be located inside the event horizon of a black hole because all transition radii are much smaller than the radius of the event horizon.  There are three ways that  TRM may influence particle  masses  (energies) exterior to the event horizon. If virtual particles inside the event horizon can tunnel out to   influence   the  particle  energies  of   Hawking radiation, then TRM shows how the pattern of rest masses and energies of  fundamental particles can be determined by Schwarzschild black holes.   A second way for TRM to influence the pattern of particle masses in our universe is for the black holes influencing this pattern to have event horizons which are smaller than the transition radius of most particle pairs; an event horizon radius less than about 3 x 10\textsuperscript{5 } Planck lengths would be smaller than the transition radius of the boson associated with a top quark/anti top quark pair.   The third way that the pattern of  rest masses and energies of fundamental particles throughout our universe can be determined by TRM is if our universe influences the fundamental particles the same way that Schwarzschild black holes influence fundamental particles in their interiors, see section 9.% (see pages 25 and 26).  

%\indent Both the extended TRM version and the previous TRM version show that the pattern of  rest masses and energies for fundamental particle types is determined by extra dimensional metric terms intersecting the (3 + 1 ) dimensional Schwarzschild metric term  (1 – 2M/r)  at the same transition radius, where particle rest masses and energies  are determined by expression 5.28 on page 11.  This TRM predicted pattern of fundamental particle rest masses and energies  will apply over a range of black hole masses from 2.2 solar masses up to the mass of our universe (see section 7).

%\indent TRM might be merged with other particle theories since the transition radii found per TRM may correspond to or limit the length of strings, the size of brane particles, the size of loop quantum gravity particles,  or the size of other  particle types based on different theories.   Therefore,  a merger between TRM and string theory, a merger between TRM and loop quantum gravity, or a merger between TRM and  some other particle theory may advance our understanding of  fundamental particles.

%\indent  A major benefit provided by both versions of TRM is that, for each extra dimension, once the mass or energy of a known particle type is fit by TRM  it anchors the mass or energy of a second different particle type for most extra dimensions   (4 - Ds and 6 - Ds  are exceptions).
         
%\indent Looking back over this model,  I find that the beauty of TRM is that it is simpler than other models, and it extends classical general relativity in a way that connects general relativity to annihilation boson energies and fundamental particle masses.  Both versions of TRM fit the fundamental particle masses and energies with  the smooth change of the single parameter Nsx when Nsx is plotted versus odd Ns and even Ns values separately.  

%\textbf{Acknowledgments }

\section*{Acknowledgements}

I received no advice or input during the development of this method, but I would like to thank the following authors for works which inspired or helped these efforts:  Robert M. Wald for \textit{General Relativity,} Alan Guth for \textit{The Inflationary} \textit{Universe}, John Gribbin for \textit{In Search of the Edge} \textit{of Time}, Martin Rees for \textit{Just Six Numbers}, Kip Thorne for \textit{Black Holes $\&$ Time} \textit{Warps}, Matt Visser for \textit{Lorentzian Wormholes}, Lee Smolin for \textit{Three Roads to Quantum Gravity}, Michio Kaku for \textit{Beyond Einstein}, and Lisa Randall for \textit{Warped Passages}.

This manuscript was typeset and submitted posthumously with the assistance of B.L. and editing for style and audience by R.G.  The original manuscript was dated 23 September 2006 and came into our hands shortly after the author passed away in 2019 and this version completed in 2023.  It was a revision and extension of an earlier self-published work called Transmission Radius Method from 2003.  References to the earlier version have been edited such that this manuscript stands on its own.  A special request for posthumous electronic publication on arXiv.org was declined.

As part of the posthumous editing process we did a simplified literature search and found eight other works (references therein and later citing works) relating to predicting the masses of fundamental particles and may be of interest to readers of this manuscript.

Alejandro Rivero and Andre Gsponer, "The strange formula of Dr. Koide", arXiv:hep-ph/0505220
doi 10.48550/arXiv.hep-ph/0505220

Alexander Kyriakos, "On calculation of elementary particles' masses", 
\\arXiv:physics/0604063 (2006)  doi 10.48550/arXiv.physics/0604063

Ding-Yu Chung, "The Accurate Mass Formulas for Leptons, Quarks, Gauge Bosons, the Higgs Boson, and Cosmic Rays", J.Mod.Phys v.7 p.1591 (2016) 
\\doi 10.4236/jmp.2016.712144 (and earlier arXiv.org versions dating to 2000)

Karl Otto Greulich, "Calculation of the Masses of All Fundamental Elementary Particles with an Accuracy of Approx. 1\%", J.Mod.Phys v.1 p.300 (2010)
\\doi 10.4236/jmp.2010.15042

Dao Duc and Nguyen Giao, "Vector Boson Mass Spectrum from Extra Dimension", J. Mod.Phys v.4 p.991 (2013) 
\\doi 10.4236/jmp.2013.47133

Klaus Paasch, "On the Calculation of Elementary Particle Masses", self-published on hal-online (2017)
https://hal.science/hal-01368054

ZH Mei, "B. Feng's Theory:  The Prediction of Mass Spectrum of Elementary Particles and the Confidence of at Least 4-D Space-Time (Part I)", J.Phys.Astron. v.5 p.126 (2017)
available through tsijournals.com  

Nathalie Olivi-Tran, "Theoretical Calculations of the Masses of the Elementary Fermions", chapter 5 in Accelerators and Coliders, IntechOpen (Rijeka, Croatia), editor Ozan Artun (2020) 
doi 10.5772/intechopen.91846





\bibliographystyle{plain}

\bibliography{references}


%\textbf{Appendix A}
\section*{Appendix A}



\begin{table}[!htbp]
%\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{|c|c|c|c|}

\hline 

  Even &Nsx for& Odd &Nsx for \\
    Ns &Even Ns& Ns&	 Odd Ns\\ \hline 
     
     
      4 & 4 & 5 & 4 \\
      6& 4 &7 & 4 \\
      8	 &3.5 ,   4 ** &9 &3.875  ,  4 ** \\  \hline 

    
     10& 3.25 ,  4 &11 &3.75 ,   4 \\
     12&3 ,   4 ‡ & 13 &3.668 ,   4  ‡ \\
     14	& 1.969,   4	  &15	 &2.758 ,   4 \\  \hline   
     
     16& 1. 13 ,   4 &17 & 2.27 ,   4\\
     18& 1.781 ,   4 &19 & 4.024 ,   4 \\
     20	 &3.49 ,   4 & 21 &4.032 ,   4 \\\hline 
     

   
     22&   4 &23 &4 \\
     24&   4 &25 &3.999 † ,  4  \\
     26  &   4 & 27	 &4 \\
     
    


\hline 
\end{tabular}
%\end{adjustbox}
\end{table}
\textbf{Table 5".}  Even Ns and Odd Ns versus Nsx for %New Extended
TRM (2M = $4.076 \times 10^{38}$)

$\ast$$\ast$  The Nsx distributions where  Nsx $=$ 4  for all values of  Ns satisfying  8 $\leq$ Ns $\leq$  16  are very speculative 

      (see tables 14  and 15) while the second Nsx distributions, over this same range of Ns , which are shown in tables 12  and 13  are far less speculative.

$\ddag$  Alternative values for  Nsx  at  Ns $=$ 12  and at Ns $=$ 13  can give smoother plots of  Nsx versus Even Ns and of  Nsx versus Odd Ns.   When Nsx $=$ 2.6 for Ns $=$ 12  a smoother plot results,  giving E\textsubscript{B}  energies  of  3.56 x 10\textsuperscript{4} GeV and  9.9 x 10\textsuperscript{8} GeV.  When Nsx $=$ 3.4 for Ns $=$ 13  a smoother plot results,  giving E\textsubscript{B} energies of  430 GeV and 1.6 x 10\textsuperscript{8} GeV.   Only the 430 GeV energy is likely to be detected at the LHC, and its detection would support the smoothest plots of  Nsx versus Even Ns and Nsx versus Odd Ns.   An E\textsubscript{B} energy of 430 GeV  may be  3 (80.3) + 2 (91.2) $=$ 423 GeV or  2 (80.3) + 3 (91.2)  $=$ 434 GeV  (Is this a Higgs boson? ).  Symmetric decay of E\textsubscript{B} $=$ 434 GeV gives particle energies at 217 GeV and/or 145 GeV.  This note was added on September 10, 2006 and refined on September 21, 2006.   

† Positive energy and negative energy particles associated with Ns $=$ 25  may yield  net cancellation energy  equal to the energy of a graviton when particle energies for Nsx near 3.999  and for Nsx $=$ 4.0  participate in a two-step cancellation as described in section 7  on page 20.

\textbf{Appendix B}

The BASIC language program used to calculate the f\textsubscript{i} (r) values follows:

10 REM  A BASIC program to find f i (r) for f4 (r) through f25 (r) for single radius inputs in units of Planck 20 REM lengths.  The function ln (r) which is a logarithm in base e is written as log (r) in BASIC.

30 REM The radius value 2 x 10\textsuperscript{9} must be input in the form 2e9.  This was written 11/26/01 by M. D. Holte

40 REM Note that the asterisk $\ast$ is a multiplication symbol in BASIC and $\string^$ means raised to the power of

50 PRINT ``WHAT IS THE BUBBLE RADIUS (PLANCK LENGTHS) WITHOUT COMMAS "; : INPUT R

60 TERM1 $=$ 2 $\ast$ (LOG(R) )

70 TERM2 $=$ $-$2 $\ast$ (LOG(R) )$\string^$2

80 TERM3 $=$ (4/3) $\ast$ (LOG(R) )$\string^$3

90 TERM4 $=$ $-$ (2/3) $\ast$ (LOG(R) )$\string^$4

100 TERM5 $=$ (4/15) $\ast$ (LOG(R) )$\string^$5

110 TERM6 $=$ $-$ (4/45) $\ast$ (LOG(R) )$\string^$6

120 TERM7 $=$ (8/315) $\ast$ (LOG(R) )$\string^$7

130 TERM8 $=$ $-$ (2/315) $\ast$ (LOG(R) )$\string^$8

140 TERM9 $=$ (4/2835) $\ast$ (LOG(R) )$\string^$9

150 TERM10 $=$ $-$ (4/14175) $\ast$ (LOG(R) )$\string^$10

160 TERM11 $=$ (8/155925) $\ast$ (LOG(R) )$\string^$11

170 TERM12 $=$ $-$ (4/467775) $\ast$ (LOG(R) )$\string^$12

180 TERM13 $=$ (8/6081075) $\ast$ (LOG(R) )$\string^$13

190 TERM14 $=$ $-$ (16384/8.718E+10) $\ast$ (LOG(R) )$\string^$14

200 TERM15 $=$ (32768/1.3077E+12) $\ast$ (LOG(R) )$\string^$15

210 TERM16 $=$ $-$ (65536/2.0923E+13) $\ast$ (LOG(R) )$\string^$16

220 TERM17 $=$ (131072/3.557E+14) $\ast$ (LOG(R) )$\string^$17

230 TERM18 $=$ $-$ (262144/6.402E+15) $\ast$ (LOG(R) )$\string^$18

240 TERM19 $=$ (524288/1.216E+17) $\ast$ (LOG(R) )$\string^$19

250 TERM20 $=$ $-$ (1048576/2.433E+18) $\ast$ (LOG(R) )$\string^$20

260 TERM21 $=$ (2097152/5.109E+19) $\ast$ (LOG(R) )$\string^$21

270 TERM22 $=$ $-$ (4194304/1.124E+21) $\ast$ (LOG(R) )$\string^$22

280 REM This concludes the term generation.

290 REM Exploring more than 25 spatial dimensions requires additional terms.

300 F4 $=$ ($-$ R/3 ) + TERM1

310 F5 $=$ F4 + TERM2

320 F6 $=$ F5 + TERM3

330 F7 $=$ F6 + TERM4

340 F8 $=$ F7 + TERM5

350 F9 $=$ F8 + TERM6

360 F10 $=$ F9 + TERM7

370 F11 $=$ F10 + TERM8

380 F12 $=$ F11 + TERM9

390 F13 $=$ F12 + TERM10

400 F14 $=$ F13 + TERM11

410 F15 $=$ F14 + TERM 12

420 F16 $=$ F15 + TERM 13

430 F17 $=$ F16 + TERM14

440 F18 $=$ F17 + TERM15

450 F19 $=$ F18 + TERM16

460 F20 $=$ F19 + TERM17

470 F21 $=$ F20 + TERM18

480 F22 $=$ F21 + TERM19

490 F23 $=$ F22 + TERM20

500 F24 $=$ F23 + TERM21

510 F25 $=$ F24 + TERM22

520 REM Output to the computer screen follows.  For printout rewrite 530 – 650 with PRINT $\rightarrow$ LPRINT.  

530 PRINT ``THE SUMMED f i (r) TERMS GIVE THE f i (r) VALUES BELOW"

540 PRINT ``RADIUS $=$ ``; R

550 PRINT ``f4 (r) $=$ ``; F4; : PRINT ``              f5 (r) $=$  ``; F5 : PRINT `` ``

560 PRINT ``f6 (r) $=$ ``; F6; : PRINT ``              f7 (r) $=$  ``;  F7 : PRINT `` ``

570 PRINT ``f8 (r) $=$ ``; F8; : PRINT ``              f9 (r) $=$  ``; F9  : PRINT `` ``

580 PRINT ``f10 (r) $=$ ``; F10; : PRINT ``             f11 (r) $=$  ``; F11 : PRINT `` ``

590 PRINT ``f12 (r) $=$ ``; F12; : PRINT ``             f13 (r) $=$ ``; F13 : PRINT `` ``

600 PRINT ``f14 (r) $=$ ``; F14; : PRINT ``             f15 (r) $=$ ``; F15 : PRINT `` ``

610 PRINT ``f16 (r) $=$ ``; F16; : PRINT ``             f17 (r) $=$ ``; F17 : PRINT `` ``

620 PRINT ``f18 (r) $=$ ``; F18; : PRINT ``            f19 (r) $=$ ``; F19 : PRINT `` ``

630 PRINT ``f20 (r) $=$ ``; F20; : PRINT ``           f21 (r) $=$ ``; F21 : PRINT `` ``

640 PRINT ``f22 (r) $=$ ``; F22; : PRINT ``            f23 (r) $=$ ``; F23 : PRINT `` `` 

650 PRINT ``f24 (r) $=$ ``; F24; : PRINT ``            f25 (r) $=$ ``; F25 

660 END


\end{document}











%
%
%  Editors note.   In preparing for posthumous publication of this manuscript
%  we have removed references to the previous TRM paper.
%  the author's discussion indicates that the extended TRM stands on its own
%  and anyway the previous paper is not readily available.
%
%  An entire chapter was devoted to the comparison, and it is removed to here
%  making it available in the LaTeX source but not in the pdf of the manuscript itself.
%  also, an appendix and a few other bits are moved here or comemented out above.
%
%  A major editorial removal also was applied to the conclusion section
%  again with the original available in the LaTeX source.
%
%

\section{Comparison of Extended TRM to the Previous TRM}

\indent When I developed the previous TRM version in a previous paper,  I thought that the SLOPE parameter would compensate for leaving the value of  2M in units of centimeters. This assumption may be flawed depending on what length scale the metric component influences the radius of fundamental particles which in turn influences both  the TRM predicted masses of fundamental particles and energies of annihilation bosons  per expression 5.28 or 5.28’ on page 11.   In this paper, the extended version of TRM, the value of 2M is 4.o76 x 10$^38$  Planck lengths at 2.2 solar masses, whereas, in the previous TRM model the value of  2M was 652,100 centimeters.  Since 2M/r  is subtracted from one in (1 – 2M/r), making 2M significantly larger by changing units will cause some shifts in transition radius values which will also shift the predicted rest mass of  some particles.  This unprompted reevaluation of the effect of 2M units was begun in late May of 2006.

\indent To see the effect of this change in units four tables are presented on the following pages. The first table for this extended version of TRM where 2M is 4.076 x 10\textsuperscript{38} will compare Nsx, EB , and EB/2 over a range of odd Ns values to the same three values for the previous version where 2M is 652,100  and the values of Nsx are allowed to vary in a way to obtain the best fit of known particle energies.  The second table will compare the same three values  (Nsx, EB , and EB/2) over a range of even Ns values for this extended version of TRM to the previous version of TRM.  The third and fourth tables repeat the first two tables except the value of Nsx  remains fixed at Nsx = 4.  Previous work suggested that there may be particle resonance  between two different Nsx values while  retaining the same particle energy; this second table will allow future analysis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Table 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\centering
\caption{Odd Ns versus Nsx, EB , and $E_{B}$/2 for Extended TRM model where 2M = 4.o76 x $10^{38}$  for columns  after Ns on the left side versus columns of Previous TRM model data for $E_{B}$/2 , $E_{B}$ , and Nsx.}

\begin{tabular}{|c|c|c|c|c|c|c|c|} 
\toprule
\textbf{Ns}                                         & \textbf{Ext *Nsx}                                                          & \textbf{Ext *EB}(GeV)                                                                   & \begin{tabular}[c]{@{}c@{}}\textbf{Ext *} \\\textbf{EB/2}(GeV)\end{tabular}                                 & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{Previous ‡} \\\textbf{EB/2} \\(GeV)\end{tabular}                             & \begin{tabular}[c]{@{}c@{}}\textbf{Previous ‡} \\\textbf{EB} \\(GeV)\end{tabular}                       & \begin{tabular}[c]{@{}c@{}}\textbf{Previous ‡} \\\textbf{Nsx}\end{tabular}  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}5\\7\end{tabular}        & \begin{tabular}[c]{@{}c@{}}4\\4\\4\\4\end{tabular}                         & \begin{tabular}[c]{@{}c@{}}3.71 (13)\\9.47 (16)\\8.50 (9)\\5.55 (15)\end{tabular}       & \begin{tabular}[c]{@{}c@{}}\textbf{1.86 (13)} \\\textbf{4.74 (16)}4.25 (9)\\\textbf{2.78 (15)}\end{tabular} & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{1.90 (13)} \\\textbf{4.71 (16)} \\5.45 (9) \\\textbf{2.98 (15)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}3.80 (13) \\9.42 (16)\\1.09 (10)\\4.96 (15)\end{tabular}                     & \begin{tabular}[c]{@{}c@{}}4 \\4 \\3.83\\3.83\end{tabular}                  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}9 \\11\end{tabular}      & \begin{tabular}[c]{@{}c@{}}3.875  \\3.875 \\3.75\\3.75\end{tabular}        & \begin{tabular}[c]{@{}c@{}}2.94 (7)\\2.69 (13) \\7.21 (4) \\9.79 (10)~~~~~\end{tabular} & \begin{tabular}[c]{@{}c@{}}1.97 (7) \\\textbf{1.35 (13)} \\3.61 (4) \\4.90 (10)\end{tabular}                & ~ & \begin{tabular}[c]{@{}c@{}}1.96 (7)\\\textbf{1.13 (13)}3.67 (4)\\4.88 (10) \\~~~\end{tabular}                   & \begin{tabular}[c]{@{}c@{}}3.91 (7)\\2.26 (13)\\7.33 (4)\\9.76 (10)\end{tabular}                        & \begin{tabular}[c]{@{}c@{}}3.75 \\3.75 \\3.75\\3.75\end{tabular}            \\ 
\hline
\begin{tabular}[c]{@{}c@{}}13 \#  \\15\end{tabular} & \begin{tabular}[c]{@{}c@{}}3.668 \# \\3.668 \# \\2.758\\2.758\end{tabular} & \begin{tabular}[c]{@{}c@{}}241 ~\# \\2.47 (8)~ \#\\2.50\\2.295 (5)\end{tabular}         & \begin{tabular}[c]{@{}c@{}}\textbf{80.3}~ †\\1.24 (8) \\\textbf{1.25} \\1.148 (5)\end{tabular}              & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{80.3}~ †\\1.21 (8) \\\textbf{1.25}1.16 (5)\end{tabular}                      & \begin{tabular}[c]{@{}c@{}}241 \\2.41 (8) \\2.50 \\2.31 (5)\end{tabular}                                & \begin{tabular}[c]{@{}c@{}}3.65 \\3.65 \\2.75\\2.75\end{tabular}            \\ 
\hline
\begin{tabular}[c]{@{}c@{}}17\\19\end{tabular}      & \begin{tabular}[c]{@{}c@{}}2.27\\2.27\\4.024\\4.024\end{tabular}           & \begin{tabular}[c]{@{}c@{}}0.0260 \\182.4 \\2.39 (-6)\\8.41\end{tabular}                & \begin{tabular}[c]{@{}c@{}}\textbf{0.0130} \\\textbf{91.2} \\\textbf{1.20 (-6)4.21}\end{tabular}            & ~ & \begin{tabular}[c]{@{}c@{}}0.0283 \\–~  \\\textbf{1.13 (- 6)4.21 }\end{tabular}                                 & \begin{tabular}[c]{@{}c@{}}0.0565 \\\textbf{91.2 }\textsuperscript{++} \\2.16 (- 6 )\\8.41\end{tabular} & \begin{tabular}[c]{@{}c@{}}1.94 \\1.94 \\4\\4\end{tabular}                  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}21\\23\end{tabular}      & \begin{tabular}[c]{@{}c@{}}4.032 \\4.032 \\4\\4\end{tabular}               & \begin{tabular}[c]{@{}c@{}}4.26 (-9) \\0.0271 \\8.85 (-12)\\8.38 (-5)\end{tabular}      & \begin{tabular}[c]{@{}c@{}}\textbf{2.13 (-9)} \\\textbf{0.0136} \\4.43 (-12) \\4.19 (-5)\end{tabular}       & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{2.13 (- 9)0.0137} \\4.88 (- 11)  \\4.87 (- 6)\end{tabular}                   & \begin{tabular}[c]{@{}c@{}}4.26 ( - 9) \\0.0273\\9.76 (- 11)\\9.74 ( - 6)\end{tabular}                  & \begin{tabular}[c]{@{}c@{}}4 \\4 \\3~ **\\3~ **\end{tabular}                \\ 
\hline
25                                                  & \begin{tabular}[c]{@{}c@{}}3.999 \\3.999~~~\end{tabular}                   & \begin{tabular}[c]{@{}c@{}}2.9434(-14) \\1.719 (-7)\end{tabular}                        & \begin{tabular}[c]{@{}c@{}}1.4717 (-14) \\8.595 (-6)\end{tabular}                                           & ~ & \begin{tabular}[c]{@{}c@{}}1.4617 (-14)\\X\end{tabular}                                                         & \begin{tabular}[c]{@{}c@{}}2.9233 (-14)\\X\end{tabular}                                                 & \begin{tabular}[c]{@{}c@{}}3.999 \\3.999\end{tabular}                       \\
\bottomrule
\end{tabular}
\end{table}

*  Ext  refers to extended TRM for which  2M = 4.076 x 1038  Planck lengths at 2.2 solar masses.
‡  Previous  refers  to  previous TRM for which 2M = 652,100 cm  at 2.2 solar masses.

†  The energies at Ns = 13  above are  241 divided by 3   instead of  241 divided by 2.

** These values should have been changed to Nsx = 4  in the previous version.

++  Previous TRM at Ns = 17 should have fit Previous EB = 182.4  so Previous Nsx = 2.27,  and when

Previous Nsx = 2.27  at Ns = 17  the lower energy gives EB/2 = 0.01295  consistent with tau neutrino.

\#  An alternative Nsx value for Ns = 13  is Nsx = 3.4  which makes a smoother plot of  Nsx versus Odd Ns, giving  EB = 430 GeV (Higgs boson?)  and EB = 1.6 x 108 GeV (see Appendix A table 5”).   

Note : Data above presented in bold type indicate known particle masses or energies.  For Ns = 5 to 9 bold type particle energies match expected X boson energies.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\centering
\caption{Even Ns vs.  Extended  and Previous  Nsx, $E_{B}$ , and $E_{B}$/2  (like  Table 12,  except Ns is even)}
\begin{tabular}{|c|c|c|c|c|c|c|c|} 
\toprule
\textbf{Ns}                                      & \textbf{Ext *Nsx}                                                  & \begin{tabular}[c]{@{}c@{}}\textbf{Ext *} \\\textbf{EB}(GeV)\end{tabular}            & \begin{tabular}[c]{@{}c@{}}\textbf{Ext *} \\\textbf{EB/2}(GeV)\end{tabular}                   & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{Previous ‡} \\\textbf{EB/2}(GeV)\textbf{~ }\end{tabular}      & \begin{tabular}[c]{@{}c@{}}\textbf{Previous ‡} \\\textbf{EB} \\(GeV)\end{tabular}  & \begin{tabular}[c]{@{}c@{}}\textbf{Previous ‡} \\\textbf{Nsx}\end{tabular}  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}8\\10\end{tabular}    & \begin{tabular}[c]{@{}c@{}}3.5 \\3.5\\3.25\\3.25\end{tabular}      & \begin{tabular}[c]{@{}c@{}}1.05 (9)\\3.27 (14)\\3.66 (6)\\9.10 (11)\end{tabular}     & \begin{tabular}[c]{@{}c@{}}5.03 (8) \\\textbf{1.64 (14)}1.83 (6)\\4.55 (11)\end{tabular}      & ~ & \begin{tabular}[c]{@{}c@{}}5.30 (8)\\\textbf{1.63 (14)}2.72 (6)\\3.51 (8)\end{tabular}           & \begin{tabular}[c]{@{}c@{}}1.06 (9)\\3.25 (14)\\5.43 (6)\\7.02 (8)\end{tabular}    & \begin{tabular}[c]{@{}c@{}}3.5\\3.5 \\3~ †\\3~ †\end{tabular}               \\ 
\hline
\begin{tabular}[c]{@{}c@{}}12  \\14\end{tabular} & \begin{tabular}[c]{@{}c@{}}3 ~**\\3~ **\\1.969\\1.969\end{tabular} & \begin{tabular}[c]{@{}c@{}}1.96 (4) ** \\1.60 (9) ** \\160.4\\1.705 (6)\end{tabular} & \begin{tabular}[c]{@{}c@{}}9800 \\8.00 (8) \\\textbf{80.2~~}8.53 (5)~~\end{tabular}           & ~ & \begin{tabular}[c]{@{}c@{}}1.00 (4)\\7.85 (8)\\30.3\\1.88 (6)\end{tabular}                       & \begin{tabular}[c]{@{}c@{}}1.99 (4) \\1.57 (9)\\60.6\\3.76 (6)\end{tabular}        & \begin{tabular}[c]{@{}c@{}}3 \\3 \\3~ †\\3~ †\end{tabular}                  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}16\\18\end{tabular}   & \begin{tabular}[c]{@{}c@{}}1.13 \\1.13\\1.781\\1.781~\end{tabular} & \begin{tabular}[c]{@{}c@{}}6.89 \\351 \\3.76 (-3) \\3.60\end{tabular}                & \begin{tabular}[c]{@{}c@{}}3.45\\\textbf{175.5} \\\textbf{1.88 (-3)1.80}\end{tabular}         & ~ & \begin{tabular}[c]{@{}c@{}}3.53 \\\textbf{175.3} \\\textbf{1.88 (- 3)1.80}\end{tabular}          & \begin{tabular}[c]{@{}c@{}}7.06 \\350.5 \\3.76 ( - 3)\\3.60\end{tabular}           & \begin{tabular}[c]{@{}c@{}}1.1\\1.1\\1.78\\1.78\end{tabular}                \\ 
\hline
\begin{tabular}[c]{@{}c@{}}20\\22\end{tabular}   & \begin{tabular}[c]{@{}c@{}}3.49 \\3.49 \\4\\4\end{tabular}         & \begin{tabular}[c]{@{}c@{}}2.55 (-7) \\0.212 \\3.13 (-10)\\0.00102\end{tabular}      & \begin{tabular}[c]{@{}c@{}}1.28 (-7)\\\textbf{0.106}1.57 (-10)\\\textbf{0.00051}\end{tabular} & ~ & \begin{tabular}[c]{@{}c@{}}1.31 (- 7) \\\textbf{0.106}1.70 (- 10)\\\textbf{0.00051}\end{tabular} & \begin{tabular}[c]{@{}c@{}}2.61 (- 7) \\0.212 \\3.39 (- 10) \\0.00102\end{tabular} & \begin{tabular}[c]{@{}c@{}}3.5 \\3.5 \\4\\4\end{tabular}                    \\ 
\hline
24                                               & \begin{tabular}[c]{@{}c@{}}4\\4\end{tabular}                       & \begin{tabular}[c]{@{}c@{}}1.16 (-13)\\1.32 (-5)\end{tabular}                        & \begin{tabular}[c]{@{}c@{}}5.80 (-14)\\6.60 (-6)\end{tabular}                                 & ~ & \begin{tabular}[c]{@{}c@{}}1.78 ( - 11)\\4.88 (- 8)\end{tabular}                                 & \begin{tabular}[c]{@{}c@{}}3.55 (- 11)\\9.76 (- 8)\end{tabular}                    & \begin{tabular}[c]{@{}c@{}}3.98\\3.98\end{tabular}                          \\
\bottomrule
\end{tabular}
\end{table}
*  Ext  refers to extended TRM for which  2M = 4.076 x 1038  Planck lengths at 2.2 solar masses.

‡  Previous  refers  to  previous TRM for which 2M = 652,100 cm  at 2.2 solar masses.

†  The energies at Ns = 13  above are  241 divided by 3   instead of  241 divided by 2.

** These values should have been changed to Nsx = 4  in the previous version.
++  Previous TRM at Ns = 17 should have fit Previous EB = 182.4  so Previous Nsx = 2.27,  and when

Previous Nsx = 2.27  at Ns = 17  the lower energy gives EB/2 = 0.01295  consistent with tau neutrino.

\#  An alternative Nsx value for Ns = 13  is Nsx = 3.4  which makes a smoother plot of  Nsx versus Odd Ns, giving  EB = 430 GeV (Higgs boson?)  and EB = 1.6 x 108 GeV (see Appendix A table 5”)


Note : Data above presented in bold type indicate known particle masses or energies.  For Ns = 5 to 9 bold type particle energies match expected X boson energies

\indent Before analyzing the data, I will redefine the parameters tabulated in this section.  The left column of each table is Ns which is the number of spatial dimensions which ranges from 4 to 25, but only  5   to 25   is  presented  because  Ns $=$ 4  and Nsx $=$ 6  have no unique transition radii.   Next, Nsx is a fit parameter determined by adding one to the number of radius decades spanned by the parameter SX when the largest consecutive  SX values are averaged to determine a SLOPE value.  Finally, E\textsubscript{B} is the annihilation energy most often associated with two particles of the same type, one matter and one antimatter, which annihilate; however, in some cases the number of particles that E\textsubscript{B} refers to is only one or sometimes more than two.

\indent Now let’s look at tables 12 and 13 on the previous two pages.  The Extended E\textsubscript{B}/2  values in these tables match the energies of six quark types, six lepton types, five or six X boson types, the W bosons, and the Z boson.  In each table the E\textsubscript{B}/2  values which match known (or likely to be found) particle masses or energies are highlighted by bold type.

\indent Comparing Extended E\textsubscript{B}/2  to Previous E\textsubscript{B}/2  in table 12,  we see that Ext E\textsubscript{B}/2  fits the tau neutrino energy at Ns $=$ 17 and Nsx $=$ 2.27,  whereas, the Previous E\textsubscript{B}/2  column of data has no energies below 0.018 GeV close enough to the tau neutrino, but that mismatch occurred because the energy of the Z boson was erroneously fit to E\textsubscript{B} $=$ 91.2 GeV for the Previous TRM instead of  E\textsubscript{B} $=$ 182.4 GeV.   As the notes below each table indicate,  the Ext data columns are for 2M $=$ 4.076 x 10\textsuperscript{38}  at 2.2 solar masses and the Previous data columns are for 2M $=$ 652,100  at 2.2 solar masses.

\indent In tables 13 we also see that for Ns $=$ 14  and Nsx $=$ 1.969  the Ext E\textsubscript{B}/2 value is 80.2 GeV.  However,  when Nsx $=$ 2 at Ns $=$ 14 the value of E\textsubscript{B} $=$ 147 GeV which might be a Higgs particle.

\indent In table 12 the largest percentage change in Nsx comparing Extended to Previous columns,  when E\textsubscript{B}/2 values are the same for both columns (not counting Ns $=$ 17 which was intentionally changed to improve the smoothness of the Ns versus Nsx distribution)  is 3.3 percent for Ns $=$ 9. This suggests there is only a slight shift in the Ns versus Nsx distribution due to changing 2M from 652,100 to 4.076 x 10\textsuperscript{38}.

\indent For table 13 the largest percentage change in Nsx comparing Extended to Previous, when excluding Ns $=$ 10  and Ns $=$ 14,  is 2.7 percent for Ns $=$ 16.  Again, the shift in Ns versus Nsx distribution when 2M units are changed is minor.

\indent Percentage change in E\textsubscript{B}/2 when Nsx is fixed at  Nsx $=$ 4  can be  determined  for  tables 14  and 15  below.  The largest percentage change in E\textsubscript{B}/2  is 83.3 percent for Ns $=$ 24.   The   second   largest percentage change in E\textsubscript{B}/2  is 27.4 percent for   Ns $=$ 23.  The remaining E\textsubscript{B}/2  values change less than 15 percent when comparing Extended to Previous.

\indent In the Previous TRM version there was a limited indication that particles may resonate between two Nsx values, without a change of energy,  where one of these two Nsx values is very close to Nsx $=$ 4.  I see this trend again in tables 14 and 15  for Ns $=$ 17,  Ns $=$ 18,  Ns $=$ 20,  Ns $=$ 21 and Ns $=$ 22.  As suggested in the Previous TRM version,  $\Delta$ Nsx $=$ 4 – Nsx  tracks many of the particle masses fit by  extended TRM.

\indent If new particle masses are discovered by the LHC,  this extended version of TRM predicts a pattern of particle masses when it is assumed that Nsx $=$ 4  for most Ns values.  From table 14  Ext E\textsubscript{B}  for Ns $=$ 17  is  2766  GeV.  Particles can decay  either symmetrically where all the decay particles have identical energies or particles can decay asymmetrically where decay particles have different energies.  It is difficult to predict the decay energies resulting from asymmetric decay so I will limit this analysis to symmetric decay. 

\indent If we assume symmetric decay for the Ext E\textsubscript{B}  of 2766 GeV, this assumption gives the following decay energies:  346 GeV, 461 GeV, 692 GeV, 922 GeV, and 1383 GeV.  From table 15  at Ns $=$ 12  Ext E\textsubscript{B}  is 3093 GeV which gives the following symmetric decay energies:  387 GeV, 516 GeV, 773 GeV, 1031 GeV, and 1547 GeV.  Combining these two symmetric decay energy patterns gives the following pattern of  particle energies:  346 GeV, 387 GeV, 461 GeV, 516 GeV, 692 GeV, 773 GeV, 922 GeV, 1031 GeV, 1383  GeV, and 1547 GeV.  Extended TRM predicted particle masses (or energies) may not all be detectable since some of these particles may be noninteracting like dark matter.  This pattern of particle masses is predicted assuming that Nsx $=$ 4  for all Ns values is a valid Ns versus Nsx distribution, and it assumes symmetric particle decay.

\indent A different pattern of particle masses,  due to symmetric decay,  resulted for Previous E\textsubscript{B}  values of 2820 GeV  at Ns $=$ 11  and  2980 GeV at Ns $=$ 17.  The combined predicted symmetric decay particle energies for Previous TRM are:  350 GeV, 373 GeV, 470 GeV, 497 GeV, 705 GeV, 745 GeV, 940 GeV, 993 GeV, 1410 GeV, and 1490 GeV.  These Previous TRM version predicted symmetric decay particle masses (or energies) may not all be detectable since some may be noninteracting.

\indent Results of LHC experiments may match either the Extended TRM predicted pattern or the Previous TRM predicted pattern more closely so such a match is one possible  way to determine the best value for 2M. 

\indent Trying to predict the pattern of particle masses  resulting from asymmetric decay of  the  energies 2766 and 3093 GeV  or  2820 and 2980 GeV  will be more difficult.  A good assumption to make about asymmetric decay might be that a maximum  number of known particle masses (or energies) will result during a decay so that a minimum  number of  new particle types result.  This is a good starting assumption because it would be very difficult to explain a huge number of  new particle types that have not been detected.
\indent I will not add such an analysis to this paper.  This speculation is mentioned to note that other asymmetric decay particle masses (or energies) may be detected during LHC experiments which are different from those predicted above.

\indent Finally,  I will examine  energies  at Ns $=$ 25  to see  if  it is possible for the graviton energy to be explained within a model having 25 spatial dimensions.   To test this possibility of fitting a graviton’s energy the particle energies at Ns $=$ 25 for Nsx $=$ 3.999  and  for  Nsx $=$ 4  in tables 12 and 14 are  determined to see if  particle energy cancellation  is  a way to explain the extremely low energy per graviton.

\indent From table 12,  Ext E\textsubscript{B}/2  at Ns $=$ 25  and  Nsx $=$ 3.999  is 1.4717 x 10\textsuperscript{-14} GeV  or 1.4717 x 10\textsuperscript{-5}  eV.   From table 14  below,  Ext E\textsubscript{B}/2  at Ns $=$ 25 and Nsx $=$ 4  is 1.4682 x 10\textsuperscript{-14}  GeV  or 1.4682 x 10\textsuperscript{-5} eV.  If each of these two particle energies can exist as both positive energy particles and negative energy particles which can cancel each other out to leave a residual energy, then the residual energy particle that results will be either  (1.4717 – 1.4682) x 10\textsuperscript{-5} eV which is    +3.5 x 10\textsuperscript{-8} eV  or  (1.4682 – 1.4717) x 10\textsuperscript{-5} eV or   – 3.5 x 10\textsuperscript{-8} eV.  One might assume that these residual particles do not interactive with matter and are very difficult to detect. If these two residual energy particles were to combine in a second step cancellation the final residual energy would be zero, assuming these two residual energies exhibit no slight asymmetry in energy.  Another possibility is that there is a slight energy asymmetry between these two residual particles which only occurs for particles of very low energy; this slight energy asymmetry is different from matter/antimatter asymmetry because matter/antimatter asymmetry is a particle count asymmetry, whereas, this slight energy asymmetry is an asymmetry in residual particle energy between positive energy and negative energy particles of the same type.   Let’s assume that this slight energy asymmetry is such that each + 3.5 x 10\textsuperscript{-8} eV particle has about one part per trillion more energy than the – 3.5 x 10\textsuperscript{-8} eV particle.  

\indent In this case, the  two-step residual energy after a second step of particle cancellation would be:

%Maximize: 5.00 x_{1} + 7.50 x_{2} \\\\ 
%Subject\ to: \\\\
%\begin{align}
%x_{1} \leq& 200 \\\\
%x_{2} \leq& 300 \\\\
%1.0x_{1} + 1.5x_{2}\leq& 650 \\\\
%x_{1}, x_{2} \geq& 0
%\end{align}

\begin{equation}
             + 3.5000 000 000 018  x 10\textsuperscript{-8}  eV
\\             –  3.4999 999 999 982  x 10\textsuperscript{-8}  eV           
\\ \textsuperscript{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}
\\                 0.0000 000 000 036  x 10\textsuperscript{-8}  eV
\end{equation}

\indent This  second step residual is 3.6 x 10\textsuperscript{-12} x 10\textsuperscript{-8}  or  3.6 x 10\textsuperscript{-20} eV, and this energy per particle is very close to the energy expected for a graviton. And we can assume that only this two-step residual is able to interact with matter. However, an explanation is needed for this bizarre energy asymmetry,  and  there is a need to accept the existence of negative energy particles capable of interacting with positive energy particles at energies below that of neutrino energy to make this mechanism plausible.  Such a two-step particle cancellation mechanism is highly speculative.

\indent An interesting side analysis would be to use the best estimates of graviton energy from other theories to determine the transition radius of a graviton from expression 5.28 on page 11,  then compare this radius to the size of a proton, an up quark,  and a down quark.   Using the graviton energy shown above gives a transition radius of   7 x 10\textsuperscript{15} Planck lengths or  1.1 x 10\textsuperscript{-17} centimeters.

\indent There are other excellent explanations for the extremely low gravitational effect resulting from graviton interactions, and the above two-step cancellation mechanism may work in conjunction with other mechanisms.  For example, Lisa Randall, in her recent book, explains the weakness of gravitational force as being due to gravitons having  ``no ends to pin down on a brane" [so they can exit our brane with ease because] ``there is no way to confine gravity to lower dimensions." \cite[p.327]{randall2005warped}   particle cancellation mechanism I have suggested above has nothing to do with the extremely low gravitational effect resulting from graviton interactions with matter, making merger of extended TRM with another theory that explains weak  gravity  necessary.

\indent Extended TRM does not need to explain the weakness of gravity if extended TRM can explain a unique pattern of particle energies in conjunction with another theory which explains the weakness of gravity.   The advantage of extended TRM (or TRM) is that it explains the pattern of particle masses (and energies) using extra dimensions in a way that is consistent with general relativity, allowing the prediction of new particle energies.  Extended TRM should not be rejected if  extended TRM on its own presently falls short of giving a definitive explanation for the weakness of gravity.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 14 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering
\caption{Odd Ns versus Extended  and Previous Nsx, E$_{B}$ , and E$_{B}$/2  (like Table 12 , except Nsx = 4)}
\begin{tabular}{|c|c|c|c|c|c|c|c|} 
\toprule
\textbf{Ns}                                     & \textbf{Ext *Nsx}                                          & \textbf{Ext *EB}(GeV)                                                              & \begin{tabular}[c]{@{}c@{}}\textbf{Ext *} \\\textbf{EB/2}(GeV)\end{tabular}                                    & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{Previous ‡} \\\textbf{EB/2}(GeV)\end{tabular}                        & \begin{tabular}[c]{@{}c@{}}\textbf{Previous ‡} \\\textbf{EB}(GeV)\end{tabular}            & \textbf{Previous ‡Nsx}                                      \\ 
\hline
\begin{tabular}[c]{@{}c@{}}5\\7\end{tabular}    & \begin{tabular}[c]{@{}c@{}}4 \\4~~  \\4\\4\end{tabular}    & \begin{tabular}[c]{@{}c@{}}3.71 (13)\\9.47 (16)\\8.50 (9)\\5.55 (15)\end{tabular}  & \begin{tabular}[c]{@{}c@{}}\textbf{1.86 (13)} \\\textbf{4.74 (16)} \\4.25 (9)\\\textbf{2.78 (15)}\end{tabular} & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{1.90 (13)4.71 (16)} \\4.23 (9)\\\textbf{2.78 (15)}\end{tabular}      & \begin{tabular}[c]{@{}c@{}}3.80 (13) \\9.42 (16) \\8.46 (9)\\5.55 (15)\end{tabular}       & \begin{tabular}[c]{@{}c@{}}4\\4 \\4\\4\end{tabular}         \\ 
\hline
\begin{tabular}[c]{@{}c@{}}9 \\11\end{tabular}  & \begin{tabular}[c]{@{}c@{}}4\\4\\4\\4\end{tabular}         & \begin{tabular}[c]{@{}c@{}}~2.28 (7)\\3.15 (13)\\4.06 (4)\\1.44 (11)\end{tabular}  & \begin{tabular}[c]{@{}c@{}}1.14 (7)\\\textbf{1.58 (13)} \\2.03 (4) \\7.20 (10)\end{tabular}                    & ~ & \begin{tabular}[c]{@{}c@{}}1.15 (7)\\\textbf{1.56 (13)} \\2.08 (4)\\7.15 (10)\end{tabular}              & \begin{tabular}[c]{@{}c@{}}2.30 (7) \\3.11 (13) \\4.15 (4)\\1.43 (11)\end{tabular}        & \begin{tabular}[c]{@{}c@{}}4 \\4 \\4 \\4\end{tabular}       \\ 
\hline
\begin{tabular}[c]{@{}c@{}}13\\15\end{tabular}  & \begin{tabular}[c]{@{}c@{}}4\\4\\4\\4\end{tabular}         & \begin{tabular}[c]{@{}c@{}}132 \\3.83 (8) \\0.383\\1.00 (6)\end{tabular}           & \begin{tabular}[c]{@{}c@{}}66.0\\1.92 (8)\\0.192\\5.00 (5)\end{tabular}                                        & ~ & \begin{tabular}[c]{@{}c@{}}65.5 \\1.92 (8)\\0.190\\5.01 (5)\end{tabular}                                & \begin{tabular}[c]{@{}c@{}}131 \\3.83 (8) \\0.379\\1.02 (6)\end{tabular}                  & \begin{tabular}[c]{@{}c@{}}4\\4\\4\\4\end{tabular}          \\ 
\hline
\begin{tabular}[c]{@{}c@{}}17\\19\end{tabular}  & \begin{tabular}[c]{@{}c@{}}3.978~~  \\4\\4\\4\end{tabular} & \begin{tabular}[c]{@{}c@{}}0.00102\\2766\\2.25 (-6)\\8.11\end{tabular}             & \begin{tabular}[c]{@{}c@{}}\textbf{0.00051} \\1383\\\textbf{1.13 (-6)}4.06\end{tabular}                        & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{0.00051}1410 \\\textbf{1.08 (-6)}4.23\end{tabular}                   & \begin{tabular}[c]{@{}c@{}}0.00102 \\2820\\2.16 (- 6)\\8.46\end{tabular}                  & \begin{tabular}[c]{@{}c@{}}3.977~~  \\4\\4\\4\end{tabular}  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}21 \\23\end{tabular} & \begin{tabular}[c]{@{}c@{}}4\\4\\4\\4\end{tabular}         & \begin{tabular}[c]{@{}c@{}}4.49 (-9) \\0.0260 \\8.85 (-12)\\8.38 (-5)\end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{2.25 (-9)0.0130} \\4.43 (-12)\\4.19 (-5)\end{tabular}                       & ~ & \begin{tabular}[c]{@{}c@{}}\textbf{2.13 (- 9)} \\\textbf{0.0137} \\6.10 (- 12)\\4.42 (- 5)\end{tabular} & \begin{tabular}[c]{@{}c@{}}4.26 (- 9) \\0.0273 \\1.22 (- 11)\\8.83 (- 5)~ ~~\end{tabular} & \begin{tabular}[c]{@{}c@{}}4\\4\\4\\4\end{tabular}          \\ 
\hline
25                                              & \begin{tabular}[c]{@{}c@{}}4\\4\end{tabular}               & \begin{tabular}[c]{@{}c@{}}2.9363(-14)\\1.723 (-7)\end{tabular}                    & \begin{tabular}[c]{@{}c@{}}1.4682(-14)\\8.62 (-8)\end{tabular}                                                 & ~ & \begin{tabular}[c]{@{}c@{}}1.45756 (-14)\\X\end{tabular}                                                & \begin{tabular}[c]{@{}c@{}}2.9151 1 (-14)\\X~ ~~\end{tabular}                             & \begin{tabular}[c]{@{}c@{}}4\\4\end{tabular}                \\
\bottomrule
\end{tabular}
\end{table}

*  Ext  refers  to extended TRM for which  2M = 4.076 x 1038  Planck lengths at 2.2 solar masses.

‡  Previous  refers  to  previous TRM for which 2M = 652,100 cm  at 2.2 solar masses.
Note :  Data above presented in bold type indicate known particle masses or energies.

For Ns = 5 to 9 the 
             bold type particle energies match expected X boson energies.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table 15 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
\centering
\caption{Even Ns versus Extended  and Previous Nsx, $E_{B}$ , and $E_{B}$/2  (like Table 13 , except Nsx = 4)}
\begin{tabular}{|c|c|c|c|c|c|c|c|} 
\toprule
\textbf{Ns}                                       & \textbf{Ext *Nsx}                                    & \textbf{Ext *EB}(GeV)                                                               & \textbf{Ext *EB/2}(GeV)                                                                          & ~ & \textbf{Previous ‡EB/2}(GeV)\textbf{~ }                                                                 & \textbf{Previous ‡EB}(GeV)                                                               & \textbf{Previous ‡Nsx}                                \\ 
\hline
\begin{tabular}[c]{@{}c@{}}8\\\\10\end{tabular}   & \begin{tabular}[c]{@{}c@{}}4\\4\\\\4\\4\end{tabular} & \begin{tabular}[c]{@{}c@{}}4.42 (8) \\5.38 (14)\\\\1.24 (6)\\1.86 (12)\end{tabular} & \begin{tabular}[c]{@{}c@{}}2.21 (8)\\\textbf{2.69 (14)} \\6.20 (5) \\9.30 (11)\end{tabular}      & ~ & \begin{tabular}[c]{@{}c@{}}2.03 (8)\\\textbf{3.07 (14)}\\\textbf{}6.05 (5)\\8.90 (11)\end{tabular}      & \begin{tabular}[c]{@{}c@{}}4.06 (8) \\6.13 (14)\\\\1.21 (6) \\1.78 (12)\end{tabular}     & \begin{tabular}[c]{@{}c@{}}4\\4\\\\4\\4\end{tabular}  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}12\\14\end{tabular}    & \begin{tabular}[c]{@{}c@{}}4\\4\\\\4\\4\end{tabular} & \begin{tabular}[c]{@{}c@{}}3093 \\6.25 (9) \\\\6.56\\2.11 (7)~~\end{tabular}        & \begin{tabular}[c]{@{}c@{}}1547 \\3.13 (9) \\\\3.28\\1.06 (7)\end{tabular}                       & ~ & \begin{tabular}[c]{@{}c@{}}1490 \\3.13 (9) \\\\3.28 \\1.05 (7)\end{tabular}                             & \begin{tabular}[c]{@{}c@{}}2980 \\6.25 (9) \\\\6.56 \\2.10 (7)\end{tabular}              & \begin{tabular}[c]{@{}c@{}}4\\4\\\\4\\4\end{tabular}  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}16 \\\\18\end{tabular} & \begin{tabular}[c]{@{}c@{}}4\\4\\\\4\\4\end{tabular} & \begin{tabular}[c]{@{}c@{}}0.0122 \\7.58 (4) \\\\3.70 (-5)\\182.3\end{tabular}      & \begin{tabular}[c]{@{}c@{}}0.0061 \\3.79 (4) \\\\1.85 (-5) \\\textbf{91.2}\end{tabular}          & ~ & \begin{tabular}[c]{@{}c@{}}0.0061 \\3.94 (4) \\\\0.00019 \\86 **\end{tabular}                           & \begin{tabular}[c]{@{}c@{}}0.0122 \\7.88 (4) \\\\0.00038 \\\textbf{172}~ **\end{tabular} & \begin{tabular}[c]{@{}c@{}}4\\4\\\\4\\4\end{tabular}  \\ 
\hline
\begin{tabular}[c]{@{}c@{}}20 \\\\22\end{tabular} & \begin{tabular}[c]{@{}c@{}}4\\4\\\\4\\4\end{tabular} & \begin{tabular}[c]{@{}c@{}}1.13 (-7) \\0.418 \\\\3.13 (-10)\\0.00102\end{tabular}   & \begin{tabular}[c]{@{}c@{}}5.65 (-8) \\\textbf{0.209}\\1.57 (-10)\\\textbf{0.00051}\end{tabular} & ~ & \begin{tabular}[c]{@{}c@{}}5.35 (- 7) \\\textbf{0.220} \\1.70 (- 10) \\\textbf{0.00051 ~~}\end{tabular} & \begin{tabular}[c]{@{}c@{}}1.07 (- 7) \\0.440 \\\\3.40 (- 10) \\0.00102~\end{tabular}    & \begin{tabular}[c]{@{}c@{}}4\\4\\\\4\\4\end{tabular}  \\ 
\hline
24                                                & \begin{tabular}[c]{@{}c@{}}4\\4\end{tabular}         & \begin{tabular}[c]{@{}c@{}}1.16 (-13) \\1.32 (-5)\end{tabular}                      & \begin{tabular}[c]{@{}c@{}}5.80 (-14) \\6.60 (-6)\end{tabular}                                   & ~ & \begin{tabular}[c]{@{}c@{}}3.47 (-13)\\1.41 (-6)\end{tabular}                                           & \begin{tabular}[c]{@{}c@{}}6.94 (-13) \\2.82 (-6)~~  \\~~~~~~~\end{tabular}              & \begin{tabular}[c]{@{}c@{}}4\\4\end{tabular}          \\
\bottomrule
\end{tabular}
\end{table}
*  Ext  refers  to extended TRM for which  2M = 4.076 x 1038  Planck lengths at 2.2 solar masses.
‡  Previous  refers  to  previous TRM for which 2M = 652,100 cm  at 2.2 solar masses.
Note :  Data above presented in bold type indicate known particle masses or energies.  For Ns = 5 to 9 the bold type particle energies match expected X boson energies.



\textbf{Appendix C}

       In column two on page 24 possible values for Nb were derived.  For that analysis I assumed that all 39 transition radii have only positive energy  boson types for each transition radius value or only one possible sign per boson energy.  If this assumption is changed, it is possible to get much lower maximums for the number of positive energy boson types, and it is possible to get much higher maximums.  The most obvious change to consider is the possibility that only the odd numbers of spatial dimensions give positive energy boson types and only the even numbers of spatial dimensions give negative energy boson types.  If we also assume that each boson type can result from both a single transition radius and from combining the energies of transition radii in pairs, assuming probability limits three boson types combining, then we can find the new lower maximums.  Ignoring bosons that may be composites of three or more boson types is likely a flawed assumption since the $\tau$-neutrino/anti-$\tau$-neutrino annihilation boson appears to be an exception so the following Nb estimates may  slightly underestimate actual values.  Making these new assumptions gives [(21)(20)/2] + 21  or 231 positive energy boson types for 21 transition radii and [(18)(17)/2] + 18  or 171 negative energy boson types for 18 transition radii when positive – positive and negative – negative are the only pairings allowed.   So for this case Nb $=$ 231 + 171 + 1  $=$ 403  versus the Nb $=$ 781 value found previously.  One is added to account for the boson with an infinite number of spatial dimensions.

\indent It is also possible that negative energy boson types also exist for each of the 39 transition radii so the total number of boson types, positive energy plus negative energy, is 78.  In this case,  the value of Nb should be Nb $=$ [78(77)/2] - 39 + 78 + 2   or Nb $=$ 3044  (two is added to account for boson with infinite number of dimensions and its negative energy counterpart) .  Of course, if  negative energy bosons can not pair with any other boson, then Nb $=$ 403 or Nb $=$ 781  (see page 24).  If no pairing of boson types of any kind occurs in nature (which is unlikely),  then the lowest value of Nb possible is 2(39) + 1  or 79. 

\indent One of the two reasons we should observe fewer bosons types than suggested above is because totally negative energy boson types are likely unobservable.  The number of positive energy boson types, Nbp, that we observe is given by Nbp $=$ Nb – Nbn where Nbn is the number of totally negative energy boson types. This expression for Nbp gives Nbp $=$ 403 – 171 $=$ 232  when Nb $=$ 403,  gives Nbp $=$  3044 – 781 $=$ 2263  when Nb $=$ 3044,  and gives Nbp $=$  40  when Nb $=$ 79.  I have not decided which Nbp value is most likely.     

\indent The second reason we should observe fewer boson types, so that the number of observable positive energy boson types (Nobs) will always satisfy Nobs < Nb and may also satisfy Nobs < Nbp, is because the energy difference between some combining boson types will be so large that the energy of the combined boson will be indistinguishable from the energy of  the higher energy boson of the initial pair.   Many of these particle types may not interact to form composite particles due to other factors which prevent interactions.  

\indent Asymmetric decay of high mass particles may add to the complexity of the observed pattern of masses.
