Genesis of the Reciprocal Fine Structure Constant

This article presents hypotheses about space, time, matter and energy that could function beneath the theories of General Relativity and Quantum mechanics.  Based on these hypotheses, several critical dimensionless numbers of physics/mathematics are predicted from the first twelve characters of the first verse of Genesis.  For the purpose of illustration, the Reciprocal Fine Structure Constant (α-1 = 137.035999) is given particular attention.

 

Introduction

In this paper, ten critical dimensionless numbers of physics are predicted and functionally related by a simple 12 by 12 matrix comprising the elements 0, 1 and 2.  This matrix is based on the first twelve Hebrew characters from the first verse of Genesis.  The Reciprocal Fine Structure Constant, the electron-proton electrical to gravitational force ratio, and the fraction of mass converted to energy in stars, for example, can be accurately predicted starting from this 12 by 12 matrix.  Similarly, the dimensionless numbers pi and phi are encrypted in a 3 by 3 matrix based on the first three characters of the first verse of Genesis.  All these findings could represent a completely serendipitous concurrence of events.  Alternately, they could suggest a causal connection between God and the critical dimensionless numbers of science and mathematics. 

The Reciprocal Fine Structure Constant is given particular attention because attempts to predict it, based on the principles of physics, have failed for a hundred years.  Attempts at prediction, based on mathematics, have also failed.  This paper presents an eclectic explanation based on a syncretism of science, mathematics and the Hebrew Bible.

Background

Imagine the universe is constructed around a deformable cubic lattice such that the smallest identifiable physical entity is a region called a “voxel” which is bounded by eight vertices, twelve edges and six flat faces.  The universe is thereby divided into an ordered (numbered) array of volumes.  The volume enclosed by a voxel is on the order of the cube of the local Planck Length (e.g. 1.6 x10-35 meter).  Assume the state of each voxel is 1 or 0.  Then the state of the entire universe is defined by a countably infinite sequence of binary digits or, in other words, a single real number.  The possible 3-D geometric arrangements of 1’s and 0’s would be more than sufficient to accommodate the complexity of any “Theory of Everything” that might be devised.  Now imagine the state of each voxel changes at time intervals equal to the Planck Time (5.4 x 10-44 seconds).  At the end of each time interval, a new real number defines the state of the universe.  By this view, the voxels may be deformed somewhat by matter and energy but are essentially fixed in space.  They simply assume a new identity at the end of each time step much like the pixels on a television screen.  This concept comprises a “pixelated” universe as suggested by Quantum Mechanics.

This hypothetical exercise may seem far-fetched given the equations of Continuum Mechanics and General Relativity which treat space and time as continuums.  But voxels are merely the three dimensional cellular automata defined in A New Kind of Science (Wolfram, 2002, 179). We only need a “rule of transformation” for defining what takes place at each time step.  Concepts of “state” and “rule of time transformation,” more complex than those governing the 256 elementary cellular automata, should certainly be considered.  Some concepts might be inspired by certain aspects of Judeo-Christian tradition.  For example, one tradition is that the 304,805 character string of concatenated words in the Torah encodes fundamental information about the beginning of the universe and its development over time as well as events in human history (Satinover, 1998; Drosnin, 1998).

Genetic Code for the Universe?

Perhaps relatively short strings of Hebrew characters from the first chapter of Genesis provide a shorthand for codons undergirding all space, time, matter and energy.  Various arrangements of these codons could provide the genetic microstructure and macro-characteristics of magnetism, electricity, normal matter, normal energy, dark matter, dark energy, electromagnetic force, gravitational force, strong force and weak force.  However, these codons do not reveal their secrets when read merely as Hebrew character sequences.  They must be viewed as mathematical entities as illustrated by the following.

First invert the character sequence, in the first chapter of Genesis, so the characters are read from left to right.  Next connect the first three contiguous characters in the first verse of Genesis to form a string.  Now represent each of the Hebrew characters by a base-3 triplet (column vector) according to a rule that starts with Aleph as (000) and ends with Tsadey Final as (222).  The three column vectors are (001,201,000).  These three base-3 triplets create a 3 by 3 matrix having the elements 0, 1 and 2.  Such groups of three Hebrew characters and the associated 3 by 3 matrix could represent fundamental codons for the structure of the universe.

Increasing complexity is created if groups of three contiguous characters are connected to one another.  For example, four groups of three contiguous characters could be arranged like tiles on a floor to produce a 6 by 6 matrix.  Continuing in this fashion, the individual 3 by 3 matrices from 1, 4, 9, 16, 25, 36, 49 ... groups of three contiguous characters could be arranged to form square matrices of dimension 3, 6, 9, 12, 15, 18, 21 … respectively.  Alternately, 3 groups of three characters can produce a Rubik Cube type three dimensional matrix containing 15 individual 3 by 3 matrices.

Cosmological information encoded in the Torah would probably not take the form of the fundamental equations of quantum mechanics, general relativity, thermodynamics or any other patterns observed by the human mind.  Information would most likely be revealed in a “Bible Code” format.  Such a format is so incomprehensible to the human thought process that we are able to find information only if we first know what to look for!  The following question serves to illustrate.  Can a snapshot of critical parameters, for the early universe, be found within the first few concatenated characters comprising the first verse of Genesis?

A Fundamental Particle?

If relatively short strings of Hebrew characters from the first chapter of Genesis provide the codons for all space/time and matter/energy then the universe itself should be constructed based on these codons.  In other words, the same 27 characters of the Hebrew alphabet used to define the genetic code would ultimately define the structure of all space, time, matter and energy.

Consider the possibility that the assignment of one of the 27 Hebrew characters -- instead of 1 or 0 -- to a voxel can transform a mere volume of three dimensional space into the most basic building block of matter/energy.  Furthermore, the state of the entire universe is still defined by a countably infinite sequence of digits representing a single real number -- now base-27 (with Aleph as zero) instead of base-2.  Now connect three contiguous voxels together to generate a string.  The three corresponding base-3 triplets generate a 3 by 3 matrix with the elements 0, 1 and 2.  Such groups of three voxels and the associated 3 by 3 matrix could represent fundamental codons for the structure of the universe; a string of Hebrew characters now becomes a string of voxels in space. 

Increasing complexity can be created by joining together groups of three contiguous voxels (codons) to create various geometric shapes.  Such geometric constructions might represent fundamental particles of matter and energy.  However, the arrangement of the 3 by 3 matrices does not necessarily coincide with the arrangement of the voxels.  For example, four groups of three contiguous voxels could be arranged end-to-end to form a 12 voxel string.  But the four 3 by 3 matrices representing these four codons could be arranged like tiles on a floor to create a single matrix of dimension six.

To illustrate a possible connection with Quantum Mechanics, a large number of groups of three contiguous voxels -- e.g. (1010)2 -- could be arranged like tiles to produce a corresponding square matrix of dimension (3)(1010).  This matrix could be used to compute eigenvalues (observables) and eigenvectors (states) for the various particles of physics.  To illustrate a possible connection with General Relativity, consider the gravitational field equations Gij = Rij – (1/2)gijR and Gij + αPij = 0.  The Ricci tensor (Rij) and scalar curvature (R) are derived from the Riemann-Christoffel tensor Rhirs which is, in turn, connected to the rotation required to pass from one reference frame to another when two nearby points are joined by two different paths in a non-Euclidean space: &hi = Rhirsdyrδys .  If the Riemannian space was simply a three dimensional space instead of four dimensional space-time, then &hi could be viewed as a rotation matrix defining both axis and angle.

Recalling that certain dimensionless numbers are associated with every portion of space, is it reasonable to surmise that some primitive particle, reflecting these numbers might provide a pervasive and structural influence throughout the entire universe?  One of the first particles, that can be envisioned, comprises a string of three groups of three contiguous voxels with the corresponding three 3 by 3 matrices arranged in a Rubik Cube.  The Hebrew characters assigned to the first, second and third groups of three voxels would most logically be the first three, second three and third three characters respectively from the first verse of Genesis.  The three matrices are therefore defined as:

 

            020                  212                  020

Ayz   = 000      Byz  =  001      Cyz =   000

            110                  200                  110

 

which denote the three planes parallel to the yz axis.  Notice Ayz and Cyz are identical so we really have only two independent matrices with 0, 1 and 2 as elements.  These three matrices define all 15 slices such that each slice produces a 3 by 3 matrix.  Can a Rubik Cube constructed from these trivial matrices be expected to conceal such numbers as the Fine Structure Constant and the mass ratios of sub-atomic particles?  That question is answered in the remainder of this paper.  The number of two dimensional 3 by 3 matrices that can be sliced from the cube and arranged like tiles on a floor together with the number of possible arrangements of these tiles is too large to examine.  So we will simply make a reasonable guess.

Each of fifteen slices through the cube produces up to 64 matrices by successive rotations about various axes.  In this trial, only one matrix will be selected for each slice.  The fifteen selected 3 by 3 matrices can then be permuted in 15! ways.  In this trial only one permutation will be chosen.  After these assumptions, we still find ourselves short by one 3 by 3 matrix in order to construct a single matrix of dimension 12.  So we will add a single 3 by 3 matrix, at the end, to complete the square.  That matrix is

 

             0 1 0

XXX =   0 0 1

             0 2 1

 

which represents the fourth three characters from the first verse of Genesis and is also a Hebrew name for God (Elah).  The final 12 by 12 matrix is constructed by the operation

 

M=ArrayFlatten[{{Ayz,Byz,Cyz,Xyz},{Azx,Bzx,Czx,Xzx},{Axy,Bxy,Cxy,Xxy},{Xxz,Xzy,Xyx,XXX}}]

where Xyz, Xzy, Xzx, Xxz, Xxy and Xyx represent diagonal slices.

Could anything of substance be obtained from this smorgasbord of assumptions?

Here is the final 12 by 12 matrix, designated as M which characterizes our fundamental particle called, for convenience, the Trinity Particle.

 

0 2 0 2 1 2 0 2 0 1 2 1

0 0 0 0 0 1 0 0 0 0 0 0

1 1 0 2 0 0 1 1 0 0 2 0

0 2 0 2 1 2 0 2 0 0 2 0

0 1 0 0 0 0 0 0 0 0 0 1

0 0 0 1 0 1 1 2 1 1 1 0

0 2 0 0 0 0 1 1 0 1 0 0

2 1 2 0 0 1 2 0 0 0 0 1

0 2 0 0 0 0 1 1 0 0 0 0

1 1 0 0 2 0 1 0 0 0 1 0

0 0 1 0 0 0 0 0 1 0 0 1

0 2 0 0 0 0 0 0 0 0 2 1

 

Now define X = MM*, Y = X2 and Z = X3 where M* is the transpose of M.  X, Y and Z are each 12 by 12 matrices and each will have a characteristic equation with twelve coefficients.  Let the coefficients for X, Y and Z be denoted by J1…J12, JJ1…JJ12 and JJJ1…JJJ12 respectively.

(JJ6)(10-15) = 0.00704. The fraction of mass converted to energy when helium is formed from hydrogen in stars is currently 0.0071 for a 0.8% deviation (Rees, 1999, 54).

Abs{JJ3/JJ2} = 137.705 and Abs{JJ2/JJ3} = 0.007262. The current values for the Reciprocal Fine Structure constant and the Fine Structure Constant are 137.036 and 0.007297 respectively for a 0.5% deviation (Parker, 2018, 191).

Abs{(J5)(10-3)} = 3675.9. The total mass of a proton and a neutron compared to that of an electron is 1836.2 + (1.0014)(1836.2) = 3674.9 for a 0.03% deviation (Borrow, 2002).

1/(JJ2) = 4.4(10-7); JJ10/JJJ10 = 7.3(10-7); J10/JJ10 = 9.8(10-7). These are all possible values for the weak force coupling constant which is currently 10-6 to 10-7  (Rohlf, 1994).

1/[(JJJ6)(JJJ8)] = 0.47(10-39). The current value for the proton-electron gravitational to electrical force ratio is 0.435(10-39) for an 8.0% deviation (Bondi, 1961, 60).

J11/JJ11 = 8.5(10-6). This is essentially equal to fraction of rest mass energy needed to disperse large cosmic structures such as galaxies and clusters (Rees, 1999, 118).

(JJJ7)2(JJJ9)2 = 3(1080) which is nominally the number of elementary particles in the universe (Bondi, 1961,60).

(JJJ11/JJ11)(2) = 247168. The mass of a Higgs Boson compared to that of an electron is 246037 for a 0.46% deviation (Eichten, 2013).

1/(JJ2) = 4.4(10-7). The mass of a Majorana neutrino compared to that of an electron is 4.7(10-7) for a 6% deviation (Klapdor-Kleingrothaus, 2001, 2409).

(JJJ8)(JJJ9) = 3.43(1040). The characteristic length of the universe (velocity of light divided by Hubble’s constant) divided by the classical radius of an electron is 4(1040) for a 14% deviation (Bondi, 1961,60).

No more than two coefficients are used for any dimensionless number.  The coefficients are combined by simple multiplication and division with an occasional power of 10 or factor of 2.  Better estimates may be possible using different combinations of coefficients.  Are these remarkable agreements just a coincidence?

The 12 by 12 matrix M, whose elements are 0, 1 and 2, was extracted from the first 12 characters of the first verse of Genesis and appears to conceal fundamental dimensionless numbers, characteristic of the early universe, as combinations of coefficients from the three characteristic equations for X, Y and Z.  If M is, in fact, a ‘magic matrix’ then the same dimensionless numbers may be concealed at deeper levels of matrix manipulation.  The Reciprocal Fine Structure constant can be used for illustration.  Its current value is 137.036.  We found that |JJ3/JJ2| = 137.705 for a 0.5% deviation.  But what if we form the following 6 by 6 matrix, designated as j1, from the 36 coefficients of the three characteristic equations?

 

JJJ1   J1       J2      JJJ12    JJJ11    J12

JJJ2   J6       J3      JJ7       JJ8       JJ9

JJJ3   JJ1      J7      JJ6       J10      JJ10

JJJ4   JJ2      J4      J8         J5       JJ11

JJJ5   JJ3      JJ4     JJ5       J9        JJ12

JJJ6   JJJ7     JJJ8   JJJ9      JJJ10    J11

 

This particular matrix is just one of 36! matrices that can be formed from the 36 coefficients. The trace of this square matrix is equal to the sum of its eigenvalues (Heading, 1958, 48); any other arrangement of elements having the same trace will have the same eigenvalue sum.  A connection between the sum of eigenvalues and a particular dimensionless number might reveal a deeper relationship than one between a simple ratio of coefficients and that dimensionless number.  Consider, for example:

 

Tr (j1)/{(23)(3)(10-3)(Abs[J11])} = 136.932

 

for a 0.07% deviation from the current value for the Reciprocal Fine Structure constant.  Similarly, the determinant of this square matrix is equal to the product of its eigenvalues and

 

|j1|/J5 = 1.4(1080)

 

This is nominally the number of elementary particles in the universe.

The matrix j1 comprises a relatively random arrangement of coefficients from the three characteristic equations.  Could a different matrix also be expected to conceal dimensionless numbers?  Consider the following matrix designated as j2.

 

JJJ6   JJ3     J4      JJ6       J5        JJ11

JJJ1   JJJ7   JJ4      J8        J9        JJ12

JJJ2   J1      JJJ8    JJ5       JJJ10    J11

JJJ3   J6      J2      JJJ9      JJJ11    J12

JJJ4   JJ1     J3      JJJ12    JJ8       JJ9

JJJ5   JJ2     J7      JJ7       J10       JJ10

 

The total mass of a proton and a neutron compared to that of an electron is 1836.2 + (1.0014)(1836.2) = 3674.9.  Using j2 we find

 

Abs[Tr (j2)/{(33)(104)(J4)(Abs[J11])1/2}] = 3672.2

 

for a 0.07% deviation.  Similarly, the current value for the electron-proton electrical to gravitational force ratio is 0.23(1040).  Using the determinant of j2 we find

 

|j2|/{(JJJ8)3(Abs[J11])} = 0.23(1040)

 

for no deviation from the current measured value.  It would appear that predictions of dimensionless numbers, using the sum and product of eigenvalues from j1, j2 and undoubtedly other comparable 6 by 6 matrices, may be even closer to the current measured values of these dimensionless numbers than predictions based on simple products and ratios of coefficients from the characteristic equations. 

The dimensionless numbers of physics reflect various properties of space, time, matter and energy.  The universe was created with such precision that any significant variation, in a single, critical dimensionless number, would preclude our existence.  However, simultaneous changes -- in the hundredths of a percent range -- covering several different dimensionless numbers may have occurred over the 13.7 billion years since the beginning.  But if a predicted value deviates by more than a few hundredths of a percent from the current value, then perhaps the dimensionless number is not critical or the best predicted value has not yet been determined.

Let us re-examine the 10 dimensionless numbers which were previously predicted using simple products and ratios of the 36 coefficients from the characteristic equations.  But this time the eigenvalue sum (trace) and eigenvalue product (determinant) for j1 and j2 will be employed.

Tr (j1)/{(10-25)(JJ10)(JJJ8)} = 0.0071. The fraction of mass converted to energy when helium is formed from hydrogen in stars is currently 0.0071 for no deviation.

Tr (j1)/{(23)(3)(10-3)(Abs[J11])} = 136.932 and the reciprocal is 0.007303. The current values for the Reciprocal Fine Structure constant and the Fine Structure Constant are 137.036 and 0.007297 respectively for a 0.07% deviation.

Abs[Tr (j2)/{(33)(104)(J4)(Abs[J11])1/2}] = 3672.2. The total mass of a proton and a neutron compared to that of an electron is 1836.2 + (1.0014)(1836.2) = 3674.9 for a 0.07% deviation.

Tr (j1)/Abs[JJ5] = 8(10-7) . The weak force coupling constant is currently 10-6 to 10-7 for no deviation.

|j2|/{(JJJ8)3(Abs[J11])} = 0.23(1040). The current value for the electron-proton electrical to gravitational force ratio is 0.23(1040) for no deviation.

JJ10/Abs[Tr (j2)] = 9.8(10-6). This is essentially equal to fraction of rest mass energy needed to disperse large cosmic structures such as galaxies and clusters.

|j1|/J5 = 1.4(1080) which is nominally the number of elementary particles in the universe.

Abs[Tr (j2)/{(32)(10-8)(J8)(J11)(J5)}] = 245934. The mass of a Higgs Boson compared to that of an electron is 246037 for a 0.04% deviation.

Tr (j1)/{(2)(102)(Abs[J9])(J12)} = 4.7(10-7) . The mass of a Majorana neutrino compared to that of an electron is 4.7(10-7) for no deviation.

{(Abs[JJ1])1/3(|j2|)}/{(JJJ8)3(Abs[J11])} = 4(1040). The characteristic length of the universe (velocity of light divided by Hubble’s constant) divided by the classical radius of an electron is 4(1040) for no deviation.

 

Aside from the sum and product of eigenvalues, only ten J values total were used to compute the ten dimensionless numbers.  These J values are fixed just as the dimensionless numbers are fixed.  The only allowed variation is the functional form of a particular equation.  The equations are all simple products and ratios of numbers from j1 and j2 along with powers of 2, 3 and 10.  J11, which is -131100, appears in five equations.  JJJ8, which is 257150395023074017825, appears in three equations.

Probing deeper into this mystery, the dimensionless numbers of physics are not the only type of dimensionless numbers.  The dimensionless numbers of pure mathematics are another type.  Such numbers cannot vary over the life of the universe.  The most pervasive, structural numbers are Pi (π = 3.1415926535…) and Phi or the Golden Ratio (φ = 1.6180339887…).  These numbers are related by

 

π = (5) cos-1(φ/2)

 

Could either one of these numbers be extracted from our Rubik Cube?  Consider that each individual slice through the cube, not just the 12 by 12 matrix may contain information.  One such slice is the 3 by 3 matrix Ayz representing the first three characters from the first verse of Genesis.  The first eigenvector of AyzAyz* -- Ayz times its transpose -- is (φ, 0, 1).  The Golden Ratio, and therefore Pi, are concealed in this matrix.

The author’s conjecture is that every fundamental dimensionless constant of physics can be generated using the approach described herein.  Assemble a string of three groups of three contiguous voxels.  Assign, to each of the three groups, a letter sequence from the first verse of Genesis and arrange the corresponding matrices into a Rubik Cube configuration of 27 numbers.  Then create multiple 3 by 3 matrices by slicing through the cube.  Finally, arrange these slices like tiles on a floor to produce multiple 2-D matrices of dimension 3 to 12 or higher.  This may require the use of the Elah matrix to complete the square.  Perhaps these strings of three groups of three contiguous voxels exercise a pervasive, structural influence throughout the universe.

 

Reciprocal Fine Structure Constant

 

A string of three groups of three contiguous voxels can be assigned the first nine characters from the first verse of Genesis with the corresponding three 3 by 3 matrices arranged in a Rubik Cube.  Adding characters 10, 11 and 12 permits the creation of the 12 by 12 matrix M. Such a string of voxels is postulated to conceal the critical dimensionless numbers of physics.

These dimensionless numbers can be estimated from X = MM*, Y = X2 and Z = X3 where M* is the transpose of M.  X, Y and Z are each 12 by 12 matrices and each will have a characteristic equation with twelve coefficients.  The coefficients for X, Y and Z are denoted by J1…J12, JJ1…JJ12 and JJJ1…JJJ12 respectively.  Critical dimensionless numbers appear as ratios of these J values and also in expressions involving the sum and product of eigenvalues from various 6 by 6 matrices formed from the J values.  Dimensionless numbers, estimated in this way, do not always match current measured values.  This may suggest that each estimate is a member of a statistical distribution.  Perhaps critical dimensionless numbers are encoded as statistical distributions and not precise numerical values.  For example, the number 137.036 could be the mean value over time for the reciprocal fine structure constant but not the specific value at any particular time.  Within each Trinity Particle, values of critical dimensionless numbers may change at each time step following some statistical distribution.  The range of values, for a dimensionless number encoded in a Trinity Particle, might simply fix the mean and standard deviation.  This paradigm shift in analysis will be illustrated using the reciprocal fine structure constant.

The origin of the Reciprocal Fine Structure Constant (α-1=137.035999) has remained a mystery for over a century.  Perhaps a statistical distribution of α-1 values is encoded in the matrix M at various levels of manipulation.  We have seen that |JJ3/JJ2| = 137.705 for a 0.5% deviation and Tr (j1)/{(23)(3)(10-3)(Abs[J11])} = 136.932 for a 0.07% deviation.  But what if additional encryption exists at deeper levels of matrix manipulation?

Each Trinity Particle may be a fluctuating entity which could be modeled by Fourier series based on certain integral identities (mathworld.wolfram.com/FourierSeries.html) involving Cos, Sin, Cos2, Sin2, and (Sin)(Cos).  These five functions will be utilized without the integral identities.

First form the symmetric matrices k1 = j1 j1* and k2 = j2 j2*.  Now evaluate cosk1 = Cos[k1], sink1 = Sin[k1], p1 = Cos2[k1], q1 = Sin2[k1], r1 = Sin[k1]Cos[k1] and cosk2 = Cos[k2], sink2 = Sin[k2], p2 = Cos2[k2], q2 = Sin2[k2], r2 = Sin[k2]Cos[k2].  These ten expressions use the power series for Sin and Cos with ordinary powers replaced by matrix powers.  The result is ten, 6 by 6 symmetric matrices.  The matrix elements themselves do not represent critical dimensionless numbers.  So, how might such numbers be encoded?  One way is to combine no more than two selected matrix elements using basic arithmetic – addition, subtraction, multiplication and division – while viewing the elements themselves as floating point numbers; in other words, the decimal point can be shifted to the right or left.  Here are some sample Reciprocal Fine Structure Constant values, obtained in this fashion by addition and subtraction, from the ten symmetric matrices.

 

α-1 (1) = 107sink1[[1,4]] - 102sink1[[2,3]] =                  137.007

 

α-1 (2) = 1010cosk1[[2,6]] + 1010cosk1[[3,6]] =             136.960

 

α-1 (3) = -109sink1[[4,6]] - 1010cosk1[[3,6]] =               137.094

 

α-1 (4) = 108sink2[[3,4]] - 103sink2[[4,4]] =                  136.693

 

α-1 (5) = 1013cosk2[[2,4]] + 108cosk2[[4,6]] =               136.972

 

α-1 (6) = 1013sink2[[2,3]] - 1013cosk2[[1,2]] =               136.951

 

α-1 (7) = -106p1[[2,4]] - p1[[4,4]] =                               137.039

 

α-1 (8) = 1010q1[[1,6]] - 106r1[[5,6]] =                          137.037

 

α-1 (9) = 1010r1[[1,6]] + 1010r1[[3,6]] =                         137.060

 

α-1 (10) = 108r1[[2,5]] - 103r1[[1,3]] =                           137.106

 

α-1 (11) = -1014q2[[3,6]] + 103q2[[1,6]] =                      137.083

 

α-1 (12) = 102p2[[2,2]] + 103r2[[4,4]] =                         137.024

 

α-1 (13) = 1013r2[[3,5]] + 1015r2[[2,6]] =                       137.013

 

α-1 (14) = -104r2[[5,6]] + 103r2[[5,6]] =                        137.349

 

α-1 (15) = -1016r2[[2,6]] - 1012r2[[2,4]] =                       137.152

 

α-1 (16) = 1014r2[[1,2]] + 102r2[[1,1]] =                        136.943

 

α-1 (17) = -1010r1[[2,6]] + 104r1[[3,4]] =                       137.138

 

Multiplication and division can also be used.  For example,

 

α-1 (18) = 10-2q1[[1,6]]/(r1[[1,6]]r1[[2,3]]4) =                137.050

 

The mean of these 18 estimated values is 137.037 and the standard deviation is 0.12.  

Estimates can also be made using matrix element pairs from the matrices k1 and k2 themselves without passing them through a trigonometric function.

 

α-1 (k1a) = -10-30k1[[4,5]] - 10-31k1[[4,5]] =                  136.891

 

α-1 (k1b) = 10-24k1[[3,3]] + 10-28k1[[4,4]] =                  137.775

 

α-1 (k1c) = -10-34k1[[5,6]] + 10-33k1[[5,5]] =                 136.430

 

α-1 (k1d)  = 10-29k1[[1,1]] + 10-39k1[[6,6]] =                 137.452

 

α-1 (k1e)  = -10-27k1[[1,2]] - 10-25k1[[2,5]] =                 136.716

 

α-1 (k2a) = -10-30k2[[5,6]] - 10-31k2[[5,6]] =                  136.892

 

α-1 (k2b) = -10-34k2[[1,6]] + 10-33k2[[6,6]] =                 136.438

 

α-1 (k2c) = -10-31k2[[1,4]] + 10-27k2[[1,2]] =                 137.062

 

α-1 (k2d) = -10-30k2[[5,6]] - 10-28k2[[4,5]] =                  136.660

 

α-1 (k2e) = -10-27k2[[4,5]] + 10-27k2[[1,3]] =                 138.118

 

 

The mean of these 10 estimated values is 137.043 and the standard deviation is 0.57 which is nearly 5 times the standard deviation of the values estimated after passing k1 and k2 through the Fourier functions.

In like manner, statistical distributions can generated for the other nine critical dimensionless numbers highlighted in this paper; the Reciprocal Fine Structure constant was utilized merely for the purpose of illustration.  To keep these findings in perspective, remember that j1 and j2 are only two of 36! matrices that can be formed from the coefficients of the characteristic equations.

 

Conclusions

 

The 12 by 12 matrix M, whose elements are 0, 1 and 2, was extracted from the first 12 characters of the first verse of Genesis.  It appears to conceal statistical distributions of critical dimensionless numbers at many levels of encryption including:

Products and quotients of coefficients (J values) from the three characteristic equations for X, Y and Z where X = MM*, Y = X2 and Z = X3 and M* is the transpose of M.

Expressions involving the sum and product of eigenvalues from various 6 by 6 matrices (e.g. j1 and j2) formed from the J

Sums and differences using the elements from k1 = j1 j1* and k2 = j2 j2*.

Sums and differences using the elements from cosk1 = Cos[k1], sink1 = Sin[k1], p1 = Cos2[k1], q1 = Sin2[k1], r1 = Sin[k1]Cos[k1] and cosk2 = Cos[k2], sink2 = Sin[k2], p2 = Cos2[k2], q2 = Sin2[k2], r2 = Sin[k2]Cos[k2].

Many other avenues of investigation have not been pursued.  For example, letting DSTk21 = FourierDST[k2,1] represent the first Fourier Discrete Sine Transform of k2 (Martucci, 1994), it can be shown that

 

10-1DSTk21[[2,3]] - DSTk21[[2,3]] = .23(1040)

 

which is the electron-proton electrical to gravitational force ratio. 

What are the implications of all these findings?  Do they represent a completely serendipitous concurrence of physics, mathematics and Hebrew text?  Or did our transcendent, immanent, infinite, eternal and immutable God provide us with a user manual encrypted within His inspired, inerrant and infallible Word.  The first verse of Genesis says: “In the beginning God created the essence of the heavens and the essence of the earth.”  What a perfect place to encrypt a user manual revealing the physics and mathematics of this new creation from nothing (bara).  But why would God bother to encrypt a message within a message?  A partial answer might be found in the New Testament. 

“For since the creation of the world, God’s invisible qualities – His eternal power and divine nature – have been clearly seen, being understood from what has been made, so that men are without excuse” (Romans 1:20).

 

References

 

Barrow, John. 2002. “The Constants of Nature: From Alpha to Omega--the Numbers That Encode the Deepest Secrets of the Universe.” London. Vintage.

Bondi, Hermann. 1961. “Cosmology.” Cambridge. University Press.

Drosnin, Michael. 1998. “The Bible Code.” New York. Touchstone. 

Eichten, Estia. 2013. “The Higgs Boson and Naturalness.” SDU Odense Denmark.  Fermilab - Origin of Mass Conference CP3 Origins.

Heading, J. 1958. “Matrix Theory for Physicists.” London. Longmans, Green and Co.

Klapdor-Kleingrothaus, Hans V. et.al. 2001. “Evidence for Neutrinoless Double Beta Decay.” Modern Physics Letters A. Vol 16. No.37. pp 2409-2420.

Martucci, Stephen. 1994. "Symmetric convolution and the discrete sine and cosine transforms." IEEE Trans. Signal Process. SP-42, 1038–1051.

Parker, Richard H.; Yu, Chenghui; Zhong, Weicheng; Estey, Brian; Müller, Holger. 13 April 2018. "Measurement of the fine-structure constant as a test of the Standard Model." Science. 360 (6385): 191–195.

Rees, Martin. 1999. “Just Six Numbers.” New York. Basic Books.

Rohlf, James William. 1994. “Modern Physics from a to Z0.” Wiley.

Satinover, Jeffrey. 1998. “Cracking the Bible Code.” New York. William Morrow.

Wolfram, Stephen. 2002, “A New Kind of Science.” 100 Trade Center Drive. Champaign, IL. Wolfram Media Inc.