II) The Fundamental Physical Parameters

Either natural structures stabilize in multiples of 22 as a random occurrence (in the absence of God) or as the result of an underlying order (by the providence of God). As there can be no middle ground, then to determine whether the observed pattern of stability is significant or not we only have to compare the measurements that define the fundamental differences between the indivisible parts of matter. If these parameters exhibit special proportionality with multiples of 22 then we will have a clear indication that the harmonic order evident in stable natural structures has an underlying physical basis, which would affirm our notion that God exists. If on the other hand, multiples of 22 prove to be irrelevant to our understanding of the fundamental physical parameters then we can be assured that the apparent pattern among stable natural structures is merely coincidental, and thus lacking physical evidence we may reasonably doubt the existence of God.

Given the stakes then, it would be wise to begin from the most secure foundation and proceed carefully through each step before reaching a conclusion. So first let us compare two distances derived from the geometry of a circle. The distance around a circle is known as the circumference C while the longest distance across is its diameter, which is twice as long as its radius r. In a perfect circle, the circumference is always about 3.14159 times longer than the diameter. The constant ratio between circumference and diameter is known as pi and is symbolized as π.

If we wish to calculate the circular distance from one end of a circle to the other then we just multiply the radius by π. If however we wish to calculate the total space enclosed by a circle A_d, which is its 2-dimensional area, then we must multiply the square of the radius by π.

A circle moving in a larger circle produces a 3-dimensional object known as a torus, which generally looks like a ring. The surface area of the torus A_t is calculated by multiplying the circumference of the smaller circle 2πr by the circumference of the larger circle 2πR.

Alternatively, we can see that dividing the surface area of the torus by the square of 2π yields a quantity equivalent to the small radius r multiplied by the larger one R.

Shifting now from timeless geometry to time-dependent astronomy, if we track the movement of planets relative to the fixed stars then it will become apparent that they move in nearly circular paths not around us but around the Sun. If their maximum distance from the Sun is measured as S and their orbital period is measured as T, then we will find that the cube of S divided by the square of T will be approximately the same for every planet in the solar system.

This proportionality, known as Kepler's third law of planetary motion, is the result of the Sun's mass being so much greater than the mass of its orbiting planets. At large scales, the mass of an object is directly proportional to the quantity of matter it contains and, in general, greater mass always implies a greater resistance to acceleration. So if the mass of the Sun is proportional with G_M, then dividing this standard gravitational parameter by the square of 2π will yield the cube of S in the same ratio with the square of T for each of the planets.

                                

More precisely we can see that Newton's gravitational constant G multiplied by the sum of a large stellar mass M and a smaller planetary mass m is equivalent to the planet's maximum distance from the Sun S multiplied by the square of its average velocity v.

From this it is clear that when the masses, distances, and velocities of each of the planets are stable then the solar system will be in equilibrium. Moving from the scale of the solar system to the scale of the indivisible parts of matter, we must acknowledge that the maximum velocity of any particle, as with all massive objects, is limited by the universal speed of light c. As Einstein observed, every particle has a special measure derived from its rest-mass energy E, which is equivalent to its mass at rest multiplied by the square of the speed of light.

Planck had previously discovered that changes in energy are discrete rather than continuous, so that energy must also be proportional to a constant h multiplied by the speed of light c and divided by the observed wavelength of light λ.

Combining Einstein and Planck's relations, Compton found that each particle has a basic size, known now as its Compton wavelength, which corresponds to the wavelength of light with energy equivalent to its rest-mass energy. The Compton wavelength of the electron λ_e is thus defined as follows:

We have already seen that the most stable configuration of matter consists of a heavy interior mass known as a proton m_P and a lighter exterior mass known as an electron m_e separated by an average distance known as the Bohr radius a₀, which is about 22 times greater than the electron's Compton wavelength λ_e. We then went on to find that ever larger natural structures attain critical stability with the same multiple. By comparing the parameters that define the fundamental components of matter we may now see if they also exhibit significant proportionality with multiples of 22.

The most stable distance between a proton and electron is established by the fine-structure constant α whose inverse α^-1 is nearly equal to 137.036. Dividing this dimensionless number by 2π yields the ratio of a₀ to λ_e. So if λ_e is proportional with 2π then a₀ will be similarly proportional with α^-1. We know that a large radius and small radius can be multiplied together with the square of 2π to obtain the surface area of a torus A_t, and the same square of 2π can be multiplied by the ratio of the cube of a planet's average distance from its star and the square of its orbital period to obtain the standard gravitational parameter of the star G_M. If we now multiply the ratio of a₀ to λ_e by the square of 2π we can obtain a new proportion that defines the elementary charge e in terms of h and c as follows:

The elementary charge describes the magnitude of the electromagnetic force between protons and electrons. This relation may be expanded to include the proton mass (in Gev/c²) simply by incorporating a multiple of (135)(136) and a divisor of 20.

^{a = 1.000361386          b = 1.000262767}

Here we find an extraordinary correlation with two known atomic limits. First, notice the remarkable similarity between the multiple in the numerator (135)(136) and α^-1 which is slightly larger than 137. It will not be immediately clear how to interpret this connection but for now it is enough simply to recognize the numerical proximity. It is also noteworthy that the denominator corresponds with the neutron-proton ratio, which exceeds one in all stable atoms containing more than 20 protons and is equal to one for most atoms with 20 or fewer protons. We will consider the implications of this later.

Continuing the current line of analysis, we find the electron mass can be broadly constrained using the same terms as the previous relation by simply dividing it by (135)(136):

^{c = 1.000083153          d = 1.000278210}

Multiplying this relation by 20² then yields an even more precise correspondence between the electron mass and the golden ratio Φ, a geometric constant describing two lengths in the same ratio as the larger is to the sum of both lengths. Aside from the golden ratio, the same terms are found in this relation as the previous one.

^{e = 1.000080415          f = 1.000002738}

Now, to obtain an extremely narrow constraint on the proton mass, it is only necessary to divide the above relation by a factor of 10 and define upper and lower bounds as follows:

^{g = 1.000000563          h = 1.000000049}

What makes this constraint so remarkable isn't merely its precision, but the fact that the upper bound employs the same terms as those from the previous relations while also incorporating the difference between 2 and 0.1, which matches the operation described by the lower bound (2–0.123456). Still, we might be inclined to dismiss this proportionality if not for the following relation that brings us back full circle:

The significance of this equation in our quest to uncover traces of divinity cannot be overstated. It involves only a numerical ratio (135)(135)/20² derived from a pair of atomic limits, and the two most stable masses alongside the harmonic quantity 22, which evidently confers stability to natural structures at various scales of space-time. However, let us not be confused into thinking that God is merely a number, nor that the Almighty is nothing more than a form of matter. We are searching for evidence of a subtle omnipresent harmony that may indicate to us that God exists and that chaos is a side-effect rather than the cause of the order we observe. To this end, we are simply examining the proportionality evident among the fundamental physical parameters.

Let us continue the analysis by demonstrating the supreme usefulness of the ratio (135)(136)/20² in deriving the muon mass. When electrons collide at speeds near the speed of light, their mass may temporarily increase in two discrete stages corresponding with a muon and a tau particle. The mass of the muon m_μ is known with a precision on par with that of the electron, so the ratio of both masses is exceptionally well-defined. Then dividing this ratio by the square of π yields a remarkably close approximation of the rational number 20.95.

^{i = 1.000000301          j = 1.000000025}

This particular quantity is notable for two reasons: first, the integer part (20) has a clear connection to the fractional part (0.95 = 1–1/20); and second, the sum of the proton mass and the electron mass multiplied by half the ratio (135)(136)/20² is very nearly equal to 0.95. In the next section we will see why this relationship is absolutely essential to understanding the basic structure of space-time.

Multiplying the muon-electron mass ratio by the proton mass and then subtracting between 20 and 22 yields the observed range for the top quark mass (172–174 Gev/c²), which is the most massive particle to emerge from near-lightspeed collisions of protons. If we set the limit of the top quark mass m_t equal to 172 and modify it slightly using a proportion with α and the electron g-factor g_e, then it's possible to obtain an extraordinarily precise expression for the number 22.

This may seem convoluted at first glance but the proportionality is unmistakable. √2 and √3 form the sides of a right triangle with a hypotenuse of √5. The linear sum of √2 and √3 is approximately equivalent to the ratio of α and g_e+2, which is a fundamental parameter derived from the electron's anomalous magnetic moment. The anomaly is produced by an inherent wobble in the electron's spin, thus yielding a value of about -2.002319 for the electron's g-factor rather than simply -2. This proportionality also extends to the ratio of the neutron mass m_N and proton mass m_P in relation to π:

^{m = 1.000028489          n = 1.000108485}

Just as the proportionality on the left side is part of a significantly more precise relation involving 4 fundamental masses and the number 22, the ratio on the right is also part of a more precise relation involving the g-factors of the atomic particles and the number 22.

^{o = 1.000395057          p = 1.000007440}

Here we find the nuclear masses and g-factors efficiently constrained by an upper bound of 22 and a lower bound of 7π. That 5 distinct but phenomenally related physical parameters can all be united in one simple proportionality is astounding. And once again it can be broken down to demonstrate the ultimate simplicity of the individual terms. While the absolute value of the electron's g-factor is a little greater than 2, the absolute difference between the neutron and proton g-factors is a little less than 3π. Multiplying the atomic g-factors together in this way yields a value that is a little less than 6π, and adding another factor of π multiplied by the neutron-proton mass ratio yields a quantity only slightly larger than 7π, which is the smallest whole number multiple of π that approximates an integer.

^{q = 1.000018823          r = 1.000221105}

The dimensionless atomic g-factors also function as a lower bound for the masses of the up and down quarks, which appear in triplets to form the nucleons constituting the central mass of atoms. While an unstable neutron consists of 1 up and 2 down quarks, the stable proton consists of 2 up and 1 down quark. The sum of the proton's quark masses (m_d+2m_u) forms a dimensionless ratio with the electron mass that is approximately equal to 6π. If an upper bound is defined as the proton-electron mass ratio divided by the square of π squared and a lower bound is defined in terms of the atomic g-factors as we have seen, then the proton's quark masses in ratio with the electron mass correspond exactly with the observed range of values. Furthermore, when this constraint on the proton's quarks is combined with the earlier constraint on the proton mass, in which the mass of 2 protons is nearly equivalent to the quantity 2–0.123456, then we have clear evidence that protons have a basic proportionality with factors of 6.

This fact is also bolstered by Stokes' law of flow whereby a frictional force F_d produced by a spherical object with a radius R, falling at a constant velocity v_s through a fluid with dynamic viscosity μ_v, is determined by experiment to be proportional with 6π. Since no other theories can account for the appearance of this factor of 6π, and we know both the mass and magnetic activity of the proton are key to this observation, it stands to reason that the proton itself has an internal structure or movement associated with its total mass, quark masses, and g-factor that in some way generates this proportionality with 6π.

* * *

We began with the most basic principles of geometry and used a series of proportions to demonstrate the intrinsic relationships between the masses and g-factors of electrons, protons, and neutrons, which form all the stable material structures around and within us. By comparing the atomic masses to those of the up, down, and top quark along with the muon, we have established a set of proportional constraints for their range of values that are relatively simple, precise, and consistent. Just as protons play a central role in the formation of matter by bonding internally with neutrons and externally with electrons, we find the proton mass has significant proportionality with each of the fundamental physical parameters. While the proton's constituent quarks and g-factor are clearly proportional with a simple factor of 6π, there is also unambiguous evidence of proportionality between the proton mass and multiples of 22. Accurate and concise definitions of the fundamental physical parameters thus accord with the following set of proportional constraints:

Derived from the experimentally determined values of the fundamental physical parameters, these constraints do not appear to be random. Multiples of 22 are clearly useful in defining the masses and g-factors of the indivisible parts of matter. Yet to prove that this proportionality implies an underlying harmonic order will require a more convincing case. We must therefore continue to explore the limits of matter and examine all the available evidence so that we may find a rational explanation that can fully account for this unique pattern.