On the Cosmic Number 3.0.5

Abstract Richard Feynman said in the 80s: “There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of G-d” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly!” (1) In fact, Armand Wyler had pointed out in 1969 that (9/16\[Pi]^3)(\[Pi]/5!)^1/4 is close to 1/137 (2), but failed to convince anyone that his explanation of the connection -that the number is a property of a 7-dimensional space-time- was viable. In this note, it is suggested that the reason for the closeness of another expression to 1/137 is, not a numerical curiosity, but -ironically- an indicator of the truth of a multi-dimensional physical theory. 

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Part 1: The Fine-Structure ConsTant

The recommended CODATA value of fine-structure constant –\alpha– is

\frac{7.29735}{10^3}=\frac{1}{137.036}=\left(e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^1 \frac{1.00009}{1^{12}}-\int_1^1 \frac{1.00009}{1^{12}} \, dn\right)}\right){}^2\right){}^2


c = the speed of light in a vacuum

\hbar = the reduced Planck constant

e = the elementary charge

\epsilon _0 = the electric constant

and it follows that

\frac{e^2}{4 \pi c \epsilon _0 \hbar }=\left(e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^1 \frac{1.00009}{1^{12}}-\int_1^1 \frac{1.00009}{1^{12}} \, dn\right)}\right){}^2\right){}^2

There is a difference of 0.0000127295 between the recommended value of \alpha and \left(e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^1 \frac{1}{1^{12}}-\int_1^1 \frac{1}{1^{12}} \, dn\right)}\right){}^2\right){}^2, but this value has changed over the decades, it is dependent on many assumptions, and positive and negative variations in the value of \alpha of one part in 100, 000 are suggested by data on quasar absorption lines.


Part ii: General RelaTivity

The General Theory Relativity in its present form says that space-time is curved by mass. It follows that, “in the beginning”, all the mass of that universe was concentrated into a zero-dimensional point. That this idea is a half-truth is suggested by the implication of multiple singularities (at the centers of black holes), the lack of a coherent mathematical framework for both large scale and small scale objects, and by the flat rotation curves of distant galaxies. A similar idea about the beginning-state that doesn’t carry any absurd consequences is the idea of the infinite compression of energy (light), and the elimination of mass. Now curvature is to be attributed, not to mass – which is a combination of light and space- but to imbalances of light and space. Mathematically, we capture what it is to be balanced, and what it is to depart from balance, thereby producing curvature, by re-expressing the tradition equation for a circle of area 1 (\pi \sqrt{\frac{1}{\pi }}^2=1) as

\lim_{x\to \infty } \left(e^{2 \gamma } \sqrt{\frac{1}{e^{2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}}\right){}^2=1

Given that \gamma is a spacial case of \zeta (s)-\frac{1}{s-1} for s = 1, we can go from \lim_{x\to \infty } \left(e^{2 \gamma } \sqrt{\frac{1}{e^{2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}}{}^2\right)=1 to the more general

\lim_{x\to \infty } \left(e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{\exp \left((s+1) \left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)}\right){}^{\frac{1}{s+1}}\right){}^{s+1}\right)=1

which -by reference to s = 12- is how we arrived at the proposed understanding of the fine-structure constant. Refining \alpha as

\alpha =\left(e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^1 \frac{1}{1^s}-\int_1^1 \frac{1}{1^s} \, dn\right)}\right){}^2\right){}^2

where s is a positive integer, we see that where s = 12  it is possible to identify the fundamental physical constants as we know them from observation with a cross-section of the number line having a minimum and maximum size given at the one extreme by the fine-structure constant and by a dimensionless quantity whose size is reminiscent of the reduced Planck constant at the other:

It is noteworthy that there are a potentially infinite number of these cross-sections, each associated to a different (positive) real value of s and to a different set of constants:

If we look beyond what we can call the “Planck radius” in the case of s = 12, we see the following series of quantised jumps that goes on forever within bounds related to the Gibbs constant (which limits the size of the overshoots of Fourier sums at jump discontinuities):

A polar plot of the denominator alone on the narrow side of the Planck radius:

Since it is the differences between the partial sums/integrals and the limit that protect against the degeneration of a cosmic spiral into a circle with no radius (a point), or the degeneration of this spiral into a circle with infinite radius (a line), we know that these differences have minimum and maximum sizes. More particularly, we know that if and only if s = 1 and

e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)}}\right)^{\frac{1}{s+1}}\right)^{s+1}=e^{2 \gamma } \sqrt{\frac{1}{e^{2 \gamma }}}^2=1

then the progression associated to

\left(\frac{e^{2 \gamma } \left(e^{\frac{1}{s-1}-\zeta (s)}\right)^2}{e^{2 \gamma } \left(e^{\frac{1}{s-1}-\zeta (s)}\right)^2-e^{2 \gamma } \left(\exp \left(-\left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)\right){}^2}\right){}^{1/s}

is potentially infinite. In the case of s = 1, the difference between the partial sum/integral and the limit is always larger than the Planck radius, but if  s is a positive real number greater or less than 1, then the difference is sometimes not less than the Planck radius where s > 1, and in both cases the progression is strictly finite. These two directions leading away from s = 1 give us two distinct notions of imbalance, and of curvature. Given that the fundamental state of the universe is infinite light, it follows that if s > 1, then the imbalance is in favour of light, and if s < 1, the imbalance is in favour of space. We might therefore model the electromagnetic and gravitational interactions (the nuclear interactions inclusive) in this way:

Combining polar plots of s = 12, 14 (white) and s = 13, 15 (red) shows that there is a still deeper level:

This data invites the following extension:


part III: THE riemann Hypothesis

Wolfram Mathworld defines a projection in this way:

A projection is the transformation of points and lines in one plane onto another plane by connecting corresponding points on the two planes with parallel lines. This can be visualized as shining a (point) light source (located at infinity) through a translucent sheet of paper and making an image of whatever is drawn on it on a second sheet of paper. 

But a “(point) light source” is the same thing as a zero-dimensional light source, which involves the infinite concentration and the zero diffusion of light. The problem with the Wolfram Mathworld definition of projection, and with every physical theory that relies on multiple zero-dimensional point-sources, is that these involve an infinite concentration and zero diffusion of light, when in our experience light is always finitely concentrated/diffused. The solution -and the theory proposed here- is that the universe is a (holographic) projection arising from combinations of artificial atomic point sources such that the gaps between the natural and artificial sources correspond to angles and positions of the planes that allow for light to arise from a finite concentration and to undergo non-zero and non-infinite diffusion according to the equation \lim_{x\to \infty } \left(e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{\exp \left((s+1) \left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)}\right){}^{\frac{1}{s+1}}\right){}^{s+1}\right)=1, an equation within which the fundamental physical constants are encapsulated.

The present universe is on this theory associated to units of a particular size, to a particular set of constants, and to a particular arithmetic/spatio-temporal -for want of a better term- “dimension”, and a plot of \sqrt{\alpha }=e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^1 \frac{1}{1^s}-\int_1^1 \frac{1}{1^s} \, dn\right)}\right){}^2 shows that these dimensions differ radically from each other:

But the key unifying feature -the jewel in the crown- of this theory is that the balance of prime/energy-density of the units belonging to each dimension is constrained by the value s = 1, and by the limit e^{2 \gamma } \sqrt{\frac{1}{e^{2 \gamma }}}^2=1, and therefore by the balance of prime-density and sparsity associated to complex zeros of L-functions if and only if the real part of these zeros is equal in every case to 1/2 (the Generalised Riemann Hypothesis). This most fundamental of constraints ensures that the progressions associated to these units involve a potentially infinite number of primes: let g = Graham’s number (if every digit in Graham’s number is considered to occupy as little as one Planck volume, it would nonetheless be too big to fit in the observable universe) and note that if s in the limit

e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)}}\right)^{\frac{1}{s+1}}\right)^{s+1}

differs from 1 by as little as


the number of primes in the progression associated to \left(\frac{e^{2 \gamma } \left(e^{\frac{1}{s-1}-\zeta (s)}\right)^2}{e^{2 \gamma } \left(e^{\frac{1}{s-1}-\zeta (s)}\right)^2-e^{2 \gamma } \left(\exp \left(-\left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)\right){}^2}\right){}^{1/s} is finite. A complex zero with real part not equal to 1/2 would upset the critical balance of density and sparsity in virtue of which an arithmetic progression associated to an L-function -and therefore the critical balance of density and sparsity in virtue of which the progression associated to the canonical Zeta function- has infinite length… This mathematics extends the inverse square law beyond the arithmetically continuous classical domains in which there is a balance of energy-density and sparsity by regarding non-classical regions as sub or super-domains. Because these non-classical domains are proto-arithmetic, their existence implies that the universe did not evolve or develop from a simple state. Note that where s > 1 the traditional value of \pi, (written as e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} where s = 1) is effectively distorted in a positive direction, thereby distorting the dimensions of a circle in a certain direction. Where s < 1, this value is distorted in a negative direction, also distorting the dimensions of a circle. In the one case, there is an imbalance of energy and space in favour of energy, while in the other, the imbalance is in favour of space. If we take the unit in which the fundamental physical constants are encapsulated and attach it to the equation \frac{S}{\pi r^2} like this:

S=\sqrt[12]{\frac{e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2}{e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^x \frac{1}{n^{12}}-\int_1^x \frac{1}{n^{12}} \, dn\right)}\right){}^2}}

\frac{S}{\pi r^2}\approx \lim_{x\to \infty } \, \frac{S}{e^{2 r^2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}=\frac{S}{e^{2 \gamma } r^2}

we see that, while the classical objects (s = 1) that are the “flux” of this energy system are governed by the inverse square law, and the non-classical objects (s != 1) that are the sources of this flux are not, both are nonetheless governed by the same equation.



1. Feynman, R (1985), QED: The Strange Theory of Light and Matter

2. Wyler, A (1969), L’espace symétrique du groupe des équations de Maxwell

3. Johnson, H (2010), Changes spotted in fundamental constant

4. Webb, J et al (2010), Evidence for the spatial variation of the fine-structure constant

5. King, J (2010), Searching for variations in the fine-structure constant and the proton-to-electron mass ratio using quasar absorption lines

6. O’Connor, J (1996), General relativity

7. Wald, R (1997), Gravitational Collapse and Cosmic Censorship

8. Wald, R (1984), General Relativity

9. Rubin, V et al (1980), Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R = 4kpc) to UGC 2885 (R = 122kpc)

10. Davenport, H (2000), Multiplicative number theory

11. Graham, R (1971), Ramsey’s Theorem for n-Parameter Sets

12. Gardner, M (1977), In which joining sets of points leads into diverse (and diverting) paths