As the non-supersymmetric Standard Model of Physics (SM) stands there are about 26 independent dimensionless constants. These are directly involved in the laws of physics and all are determined by experiment and placed such that arbritrary determined dimensional units cancel. An example of this are the mass ratios in the Lagrangian description of the SM. These ratios are natural constants which could possibly be fundamental if they are derived from some principle of math or Nature. Currently, these numbers appear to be somewhat random and just coincidental with no explanation of origin or connection with each other. In some cases this forces a procedure called 'fine tuning' which is unsatisfactory. Below is a procedure for determination of some mass ratios utilising number theory i.e. the use of integers, 'near integers' and some number theoretic objects i.e. primes, groups etc. The relating of interesting integers and 'near integers' to create these objects suggests that the Heegner numbers (163 in this case) or the imaginary quadratic fields
implies this interplay. If this is true then relations involving just integers (to those integer-philes) and the constants pi and e with volume geometry to calculate constants of Nature are not enough. It is suggested that dimension canceling ratios are possibly the only physics parameter that can be derived from a pure mathematical form (which could be fundamental) and that no non-anthropic extraction of a isolated dimensionful entity is possible. The constants of physics including the natural dimensionless ratios (using empirical values) are anthropic and this is not a good position for Science. On this page we discuss appearance of some mass ratios in some strange equations which if true would be non-anthropic values. At the worst these values are quasi-values (* see comments) which would still demand an explanation because the forms are consistent and associated with each other. As I am not competent in the matter of the famous j-invariant I will not be discussing this but please note that this is probably the major explanation for some of this. The quadratic nature of these equations and forms almost insures the involvement of the theory of modular forms.
expresses the best 'near integer' in the theory of modular forms. That this number is so close to an integer is non-coincidental and is explained by the fact that Heegner numbers (163 in this case) generate imaginary quadratic fields of Class 1 in the near integral interplay of the q expansion in the j-invariant. This 'near integer' value as a power of two approaches close to large number values in physics and as a power of two or greater retains its 'near integer' value character to a certain point. See Mathoverflow Questions, Why are powers of exp(pi sqrt163) almost integers?
In the following the mass ratios for the most part will use the neutron mass rather than the stable proton mass as the calculations seem to demand this. Perhaps, this is a degeneracy related not specifically to a neutron particle but to a group object that may represent something more fundamental, such as a gluon chain or a non-charged QCD plasma. Also, it is possible that the neutron physics may be stable under a very strong gravitational field. In the following mass ratios the Planck mass plays an important part although it is not necessarily part of the SM. The Planck mass (or energy) while not fundamental introduces gravity by its definition into the ratios with interesting implications. The Planck energy is generally considered the scale of the high-energy cutoff for the physics domain of the SM. The quadratic expressions on this page as you may notice lend to integral canceling properties of Planckian parameters which maybe suggestive of quadratic solutions. Examples of this in current physics calculations occur in semiclassical approaches to black hole entropy and this will be shown below.
The mass ratios that we are interested in are,
All the above ratios are values using 2010 Codata (except pion mass which uses Particle Data Group PDG) out to 6-8 significant figures. All of these ratios except (3) can be reformulated as a combination of other physics values. These ratios are important in that they are universal dimensionless constants which have something to do with how our World is determined. Let us compare these values to the 'number theoretic' versions in the following equation.
Solving for d,
The large number ~10^54 calculated above is the integer that reflects the number of symmetry elements in the Monster Group. Interesting aspects of the other integers in this equation will be pointed out along the way. It is very curious that almost every integer has 'number theoretic' properties. The solution d is very close to a very important constant in physics , that being the fine structure constant. It is important in a theory called Quantum Electrodynamics or QED which has implications in behavior of light in electromagnetism.
(For equations (8) and (9) see OEIS A161771 and OEIS A160515.) Equation (4) is the inverse of eq. (1) without factor 2. It is very close to the dimensionless gravitational coupling constant which is important in star formation See OEIS A162916. Equation (5) is close to the dimensionless fine structure constant which is very close to the 'number theoretic calculated d in eq. (7). The fine structure constant according to the latest value fom the NIST is,
It is probable that eq. (5) actually would approach this value if the PDG value of the charged pion mass was a more accurate and precise value. I think we can show that this is the case with the physics form of eq. (6) will actually hover near the Monster integer value if 2010 Codata is used and the charged pion mass is logically isolated in the equation. The use of the pion PDG value moves away from the solution. We will then compare the new pion values using the fine structure constant and the solution d from eq. (6) which is close to this important constant (alpha). The two new pion values compare very well and suggests consistency.
The solution d eq. (7) is very close to the fine structure constant,
This maps very well to the physics forms which will be discussed below.
Some numerologic observations about some numbers in the forms:
First, a few observations to be noted here about eq. (6) and its solution d eq. (7). In the ratio eq. (13) both numerator and denominator are 1 removed from a prime number. The number 361436250 is next to the prime 361436249 and the number 196560 is next to the prime 196561. The number 196560 is the kissing number of the 24 dimensional Leech Lattice See OEIS A198343. A hyperbolic version of the Leech Lattice (II_25,1) is involved in the hugely symmetric toroidal orbifold utilising the Monster Lie algebra. The Heegner number 163 is a prime number while the number 70 is 1 removed from the prime 71. The number 70^2 = 4900 contains two copies of the Weyl vector in 26 dimensions which can be used to construct the Lorentzian Leech Lattice II_25,1. The number 2^16 = 65536 is 1 removed from the prime 65537. In eq. (7) the number 281 is a prime number and it is found in a quantum field theory (QFT) as a largest partial sum in a perturbation expansion See Prime Curios 281. The number 11 is a prime number which has significance in M-theory. The number 38817792 is 1 removed from the prime 38817791 and 2^13 = 8192 is 1 removed form the prime 8191. The number 7008506450687690 has prime factorization of 12 (minus 3, 7, 11) of the supersingular primes related to the Monster Group. Also, if you perform the combination,
the number 438048 is set between the twin primes 438047 and 438049. Also, the ratio in eq. (12) reduces to 1721125/936 where 1721125 has 24 divisors and 936 has 24 divisors. The number 937 is prime. The number 361436250 has 160 divisors and the kissing number 196560 has 160 divisors. All of this could be coincidence (thus meaningless).
The physics forms:
We can replace the number values in eq. (6) and eq. (7) with their respective physics counterparts (as ratios only),
except for d which we assume is a solution for alpha (fine structure constant). All values used in eq. (16) are 2010 Codata and we get 6 significant figures. Later we hope to show that d is also the mass ratio of factor 2 times electron mass divided by the charged pion mass similar to eq. (5) but with a slightly corrected charged pion mass value.
Equation (16) gets even better if we use the excellent Luther-Towler result of their 1982 G (Newton constant).
This result is still considered today to be a hallmark of the empirical determination of the gravitational constant using a well thought out simple Cavendish balance. Using this result and the 2010 Codata in eq. (16) gives the value to 6 significant figures,
Compare this to the Monster Order number in eq. (6).
Here is another physics form that uses the charged pion mass eq. (17). It is a very similar quadratic form to eq. (16). In fact it is probably the same. However, the resulting solution is not as good as eq. (16) is most likely due to the use of the PDG value of the pion mass.
The PDG value for the charged pion mass used in eq. (17) is,
This has a fundamental ensemble of a chiral group of strong force carriers including anti-matter groupings. Maybe, this is a more palatable physics. We can obtain the same result as eq. (16) if the pion mass is logically isolated in eq. (17) using the 'number theoretic' value d eq. (7) for alpha and 2010 Codata and/or using all 2010 Codata we obtain to 6 significant figures.
The charged pion mass using d and 2010 Codata,
The charged pion mass using all 2010 Codata,
Both values gives the value of eq. (17),
Black hole entropy and the quadratic forms:
Here is the semiclassical form of Bekenstein-Hawking entropy of a generic black hole,
If we set it for area A = 4pi r^2 where r = gravitational length based on a Planck mass M_pl black hole then,
The gravitational length is always the same as the Planck length L_pl and the terms cancel resulting in a nice 'number theoretic' result,
For the order of the counting numbers we get number 'even square' pi entropies,
This is a 'nice' quadratic result where the counting number n represents an integer amount of Planck mass M_pl for that particular number 'even square' pi entropy. See comments in the 'even squares' sequence OEIS A016742.
A similar integral cancelation occurs in the use of the pure number form of the Bekenstein-Hawking entropy,
It appears that the Planckian elements eq. (9) also cancel while the counting number n stays integral and we get,
Compare eq. (21) with eq. (20) as they are very similar. A complex Planckian value can be calculated from eq. (21).
Here are a few numerologic observations. The number 840 is a highly composite number with 32 divisors and is the smallest number represented by 1,2,3,4,5,6,7,8 divisors (gluon octet ?). The number 840 is 1 removed from the prime 839. It is also represented by 3 x 280 where 280 is next to the prime 281 that is contained in eq. (7). Also, note that the number 279 is an important number in the Golay code which figures into the Leech lattice See OEIS A171886. The prime number 839 is the largest prime factor of 453060 where 453060 x 453060 is the largest representative matrix in the E_8 group. Three copies of this E_8 structure can be used to construct the Leech Lattice. Also, there is the number 24 which might be the 24 dimensions of the Leech Lattice. Also, the very best 'near integer' prime is represented here as being very close to the prime number 10939058860032031 See OEIS A181045.
Coincident Point Q
If equation (8) is looked at as dimensionless Planck mass squared units N we can arrive at fixed values of black hole entropy and 'degrees of freedom' of a collapsed compact object at 1.36 solar masses. These values are in excellent agreement with the semiclassical Bekenstein-Hawking entropy. Hopefully, the reader will see the Universality of these physics and pure math connections.
The 'degrees of freedom' of this object consist of,
Using the neutron mass m_n a mass value of 1.36 solar masses is obtained,
This is equivalent to,
This corresponds to a Bekenstein-Hawking entropy,
which is in excellent agreement with its semiclassical form , see equation (19).
If you look at equation (22) the complex Planckian value can be rewritten,
This can be set equal to equation (24),
in which case m_n becomes unity and the relation is pure math,
The value m_n is populating feature of relation (22) and (29) implying a probability history for the generation (amplitude) of Planck energy value M_pl. The N value = equation (8) creating the definitions of,
representing powers 1, 3/2 and 2 respectively shows a solution for the connection of the physics forms with the pure maths with solution 0 and N = equation (8) only, to be a point (Q) where they are equal (coincident). That m_n = 1 in eq. (30) also shows a universality in that any extra terrestial civilisation could have determined m_n in their own units which would give the correct Planck energy value related to eq. (8) and all following pure numbers determined by this method. The importance of relation (22) and (29) cannot be expressed enough. It is the geometric mean between two 'almost integers' (check by dividing or multiplying it by sqrt2).
Very Weak Gravity Calculation of the Fine Structure Constant: Isospin and the Figure Eight Knot
As usual there are surprises and what was formerly thought unlikely appears more likely. It may be possible to calculate the empirical value of the 'fine structure constant'. In that equation (6) only relates the extreme (but still weak) form of neutron star, black hole physics and the calculation d of the neutron-neutron 'fine structure constant' equation (6) may be modified slightly utilising some extraordinary coincidences (meaningful?) through the isopsin symmetry of proton-neutron physics and the hyperbolic volume of the figure eight knot complement. If this is true then isospin is a connecting feature of our low energy math to computable 'mathematical structure'. In addition this includes some topological features of the Jones Polynomial which may simplify some things. It should be noted at this point that if all of this is true that it does not necessarily solve things at the high-energy (short scale distance) level close to the Planck energies of Superstring where gravity is truly king. These calculations appear to be addressing the weak gravity end of things which governs solar system dynamics and possibly neutron star degeneracies and large black holes. If this is true then unification physics still has a long way to go in order to reach across the hiearchal desert.
Here we present tw0 extraordinary rational numbers which are very close and proportionaly close (and in order) to the neutron electron ratio, the square root of the proton neutron mass to the electron ratio and the proton electron ratio.
These are very close and proportionally close to their NIST Codata 2010 counterparts. The values (31),(32) are slightly fatter than the empirical derived values.
The denominators are 196560 (kissing number of the Leech Lattice), 196695 (135 plus 196560) . A guess for the 3rd ratio being that of the proton-electron ratio might be 361434570/196830 = 1836.27785398... where 196830 is 54 from the famous 196884 from modular theory. The value 54 is 27 x 2 and is related to 162 = 54 x 3 and the range from 196560 to 196884 is 27 x 12 = 324. If you divide all three of these numbers you obtain 196560/135 = 1456, 196995/135 = 1457 and 196830/135 = 1458 for the adjacent numbers 1456,1457 and 1458. Note that factorization of 361434570 = 2 x 3 x 5 x 7 x 163 x 10559 and 361434571 is prime. The spread from 361436250 to 361434570 = 1680 = 2 x 840. If there is a slight asymmetry in the proton-neutron isospin the difference is slightly larger on the proton side. This helps numerically because the use of rational numbers and geometric means are always slightly weighted asymmetrically on one side and we are using an aspect of the isospin which has this naturally. If you could not find a physical relation like this that operates in nature you would forever be adding corrections which would go on ad infinitum and you would wait for the next Codata set to correct the figures and so on. Just what we do not wish to do.
Let us look at another set of coincidences utilising a geometric mean of the proton-neutron isospin symmetry values. Here is a number which is close to relation (1) and relation(8),
Codata 2010 is only good to 6 significant figures since that is the limitation of Newton's constant. A much better value utilises the Codata 2010 'fine structure constant' value as the only determined parameter and it is good to 11 significant figures,
If we introduce a dimensionless ratio value utilising the hyperbolic volume of the figure eight knot complement V_8 = 2.0298832128... we obtain a coincidence (meaningful or meaningless),
compare to the NIST Codata 2010 value,
A pure mathematics calculation of (36) could then be using relation (8),
which presents the chance that,
The proton-neutron to electron isospin ratio (32) and the knot invariant V_8 can be introduced into the master equation (6) to obtain a value of the 'fine structure constant' that is on top of the Codata 2010 value.
Due to the limitations of the software, obtain a value of alpha to 10 pure number digits,
The mathematical structure of d of eq. (41) could be related to eq. (40) where alpha could be isolated if the relation is true. Is there an explanation why (35) maybe equivalent to (36)? From equations (6) and (41) there appears to be two 'fine structure constants' that are calculable in the weak gravity and extreme gravity both in the long distance low energy physics. The one involves the neutron mass degeneracies of neutron stars and black holes and the other involves the square root proton-neutron mass isospin degeneracy in the very weak gravity limit. This also implies two respective gravitational coupling constants. The short distance physics of strong gravity is not addressed in this regime. Additionally see OEIS A164040, A165267 and a165268.
The Isospin Symmetry and Invariants
The proton to neutron isospin symmetry has always been noted to be a very good symmetry although there is a small asymmetry to it. The proton and neutron particles almost appear to be similar particles with the exception that the neutron has a slightly higher mass than the proton. The strong force is indifferent to this and makes no distinction between the two so obviously nuclei containing both protons and neutrons can exist. It also appears that the isospin symmetry is important in the phase transitions of the conversions of stars (protonic) to dwarf stars and neutron stars. This is a calculation of the neutron mass proton mass ratio involving the small asymmetry of the transitions.
The term on the left side containing the 'fine structure constant' is a little larger than the term on the right side containing the hyperbolic volume of the figure eight complement. The number calculated is a little bit outside the error bars of the Codata value at (38) but is close. The left hand side is the transition from proton to proton neutron degeneracy mix while the right side is the transition from the proton neutron degeneracy mix (superpositions) to full neutron flavor. The constants remain invariant under this action. The knot invariant V_8 implies that SU(2) is at play here and also there is an isotopy of the figure eight knot to its mirror image maybe working through the Lie algebra translations. The symmetry of the form (43) logically implies that the 'fine structure constant' alpha is computable via a 'pure math' form since V_8 is a computable constant.
Note: Running of the Fine Structure Constant in Stronger Gravitational Fields and Its Relative Rigidity as a Constant
It has been thought for a while that the coupling constants may run to higher values in stronger gravitational fields such as that afforded by very massive compact astrophysical objects, i.e. dwarf and neutron stars, black holes. In July 2013 a paper was published in Physical Review Letters (see Reference) which checked the dependency of the 'fine structure constant' on the strong gravitational field of a white dwarf star. By observing the elemental emission lines for this white dwarf it was determined that there was no change in the constant within a sensitivity of 1 part in 10,000. The particular white dwarf star in the study has a gravitational field about 30,000 times greater that that of Earth. It should be noted that 1 part in 10,000 does not go very far decimal wise into the 'fine structure constant'. Also, 30,000 times greater than Earth's gravitational field is just not that much in regard to the gravitational strength scale from weak gravity to Planck scale gravity. It is really nothing. I am not criticising the paper as I think it is actually a cool determination and result which goes in the proper direction and may yield much in the future. The method they use cannot be applied to neutron stars as they have no elemental emission lines. Whereas, the white dwarf gravity is more Newtonian the neutron star (or in particular the neutron star that is described above at 1.36 solar masses) has relativistic effects kicking in due to its more extreme gravitational field. It has a gravitational field about 100,000,000,000 times that of Earth. Before we go any further it should be made plain as possible that this is still at the weak gravity (long distance) region of things and that there are more ungodly gravitational effects as you approach Planck energies or short distances. So forget that we are are the really super extreme end of things with the neutron star because we are not. Throw out any intuitive feel at the doormat, as there is still a mega-huge hierarchal desert before us before we get to Planckian energies. The constants do run but they have to be rigid for formation of the Universe around us. The relative rigidity of the dimensionless constants was probably set at about 10^-35 - 10^-29 seconds after the the Big Bang. There will be no variation noted in the 'fine structure constant' by looking backward in time (period). If you look at equation (7) which is solution d we have a slightly larger value of the 'fine structure constant' which may be a solution for alpha for the 1.36 solar mass neutron star of our theory. If it is true then relativistic effects start to bother the empirically derived 'very weak gravity' alpha at the sensitivity of about 1 part in 10,000,000,000. Relativistic effects start to affect the 'fine structure constant' at the neutron star gravitational field somewhat minimally but it can be seen in the computation.
From eq. (8) we can calculate (using as a semiclassical form) a compact gravitational object (neutron star or black hole) at 1.36 solar masses being a coincident point where pure math and physics meet. Using the very large symmetry of the Monster group (eq. (6)) we can calculate (in the classical limit) the 'fine structure constant" that exists in the strong gravitational field of a neutron star at 1.36 solar masses which is 0.007297352751... eq. (7). This is slightly larger that the empirically determined alpha as would be expected. The empirical 'fine structure constant' is calculated from the midpoint isospin symmetry of a collapsing star where superposition of proton neutron states could occur according to quantum mechanics. Again using the same formula but with usage of knot invariant of the figure eight knot (eq. (41)) the 'fine structure constant' in the much weaker gravitational field is 0.007297352567... eq.(42). However, there is probably not a gravitational compact object that would exist like this in nature but that this allows for a computation of classical type is phenomenal. It is interesting that these calculations are primarily classical using the Reals only. It remains to whether future Codata sets support these numbers. It is possible that these numbers could be adjusted if the integer ratios are off by very little. The mappings to the physics forms are suggestive that the forms are at the least on the right track.
For an interesting read go to P.A.M. Dirac's The Relation between Mathematics and Physics
Also see B. Mazur's Number Theory as Gadfly
Equations in Mathematica
1. (2/(d)^4) (361436250/196560)^2 E^(2 Sqrt Pi) 70^2 (((E^(2 Pi Sqrt) 70^2)^(1/65536) - 1)^(-1))^(1/2048) == 808017424794512875886459904961710757005754368000000000
2. d = (Sqrt(281/11) Sqrt(E^((Sqrt(163) Pi))))/(38817792 7008506450687690^(1/4) ( ((E^((Sqrt(163) 2Pi) )70^2)^(1/2^16))-1)^(1/2^13))
3. (for Black hole entropy) 8Pi (n Sqrt2 840 E^(Pi Sqrt163)/24)^2 (E^(2Pi Sqrt163)70^2)^-1 = 4Pi n^2
P. Mohr, B. Taylor, D. Newell, Rev. Mod. Phys. 84(4). 1527 - 1605 (2012), CODATA recommended values of the fundamental physical constants: 2010, National Institute of Standards and Technology, Gaithersburg, MD
J .C. Berengut, V. V. Flambaum, A. Ong, J. K. Webb, John D. Barrow, M. A. Barstow, S. P. Preval and J. B. Holberg, Phys. Rev. Lett. 111 010801 (2013) 'Limits on the Dependence of the Fine Structure Constant on Gravitational Potential from White Dwarf Spectra'
Stephen L. Adler, 'Theories of the Fine Structure Constant a', FERMILAB-PUB-72/059-T
Julija Bagdonaite, Mario Dapra, Paul Jansen, Hendrick L. Bethlem, Wim Ubachs, Sebastien Muller, Christian Hemkel, Karl M. Menten, 'Robust Constraint on a Drifting Proton-to-Electron Mass Ratio at z = 0.89 from Methanol Observation at Three Radio Telescopes' astro-phys arXiv:1311.3438 , published Physical Review Letters
Thibault Damour, 'Theoretical Aspects of the Equivalence Principle', arXiv: 1202.6311v1 [gr-qc] 28 Feb 2012
It is probable that if the pure number calculations produce quasi-values then they can be explained by their 'relative' adjustments to produce the numbers we look for thus rendering the calculations meaningless (basically, what Richard Feynman has said in the past). However, it is strange that the mass ratios they produce have a one to one correspondence to the physics forms.
After a long bungling period of trying to find a correlation between the Codata value of the fine structure constant and the calculated constant from the formula (7) where d = 0.00729735275109225.. (just outside error bars of Codata) I realise that this is the wrong application (entirely biased). Since the formula(s) utilise neutron physics the d value should represent alpha at a slightly larger value than Codata which it does. This would have something to do with the slightly higher energy vacuum at a certain sized neutron star or black hole that the formulation hints at (and a stronger gravitational field).
In relation (22) is the large number ~10^19 which if multiplied by the Codata 2010 neutron mass (representing a degeneracy in this case) calculates 2.176557865... x 10^-8 kg which is very close to the Codata 2010 Planck mass. The number from (22) could be complex as well and it is possible that it is involved as a probability amplitude or some kind of histories. That this number represents degrees of freedom or degenerate states may help expain why relation (21) of black hole entropy is not a cheap trick.
It appears that the neutron - electron mass ratio determined by the math (eq. (12)) is slightly fatter than the Codata 2010 value (eq. (3)). The other values are very close. It is interesting that the value (eq. 12)) determined by the math has a repeating (periodic) decimal. Is there a correction going on? Also, consider that the neutron - electron mass ratio is more important than the proton - electron mass ratio in that there is degenerate matter in dwarf, neuton stars hence the neutron - electron mass ratio has the major role and the proton is not important here. If this is correct does this imply a math- physics correspondence point where a rigid well defined mathematical structure (Monster Group) and physics meet and why not at where gravitational states and quantum mechanics start to dance at quantum degeneracies? Where most activity seems to be in finding relations between physics and maths involves the proton, this may be wrong. Outside of looking at degeneracies and into the low energy and absence of influences of strong gravitational fields (proton world) the vacuum polarisations make calculations 'mushy' and maths are then approximate. (No, the gravitational fields that stars and even the most massive stars have do not closely compare to the strong gravitational fields of neutron stars and black holes.The balancing pressures that stars have are thermal and not degenerate.) That the anthropic view is pathological is due to our looking at things at our end i.e. under the fat proton (vacuum polarisations) and the non-degenerate fat protonic gravity under which life evolves. It should be noted that the neutronic gravitational coupling constant is slightly larger than the protonic gravitational coupling constant. If all of this is true then the quasi values are those values empirically determined (and hence corrupted) while the actual values are mathematically determined from a structure at a natural correspondence point.
MYSTERY: There appears to be another relation of the proton to electron mass that is very similar to the neutron electron mass depicted in relation 12. That relation is 361435520/196830 = 1836.282680485...which is a little fatter than the Codata 2010 value 1836.15267245. The ratio has similar attributes that relation 12 has. The number 361435520 is 1 from the prime 361435519. The number 196830 is 1 from the prime 196831. The number 361435520/196830 has a period 2187 in the decimal.The number 196830 is 270 from the famous kissing number 196560. The number 270 sits in between the twin primes 269 and 271. The number 196830 is only 54 from 196884. Why is this proton electron number fat like relation 12? Obviously integer relations cannot calculate the true value and it seems that these numbers may be some kind of overcorrection to actual values. Here is why there may be something to these numbers. If you divide 361436250/196560 by the Codata 2010 value 1838.6836605 you obtain 1.0000680379. If you divide 361435520/196830 by the Codata 2010 value 1836.15267245 you obtain 1.0000708046. Look at both values together 1.0000680 and 1.0000708. They are very close in value to each other. If this is a coincidence this is still boggling because because the odds of obtaining this from the integer set is very much against such agreement. It would be interesting to know what the actual math value is supposed to be then the particle ratios could then be calculated directly.I think that the ratio 361435520/196830 shows that the neutron physics are truly what is important and not the proton intoduction into the theory. The neutron determined fine structure constant (solution for d in relation 7) is closer to the empirically derived Codata 2010 fine structured constant. It looks very hard to find any mathematical resulting structure for the Codata 2010 fine structure constant. This makes the empirically derived dimensionless constants to be something like "almost nonanthropic constants" but they still remain anthropic (uncalculable).
MYSTERY cont'd: The value 361435520/196830 was found as a result of the following formula: 2((1/(a))^4 E^(-Pi/4 1/(a))(2/E)^(0.25))^-1 where a is the fine structure constant . Placing in the Codata 2010 fine structure constant value, 2((1/(a))^4 E^(-Pi/4 1/(a))(2/E)^(0.25))^-1 = 3.38202354001 x 10^38. If you use the solutiion d from relation 7, 2((1/(d))^4 E^(-Pi/4 1/(d))(2/E)^(0.25))^-1 = 3.382014833056 x 10^38. If you divide this by relation 8, exp(2 pi sqrt 163) 70^2 you obtain 1.0013756490... . Look at this 361436250/196560 196830/361435520 = 1.0013756488... The Codata 2010 neutron proton ratio = 1.00137841917
It is looking better that the 'fine structure constant ' is 'pure math' calculable from theory. Apparently, there are two computable 'fine structure' constants related to the isospin connection between the proton and neutron in gravitational compact objects (i.e stars and neutron stars possibly black holes). The weakness of the theory now lies in the integer ratios (rational numbers ) like 361436250/196560 and whether this is a true adjustment. Since we are dealing with the geometric mean which is weighted heavier on one side the use of these ratios possibly may be justified.
I am replacing (in the above text) the proton electron mass ratio pure number form 361434555/196830 = 1836.277777 with 361434570/196830 = 1836.27785398 as I believe the spread from 361436250 -1680 = 361434570 is probably where it should be. The number 361434570 is 1 from the prime number 361434571. The number 1680 has been ubiquitous throughout and it is 840 x 2 which is important to the theory. The number 1680^2 = 2^8 3^2 5^2 7^2 which is = 24^2 70^2. However, I am not sure whether this pure number (approximate due to the geometric mean) is relevant as it does not play a role in eq.(6) and eq.(41). The establishment of pure math physics is probably (begins the computational low energy limit) at the nonphysical proton-neutron superposition object e.g. 361435488/196830 See eq. (41).
Due to some limitations of the Wolfram Alpha Mathematica platform (it is free to use) I managed to finally go around this (I had some time to do this finally) on Equation (40) and obtained a 'pure number' value for the 'fine structure constant' = 0.007297352569797892240174748... . This on top of Gabrielse's value (now Codata 2010) of 0.0072973525698 (24). On Equation (41) where d = 0.0072973525673737... this is not that far off of these values. I believe that Equation (40) is some how equivalent to Equation (41) but am not sure on how to go about proving this. The relation 361435488/196695 still seems to be the best integer ratio here but possibly could be adjusted or replaced or something. On the matter of Equation (40) computing Gabrielse's value very accurately does not necessarily mean that the computation is nailed in that there is an uncertainty of (24) involved at the tail end (which means that it is still a crapshoot). I am actually suspicious of nailing any Codata number with a 'pure math' computation. This is a very difficult business. Eventually I believe that some form of Equations (6) and (41) will represent an isopspin symmetry relation involving math/physics computations in Nature.
We will wait to see what the next Codata set of values bring (hopefully in 2014) as we seem to be in a territory, then progess as null or positive can be ascertained. It could be that equation (6) and (41) are wrong or, correct and relatively precise and that equation (40) is a fluke or, is correct and relatively precise. However, a new speculation has ocurred to me that perhaps equation (6) and (41) (utilising Monster symmetry) is approximately right and that equation (40) is maybe correct. What if equation (6) and (41) are approximate because they represent 2 'strange attractors' (and we realise that all the calculations on this page are in the classical regime and not any where near the Planck energies). Right now there is no contextual theory of the 'number line' between 196560 and 196884 and as to what that could possibly be. Maybe the Cantor Set with something involving partitioning of the number 324 (remember we have 196560 +135 = 196695 and 196695 +135 = 196830 where 196560 and 196830 are one removed from a prime and 196830 +54 = 196884. The numerators of our rational numbers e.g. 361436250, 361435488 and 361434570 span 1680 where 1680 = 24 x 70 and 24 x 70 +48 = 1728 suggests modular elliptic functions are involved here as well. A fractal set may be here which would be non-linear and represents some aspect of thermodynamics in the classical regime. Stephen Adler's has a new paper which suggests that quantum decoherence which deals with classical fields of gravitation may come about as a result of a noise component in the gravitational metric g_uv. Adler explains that this noise is in the complex plane component (best as I can explain) of g_uv and is analogous (in the classical sense) to a 'spacetime foam'. This is identified as the mechanism for state vector reduction (decoherence) in macro sytems or some line in the sand separating emerging states, classical or quantum? If this component in complex multiplication CM is there does this fractal noise have something to due with corrupting the dimensionless
cont'd : constants even at the weak gravity regime so that the requirement of these numbers not being absolute structures end the asymmetry in GR as it should? Could the noise component of Adler's be a fractal noise and then would it be a dynamical entity of a subtle type (in the complex plane) that affects quantum coherence and the concept of absolute rigidity of the dimensionless constants? I have never really thought much of the fractal maths being that important in the Standard Model and especially in the near Planck energy scheme involving superstrings, but they may really be important in the classical thermodynamic area. Even then the infinite self similarity zooming maths maybe only a tool that is approximate but very useful in application. Fractals are important in holography so maybe high energy holographic models, theory maybe include aspects of fractals but not neccessarily the overiding importance that has been given to them as they may be more emergent in the low energy (classical realm). Another interesting aspect of Adler's theory is that the complex noise component of his decohering spacetime foam does not affect the large scale structure gravitation (macro Universe structure building) of astrophysical structures. At this time (for fun only) let us say that Equation (40) actually calculates the 'fine structure constant' = 0.007297352569797892240174748... and that Equation (6) and Equation (41) calculates the low energy value of the 'fine structure constant' (close to what Equation(40) computes) and the slightly bent value of the 'fine structure constant' due to the heavy gravitational field of the 1.36 solar mass 'neutron star as a consequence of a dynamical chaotic deterministic application to the low energy fields (all originally from initial conditions near t= 0 of the Planck regime) then the two 'fine structure constants' are approximately close to the actual values but due to complexity remain as structures around a 'strange attractor' point? If so an approximate adjustment can be made as
cont'd: follows, multiply the ratio of the the two 'strange attractor' 'fine structure constant' values of Equation (6) and Equation (41) by the value obtained from Equation (40) and we have the following values, a = 0.007297352697978922....(to be verified empirically) and an adjusted value for the 1.36 solar mass neutron star at 0.00729735275317864.... A theory of isospin states between (continous Lie algebra transformation) two computable values of the low-energy 'fine structure' (empirically observed value) with an explanation of the (non-absolute) relative rigidity of dimensionless constants (including two 'gravitational coupling constant's) through a determinstic chaos ( discrete fractal maths) held even more rigid through a huge symmetry is presented. Here is an interesting mind experiment: Say you are a being at the dawn of time (at the Planck energy), can you say that there is a logi-philosophico way of determining how the Standard Model will coalesce out to the parameters after the expansion and where the points will be in the lower vacuum energies. If it is a deterministic chaos involved (after the quantum) then you would be hard pressed to make the final predictions (our Universe today). It is then kind of like predicting the weather. However, if you run the same conditions each time you get the same results ("strange attractor' points). Weird stuff, but the ultimate reality is not described by our low-energy classical physics (even though fractals appear crazy psychedlic) but some where near the origin. Here are additional references: Stephen L. Adler, 'Gravitation and the noise needed in objective reduction models' arXiv: 1401.0353v1 [gr-qc] 2 Jan 2014 Lawrence B. Crowell, 'Counting States in Spacetime' EJTP 9, No. 26 (2012) 277-292 Amanda Folsom, Zachary A. Kent, and Ken Ono (Appendix by Nick Ramsey) 'L-Adic Properties of the Partition Function' 2012 AIM Jan Hendrik Brunier, and Ken Ono, "Algebraic Formulas for the Coefficients of Half-Integral Weight Harmonic Weak Maass Forms' arXiv:1104.1182
Here is something that may apply to dynamical systems retaining invariances in emergence under action of a large symmetry group: Anatole Katok and Viorel Nitica, 'Rigidity in Higher Rank Abelian Group Actions' Introduction and Cocycle Problem, Volume I, Cambridge Tracts in Mathematics,Cambridge University Press 2011 www.cambridge.org/9780521879095 or find it under Google Books.
We will wait and see what the changes in the new Codata set will bring. In the meantime here is a powerpoint presentation on Ramanujan's remarkable intuitions 'Recent developments of Ramanujan's work' by Michel Waldschmidt 2003 (still very much relevant) http://www.math.jussieu.fr/~miw/articles/pdf/Mysore-12-2003.pdf
Well, it looks that the LHC is not discovering SUSY at the TeV level. I do not think it will. But, what is wrong with SUSY being broken at the Planck energy level? So we are faced with HOT DENSE SUSY then (at the Planck energy). So what is wrong with that. Nothing. So be it. ...... Since, we are waiting for the next Codata set (hopefully Codata 2014 which would be released about June 2015) let us declare that these calculations (here) then represent a High Energy Physics (at our low energy realm of things) experiment which has a good measure of predictabilty. What I have in mind is the use of equation (36) and its sensitivity to its values of the 'fine structure constant' and Newton constant. First understand that many of the Codata 2010 fundamental physics values are very well determined. This includes the mass values of the proton and neutron (m_p m_n) , Planck constant h and the speed of light c (which is exact by definition). Also, the 'fine structure constant is very good to 13 significant figures. Not so for the Newton constant G which has historically fluctuated quite a bit. If our coincidence of equation (8) is correct and not a coincidence then we can use the Codata 2010 to great advantage to isolate a good determined value for Newton constant G : Exp(2 Pi Sqrt163) 70^2 = hc/(Pi G m_n^2) where m_n = neutron mass we then get G = 6.67354 *10^-11 m^3/kg s^2 . We will see if the next Codata set has a new G value and if our predicted value is close. Also, as part of our ongoing experiment we will see if changes in new Codata values bring a particle physics expression closer to the value that is expressed in the OEIS 164040 http://oeis.org/A164040 of 3.820235400 *10^38 (dimensionless) if we use the current Codata 2010 we obtain hc/(Pi G m_n m_p) = 3.38187 *10^38 where m_n = neutron mass and m_p = proton mass . If we use our adjusted and predicted G = 6.67354 *10^-11 m^3/kg s^2 and assuming all the other 2010 Codata values are very good then: hc/(Pi G m_n m_p) = 3.38203 *10^38 cont'd
cont'd (since G is only good to 6 significant figures) this is very much toward the value of 3.3820235400 *10^38 and begins to look as these values are not coincidences. There appears a good chance that equation (8) and equation (36) are not coincidences and are related somehow. The idea on equation (36) is not so much a gauge unity then between gravity and the 'fine stucture constant' of electromagnetism but that of decoherence of quantum superposition in the semiclassical expressions of weak fields. Now, we then wait to see what our Codata experiment brings.
Here is an excellent reference on the historical problems of obtaining a well determined value of the Newton Constant (also known as the Big G problem) from the Eot-Wash Group. The title of the article is 'The Controversy over Newton's Gravitational Constant'. The famous Luther-Towler result is mentioned here too. http://www.npl.washington.edu/eotwash/bigG
2013 Mark A. Thomas