Hello Branko

Richard Feynman in his Nobel Acceptance Speech

(http://www.nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html)

said: "It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the relationship. And example of this is the Schrodinger equation and the Heisenberg formulation of quantum mechanics. I don't know why that is - it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature."

I too believe in the simplicity of nature, and I am glad that Richard Feynman, a Nobel-winning famous physicist, also believe in the same thing I do, but I had come to my belief long before I knew about that particular statement.

The belief that "Nature is simple" is however being expressed differently in my essay "Analogical Engine" linked to http://fqxi.org/community/forum/topic/1865 .

Specifically though, I said "Planck constant is the Mother of All Dualities" and I put it schematically as: wave-particle ~ quantum-classical ~ gene-protein ~ analogy- reasoning ~ linear-nonlinear ~ connected-notconnected ~ computable-notcomputable ~ mind-body ~ Bit-It ~ variation-selection ~ freedom-determinism ... and so on.

Taken two at a time, it can be read as "what quantum is to classical" is similar to (~) "what wave is to particle." You can choose any two from among the multitudes that can be found in our discourses.

I could have put Schrodinger wave ontology-Heisenberg particle ontology duality in the list had it comes to my mind!

Since "Nature is Analogical", we are free to probe nature in so many different ways. And each of us surely must have touched some corners of it.

Good luck and good cheers!

Than Tin

Hi Yuri,

First of all, thank you for inviting me to participate in this interesting competition.

Of course there are many ways to explain the Planck length. I am to Mr. Okinbo briefly, in one sentence pointed out what I think is most important. I think the Rudjer Boskovich seen that length in item "a" of His power curve (Figure 1) in my work. That's why; to you and others I suggest studying his work. Someone who saw the atomic orbit and Planck length, two centuries before Bohr and Planck saw many more.

I read your article first day and I shall rate it last day.

All the best in the essay contest,

Regards,

Branko

Dear Anonymous Satyavarapu,

I am convinced that the cyclicity, ceases (nor in clusters, nor in filaments). Even the whole the Universe is cyclical. There is no logical reason to stop cyclicity at any level of the organization of matter. The question of the center of the universe raises the question of shape the universe. Presentation the Universe on National Geographic TV in form of a sphere for me is more ridiculous than fear of Columbus sailors, what happens when they reach the end of the flat earth. There is no privileged center, nor the edge of the Universe.

Regards,

Branko

Dear Branco,

Still there is no response from you regarding comments on my essay and I would like to give you maximum possible rating on your innovative essay as I have promised although you care least for rating essays. So, please, rate my essay and inform me in my thread as it is very important for me.

All the best,

Sreenath

Dear Than Tin,

Something similar to you, I said in my article:

(Of course, this does not mean that I am proposing a new stereotype or that these four postulates are sufficient for explaining the functioning of the Universe. The results can be obtained in different ways.)

If you read my article you will see that Simplicity, expresed with Occams razor, is my main tool in computing my relation (2).

Plancks constant and Planck units I expresed in the table 1. in my article.

So, I expect your comment, on relation (2) and Table 1.

Regards,

Branko

Hi Branko,

I am happy to report that I did not find any typos in the formulas in the table. You wrote:

> You say (where x and y are complicated expressions). What would you say then for tensor Lie algebra ...in other articles?

I would say "very complicated expressions". :)

> Thus, the formula (2) is simple, especially if you shorten giving physical meaning of individual members.

I will take a look at your vixra papers for that.

It seems to me that all of the Planck formulas in the table follow from the definitions of the terms and the scaling factor that you use. BTW, I noticed that the scaling factor you use, i.e.

2^(-cy/4 - p/4) (2pi)^(-3/4) = 1.2512639e-61

can also be written as:

[8 pi^3 2^cy M_n]^(-1/4)

where M_n is the number of protons in the Universe (i.e. mass_universe / mass_proton). Perhaps that will give you additional insight.

I think that you might like Nassim Haramein's paper which includes a geometric derivation of the proton mass and of the gravitational coupling constant.

I wonder if there is a connection between your formulas and the hyperspherical structure that I discuss in my Software Cosmos essay.

Hugh

Hi Hugh,

Thank you for the excellent substantive discussions by taking on the occasion of my article.

Could you please send me your email so I can give you the formulas in the right form?

The form of the formula that you gave is also possible. Value p is, by definition, the logarithm to the base two (mass of the universe / proton mass ). But I avoid using the term the number of protons in the universe, because I think that this is not the same as the ratio of masses of the Universe / proton. Not all masses of the Universe are in the proton.

My approach is very similar to hyperspherical structure that you discuss in Software Cosmos essay. I think I found a simpler approach, but it is essentially the same idea.I did not finde Nassim Haramein's geometric derivation of the proton mass and of the gravitational coupling constant. Thanks to your sugestions I think that relation (2) can be written more simple and acurate as:

[math]\gamma= 2^{(cy+p+3t)/(2+2a^{2}m)}=1.0013784192[/math]

Where mathematical constant are:

[math]2\pi=6.2831853, t=log(2\pi,2)=2.6514961295, cy=e^{2\pi}= 535.4916555248 [/math]

Physical constants:

[math]a=1/\alpha=137.035999074, \mu=1836.15267245,m=log(\mu,2)=10.8424703056[/math]

Also:

[math]p=log(Mu/mp,2)=cy/2-(\mu/a+1)/(\mu/a+2)-1=265.8107668189[/math]

It is most obvious in bits.

Regards Branko

    Branco,

    An exceptional essay, not just "well organised and practical" as you wished, but incisive and inspirational. I've also discussed how we're "Overwhelmed by information overload." Of course it helps that we agree on much, though our approaches to those truths couldn't be more different. I derived the dynamic cyclic (and ~fractal) universe (Helical CMB Asymmetry) and feedback before the complexity I discuss this year. There are also analogies in Hindu and Mayan cultures.

    In terms of agreement I'm also a massive fan of Boscovich and 'non point' reality, but I learnt more from your essay, including a link with my point/circle correspondence for uncertainty. I've also found the Cartesian system inadequate for modelling evolution of interaction.

    Your agreement with Klein last year is also warming as I find an almost infinitely hierarchical and recursive series of equivalent backgrounds for discerning maximum speed (as my last 3 essays).

    Your mathematical derivations look quite brilliant. But I'm now no expert. What I am interested in is how you may find we can bridge the gap to decode the 'noise' of higher orders where uncertainty resides. I also commend the McHarris essay on chaos to read.

    I agree your work is of great value, but will it be noticed and read? It certainly gets full points from me to get it up the list (though my last two finished 7th and were overlooked!!). I hope yours may anyway help produce a recursive mathematical description consistent with my ontology and closing the gap between current binary maths and nature.

    Very well done and best wishes in rising from the information swamp.

    Peter

    Dear Branko,

    After reading your essay :

    Yes, Branko : « Of great importance in this article is, I hope the widely-accepted view, that parts are dependent on the whole (Universe) and are also an integral part of the whole, therefore, the whole is also dependent on the parts! »

    « whole, parts », two opposites.

    Bits and cycles :

    · Information

    · Cyclicity

    · Dimensionless quantities

    · The quantum nature of the Universe

    You are right. You'll be surprised one day to discover how your views are right.

    My theory begins with the first principle : eDuality.

    But I do not develop it here. A will publish a book.

    One single principle leads the Universe.

    Every thing, every object, every phenomenon

    is under the influence of this principle.

    Nothing can exist if it is not born in the form of opposites.

    I simply invite you to discover this in a few words,

    but the main part is coming soon.

    Thank you, and good luck!

    I rated your essay highly accordingly to my appreciation.

    Please visit My essay.

    Hello Peter,

    (Google translation)

    Thank you for the kind words.

    Honestly I met 2 percent Boskovich achievements until I developed my theory. I found his work to confirm my understanding. In my 61 years I have little time to study the depth of his work (very hard to read). I recommend it to younger people because of it certainly can benefit. It is very interesting that I am, McHarris article (its A = 3.82 is 3.829 for me) and your, repeatedly read and put in later to further study it. Your work, I was trying to understand, but it will be easier for me when I read your previous articles and translate it better. Then I'm going to call you on your email. Is it really, ymail.com, not gmail.com.

    In any case, you deserve a high rate.

    Regarding your concern, the work will be recognized. I quote Milutin Milankovich. Do not worry; my work is now living its own life. Now Milankovic is most cited in the world by climate change. Little is known that, during his life he was in Europe, the famous civil engineer and popular science writer.

    Regards Branko

      Branko,

      Thank you. We are the same age, and wisdom I think. But it must be true that for every Milankovich there may be 1,000 with important work buried under the information overload. It is not 'recognition' I'm worried about, only that the truth does not end up ignored and buried. 'Chance' always plays a hand, so every point now may be the one that makes the difference! Your work should also be noticed not lost.

      I'm glad you'd like to stay in touch. The best papers of mine to read are my last two essays here, and my PR quantum optics paper with J Minkowski, on open access at arXiv; Resolution of Kantor...anomalies.

      Very best wishes

      Peter

      Dear Amazigh,

      (Google translate)

      I was all the principles and postulates first applied mathematically and then I realized it was already expressed in words. But nobody, it was not able to express with simple mathematical expressions.

      Thanks for your, principled comment and rate.

      Regards Branko

      Dear Branko,

      I like your conclusion:

      "Of great importance in this article is, I hope the widely-accepted view, that parts are dependent on the whole (Universe) and are also an integral part of the whole, therefore, the whole is also dependent on the parts!"

      Good work, nice formulae and I am not surprised because I agree with you that everything is linked to the age of the Universe. Do you have a proper theory behind these formulae ? If you haven't then you will find that people will just call it numerology or coincidence (unfortunately).

      I have developped a theory that you might find of interest, it could shade some light on your formulae. I also have a lot of formulae and I am sure one could find some correlation with yours.

      Let me know what you think.

      You can also check out my essay and rate it if you like it.

      Good luck with the contest.

      Patrick

      Hi Branko,

      > Could you please send me your email so I can give you the formulas in the right form?

      Yes, I will write you.

      > But I avoid using the term the number of protons in the universe, because I think that this is not the same as the ratio of masses of the Universe / proton. Not all masses of the Universe are in the proton.

      Yes, of course, my mistake.

      > My approach is very similar to hyperspherical structure that you discuss in Software Cosmos essay. I think I found a simpler approach, but it is essentially the same idea.

      I would like to discuss this further, perhaps after the contest.

      > I did not finde Nassim Haramein's geometric derivation of the proton mass and of the gravitational coupling constant.

      Sorry, I will try again: try here.

      > Thanks to your sugestions I think that relation (2) can be written more simple and acurate as: ...

      Yes, this is very nice. My only suggestion is that we should write tau = 2pi. (See the Tau Manifesto by Michael Hartl.) Then the mathematical constants are simply tau, log_2(tau), and e^tau.

      > It is most obvious in bits.

      Yes, it suggests we are talking about information here. Thank you for this, I hope I can understand what it means for the Software Cosmos.

      Hugh

      Hi Branko,

      I used your formula to determine the uncertainty on gamma, and compared them to the CODATA 2010 uncertainty. I show them here in "concise" form with parentheses:

      gamma = 1.001378419187(17) should equal

      mn/mp = 1.001378419(89) the CODATA value.

      Notice first that the calculation for gamma is within the uncertainty of the CODATA determination, which has an uncertainty for mass of the neutron and mass of the proton of 7.4E-35 kg.

      But notice also that the gamma calculation is much more precise than the ratio determined by CODATA:

      Uncertainty in gamma = 1.734E-11

      Uncertainty in mn/mp = 8.85E-8

      In fact, the calculation is 5106 times as precise, so this is a prediction for future determinations by CODATA.

      But note that for this analysis, we have calculated the uncertainty in the CODATA mn/mp by using their final values for mn and mp. However, within the CODATA system, the ratio of mn/mp is known more accurately than either the mass of the neutron (mn) or the mass of the proton (mp), so perhaps we have been a bit unfair to CODATA.

      On the other hand, in calculating the Zivlak gamma (as I call it) we have used the mass of the electron and the mass of the proton as input values separately. Within CODATA again, their ratio is known more accurately than their values. So we might be able to improve the Zivlak calculation as well.

      I will take these considerations into account in the next post.

      Hugh

        Hi Branko,

        Using NIST website CODATA values for 2010 and previous years, I have been looking at the accuracy of your formula. I wondered, using older experimental values, how well the formula could predict the currently best known value of mn/mp (i.e. from CODATA 2010).

        First, here is the data I used:

        Year mass_e mass_p mass_n

        1969 9.1095580(540)e-31 kg 1.6726140(110)e-27 kg 1.6749200(110)e-27 kg

        1973 9.1095340(470)e-31 kg 1.6726485(86)e-27 kg 1.6749543(86)e-27 kg

        1986 9.10938970(540)e-31 kg 1.67262310(100)e-27 kg 1.67492860(100)e-27 kg

        1998 9.109381880(720)e-31 kg 1.672621580(130)e-27 kg 1.674927160(130)e-27 kg

        2002 9.10938260(160)e-31 kg 1.67262171(29)e-27 kg 1.67492728(29)e-27 kg

        2006 9.109382150(450)e-31 kg 1.672621637(83)e-27 kg 1.674927211(84)e-27 kg

        2010 9.109382910(400)e-31 kg 1.672621777(74)e-27 kg 1.674927351(74)e-27 kg

        Year 1/alpha mass_p/mass_e mass_n/mass_p

        1969 137.03602(21) 1836.1090(110) 1.001379(13)

        1973 137.036040(110) 1836.15152(70) 1.001379(10)

        1986 137.0359895(61) 1836.152701(37) 1.0013784040(90)

        1998 137.03599976(50) 1836.1526675(39) 1.00137841887(58)

        2002 137.03599911(46) 1836.152667261(85) 1.00137841870(58)

        2006 137.035999679(94) 1836.152667247(80) 1.00137841918(46)

        2010 137.035999074(45) 1836.152667245(75) 1.00137841917(45)

        In 1969 and 1973, the ratio mass_n/mass_p is not reported separately, so it is calculated from the two mass values given.

        In the next table, the "CODATA gamma" is the value given in the particular year. The "Zivlak gamma" is the value obtained by using the experimental values known at the time in the Zivlak equation.

        Year CODATA gamma Zivlak gamma

        1969 1.001379(13) 1.0013784178(53)

        1973 1.001379(10) 1.0013784162(23)

        1986 1.0013784040(90) 1.00137841927(13)

        1998 1.00137841887(58) 1.001378419181(10)

        2002 1.00137841870(58) 1.0013784191948(93)

        2006 1.00137841918(46) 1.0013784191907(19)

        2010 1.00137841917(45) 1.00137841920390(92)

        We now look at the success of the Zivlak equation in predicting the current value (CODATA 2010) for gamma.

        The "CODATA Error" is the difference between the value given at the time (i.e. in 1969 and so on) and the current 2010 value, as a proportion of the 2010 value.

        The "Zivlak error" similarily, is the difference between the value that could have been calculated at the time with the current 2010 value, as a proportion of the 2010 value.

        The C Error/Z Error column shows the ratio of the CODATA error to the Zivlak error.

        Year CODATA error Zivlak error C/Z Error

        1969 1.3144463E-5 5.790777E-9 2269.8962

        1973 1.0276465E-5 2.7340497E-9 3758.6973

        1986 1.5149118E-8 5.758918E-10 26.305494

        1998 1.0285822E-9 4.5983836E-10 2.2368343

        2002 1.0301803E-9 4.586465E-10 2.2461312

        2006 9.100147E-10 4.5128046E-10 2.016517

        2010 8.9876123E-10 4.5029516E-10 1.995938

        There are several things of note.

        (1) First, note that, for all years, the Zivlak gamma is significantly closer to the current known value than the value that was obtained at the time via the sophisticated methods of CODATA. For example, in 1973, it was 3758 times more accurate (this is partially due to the fact that the mn/mp ratio was not disclosed by CODATA).

        (2) By 1986, the Zivlak equation had produced a value that is more accurate than the value we have from CODATA even today.

        (3) Perhaps most surprisingly, even when we compare the 2010 CODATA value against itself as the gold standard, the Zivlak value is superior. This is because it predicts a (slightly different but very precise) value with very little uncertainty. The larger uncertainty in the 2010 CODATA value means its average error is higher.

        These results suggest that the Zivlak formula for the ratio of neutron to proton mass has real predictive power. Please accept my congratulations for your work on this!

        Hugh

        Thank you, Hugh. It is indeed a pleasure to talk with you.

        Regards,

        Branko

        Hi Branko,

        I am enjoyed reading your essay, and I see your position close to my own approach.

        So, I definitely welcome your work and I intended to study it further (when we will finish this battle!) Then we can examine the details and change our opinions on more reasonable ground. I have rated your work on good score!

        Best wishes,

        George

        Hi Branko,

        Let's try this again, with the tables in LaTeX...

        Using NIST website CODATA values for 2010 and previous years, I have been looking at the accuracy of your formula. I wondered, using older experimental values, how well the formula could predict the currently best known value of mn/mp (i.e. from CODATA 2010).

        First, here is the data I used:

        [math]\begin{tabular}{l l l l}

        Year &mass_e &mass_p &mass_n \\

        \hline

        1969 &9.1095580(540)e-31 kg &1.6726140(110)e-27 kg &1.6749200(110)e-27 kg\\

        1973 &9.1095340(470)e-31 kg &1.6726485(86)e-27 kg &1.6749543(86)e-27 kg \\

        1986 &9.10938970(540)e-31 kg &1.67262310(100)e-27 kg &1.67492860(100)e-27 kg\\

        1998 &9.109381880(720)e-31 kg&1.672621580(130)e-27 kg &1.674927160(130)e-27 kg \\

        2002 &9.10938260(160)e-31 kg &1.67262171(29)e-27 kg &1.67492728(29)e-27 kg\\

        2006 &9.109382150(450)e-31 kg&1.672621637(83)e-27 kg &1.674927211(84)e-27 kg\\

        2010 &9.109382910(400)e-31 kg&1.672621777(74)e-27 kg &1.674927351(74)e-27 kg\\

        \end{tabular}[/math]

        [math]\begin{tabular}{l l l l}

        Year &1/alpha &mass_p/mass_e &mass_n/mass_p\\

        \hline

        1969 &137.03602(21) &1836.1090(110) &1.001379(13)\\

        1973 &137.036040(110) &1836.15152(70) &1.001379(10)\\

        1986 &137.0359895(61) &1836.152701(37) &1.0013784040(90)\\

        1998 &137.03599976(50) &1836.1526675(39) &1.00137841887(58)\\

        2002 &137.03599911(46) &1836.152667261(85) &1.00137841870(58)\\

        2006 &137.035999679(94) &1836.152667247(80) &1.00137841918(46)\\

        2010 &137.035999074(45) &1836.152667245(75) &1.00137841917(45)\\

        \end{tabular}[/math]

        In 1969 and 1973, the ratio mass_n/mass_p is not reported separately, so it is calculated from the two mass values given.

        In the next table, the "CODATA gamma" is the value given in the particular year. The "Zivlak gamma" is the value obtained by using the experimental values known at the time in the Zivlak equation.

        [math]\begin{tabular}{l l l}

        Year &CODATA \gamma &Zivlak \gamma \\

        \hline

        1969 &1.001379(13) &1.0013784178(53)\\

        1973 &1.001379(10) &1.0013784162(23)\\

        1986 &1.0013784040(90) &1.00137841927(13)\\

        1998 &1.00137841887(58) &1.001378419181(10)\\

        2002 &1.00137841870(58) &1.0013784191948(93)\\

        2006 &1.00137841918(46) &1.0013784191907(19)\\

        2010 &1.00137841917(45) &1.00137841920390(92)\\

        \end{tabular}[/math]

        We now look at the success of the Zivlak equation in predicting the current value (CODATA 2010) for gamma.

        The "CODATA Error" is the difference between the value given at the time (i.e. in 1969 and so on) and the current 2010 value, as a proportion of the 2010 value.

        The "Zivlak error" similarily, is the difference between the value that could have been calculated at the time with the current 2010 value, as a proportion of the 2010 value.

        The C Error/Z Error column shows the ratio of the CODATA error to the Zivlak error.

        [math]\begin{tabular}{l l l l}

        Year &CODATA error &Zivlak Error &C Error/Z Error\\

        \hline

        1969 &1.3144463E-5 &5.790777E-9 &2269.8962\\

        1973 &1.0276465E-5 &2.7340497E-9 &3758.6973\\

        1986 &1.5149118E-8 &5.758918E-10 &26.305494\\

        1998 &1.0285822E-9 &4.5983836E-10 &2.2368343\\

        2002 &1.0301803E-9 &4.586465E-10 &2.2461312\\

        2006 &9.100147E-10 &4.5128046E-10 &2.016517\\

        2010 &8.9876123E-10 &4.5029516E-10 &1.995938\\

        \end{tabular}[/math]

        There are several things of note.

        (1) First, note that, for all years, the Zivlak gamma is significantly closer to the current known value than the value that was obtained at the time via the sophisticated methods of CODATA. For example, in 1973, it was 3758 times more accurate (this is partially due to the fact that the mn/mp ratio was not disclosed by CODATA).

        (2) By 1986, the Zivlak equation had produced a value that is more accurate than the value we have from CODATA even today.

        (3) Perhaps most surprisingly, even when we compare the 2010 CODATA value against itself as the gold standard, the Zivlak value is superior. This is because it predicts a (slightly different but very precise) value with very little uncertainty. The larger uncertainty in the 2010 CODATA value means its average error is higher.

        These results suggest that the Zivlak formula for the ratio of neutron to proton mass has real predictive power. Please accept my congratulations for your work on this!

        Hugh

        Hi Branko,

        Once again...

        Using NIST website CODATA values for 2010 and previous years, I have been looking at the accuracy of your formula. I wondered, using older experimental values, how well the formula could predict the currently best known value of mn/mp (i.e. from CODATA 2010).

        First, here is the data I used:

        [math]\begin{tabular}{l l l l}\\ Year &mass_e &mass_p &mass_n \\ \hline\\ 1969 &9.1095580(540)e-31 kg &1.6726140(110)e-27 kg&1.6749200(110)e-27 kg\\ 1973 &9.1095340(470)e-31 kg &1.6726485(86)e-27 kg &1.6749543(86)e-27 kg \\ 1986 &9.10938970(540)e-31 kg &1.67262310(100)e-27 kg &1.67492860(100)e-27 kg\\ 1998 &9.109381880(720)e-31 kg&1.672621580(130)e-27 kg &1.674927160(130)e-27 kg\\ 2002 &9.10938260(160)e-31 kg &1.67262171(29)e-27 kg &1.67492728(29)e-27 kg\\ 2006 &9.109382150(450)e-31 kg&1.672621637(83)e-27 kg &1.674927211(84)e-27 kg\\ 2010 &9.109382910(400)e-31 kg&1.672621777(74)e-27 kg &1.674927351(74)e-27 kg\\ \end{tabular}[/math]

        [math]\begin{tabular}{l l l l}\\Year &1/alpha &mass_p/mass_e &mass_n/mass_p\\ \hline \\1969 &137.03602(21) &1836.1090(110) &1.001379(13)\\ 1973 &137.036040(110) &1836.15152(70) &1.001379(10)\\ 1986 &137.0359895(61)&1836.152701(37) &1.0013784040(90)\\ 1998 &137.03599976(50) &1836.1526675(39) &1.00137841887(58)\\ 2002 &137.03599911(46)&1836.152667261(85) &1.00137841870(58)\\ 2006 &137.035999679(94) &1836.152667247(80) &1.00137841918(46)\\ 2010 &137.035999074(45) &1836.152667245(75) &1.00137841917(45)\\ \end{tabular}[/math]

        In 1969 and 1973, the ratio mass_n/mass_p is not reported separately, so it is calculated from the two mass values given.

        In the next table, the "CODATA gamma" is the value given in the particular year. The "Zivlak gamma" is the value obtained by using the experimental values known at the time in the Zivlak equation.

        [math]\begin{tabular}{l l l} \\ Year &CODATA \gamma &Zivlak \gamma \\ \hline \\1969 &1.001379(13) &1.0013784178(53)\\ 1973&1.001379(10) &1.0013784162(23)\\ 1986 &1.0013784040(90) &1.00137841927(13)\\ 1998 &1.00137841887(58) &1.001378419181(10)\\ 2002 &1.00137841870(58) &1.0013784191948(93)\\ 2006 &1.00137841918(46) &1.0013784191907(19)\\ 2010 &1.00137841917(45) &1.00137841920390(92)\\ \end{tabular}[/math]

        We now look at the success of the Zivlak equation in predicting the current value (CODATA 2010) for gamma.

        The "CODATA Error" is the difference between the value given at the time (i.e. in 1969 and so on) and the current 2010 value, as a proportion of the 2010 value.

        The "Zivlak error" similarily, is the difference between the value that could have been calculated at the time with the current 2010 value, as a proportion of the 2010 value.

        The C Error/Z Error column shows the ratio of the CODATA error to the Zivlak error.

        [math]\begin{tabular}{l l l l}\\ Year &CODATA error &Zivlak Error &C Error/Z Error\\ \hline \\ 1969 &1.3144463E-5 &5.790777E-9 &2269.8962\\ 1973 &1.0276465E-5 &2.7340497E-9 &3758.6973\\1986 &1.5149118E-8&5.758918E-10 &26.305494\\ 1998 &1.0285822E-9 &4.5983836E-10&2.2368343\\ 2002 &1.0301803E-9 &4.586465E-10 &2.2461312\\ 2006 &9.100147E-10 &4.5128046E-10 &2.016517\\ 2010 &8.9876123E-10 &4.5029516E-10 &1.995938\\ \end{tabular}[/math]

        There are several things of note.

        (1) First, note that, for all years, the Zivlak gamma is significantly closer to the current known value than the value that was obtained at the time via the sophisticated methods of CODATA. For example, in 1973, it was 3758 times more accurate (this is partially due to the fact that the mn/mp ratio was not disclosed by CODATA).

        (2) By 1986, the Zivlak equation had produced a value that is more accurate than the value we have from CODATA even today.

        (3) Perhaps most surprisingly, even when we compare the 2010 CODATA value against itself as the gold standard, the Zivlak value is superior. This is because it predicts a (slightly different but very precise) value with very little uncertainty. The larger uncertainty in the 2010 CODATA value means its average error is higher.

        These results suggest that the Zivlak formula for the ratio of neutron to proton mass has real predictive power. Please accept my congratulations for your work on this!

        Hugh