I suspect that many of the experimentally accurate theoretical "predictions" given in a "Grand Unified Theory" that is available online (search for the string "The calculated relations between the lepton masses") were arrived at by exhaustive numerology, i.e. by searching through a large number of simple expressions to find the ones that gave the required results. Then a "proof" of each of these results was reverse-engineered using pseudo-physical explanations rather than using rigorous maths.

Let me show you an example of what I mean.

This "GUT" gives some simple expressions for various mass ratios. There is even an expression for the ratio of the neutron mass to the electron mass, which depends

*only*on the electomagnetic coupling strength (i.e. fine structure constant) and

*not*on the strong interaction strength. How do the quarks and gluons in the neutron know how to interact in order to give this amazing result?

The expression given by this "GUT" for the muon to electron mass ratio (i.e. mμ/me) is

(α^(-2) / 2π)^(2/3) (1 + 2π α^2 / 2) / (1 + α/2)

which produces a value 206.76828 that closely corresponds to the experimentally observed value 206.76827.

Let's see whether it is possible to "derive" this result by an exhaustive search of all simple expressions of this general type. The parameterisation that I will use is the most general form that is suggested by the mass ratio quoted above

f[{a1, a2, a3, a4}, {b1, b2, b3, b4}, {c1, c2, c3, c4}, {d1, d2, d3, d4}, α]] =

(a1/a2)^(a3/a4) (α)^(b1/b2) (2π α^2)^(b3/b4) (1 + (c1/c2)(α) + (c3/c4)(2π α^2)) / (1+ (d1/d2)(α) + (d3/d4)(2π α^2))

where all of the parameters are integers which are grouped in pairs to form rational fractions. To compute numerical results I inserted specific values for these parameters (avoiding singular cases), I use α=0.00729735, and a target mass ratio mμ/me=206.76827.

I then computed f[{a1, a2, a3, a4}, {b1, b2, b3, b4}, {c1, c2, c3, c4}, {d1, d2, d3,

d4}, α]] for all parameter values in the following small ranges (I have been rather cavalier and restricted the ranges to save time):

{a1, 1, 2}, {a2, 1, 2}, {a3, 1, 3}, {a4, 1, 3},

{b1, 0, -3, -1}, {b2, 1, 3}, {b3, 0, -3, -1}, {b4, 1, 3},

{c1, 0, 2}, {c2, 1, 2}, {c3, 0, 2}, {c4, 1, 2},

{d1, 0, 2}, {d2, 1, 2}, {d3, 0, 2}, {d4, 1, 2}

There is some repetition of trial solutions here, but this doesn't matter.

I then selected from this large set of trial solutions all of the cases that predicted a value for mμ/me that lay within 0.01 of the target value, and here they are (in decreasing order of goodness of fit) with the prediction errors shown in square brackets:

(1/(2π α^2))^(2/3) (1 + π α^2) / (1 + α/2) [0.0000110213]

(1/(2π α^2))^(2/3) (1 + 2π α^2) / (1 + α/2 + π α^2) [0.000130968]

(1/(2π α^2))^(2/3) (1 + α/2 + π α^2) / (1 + α) [0.00261802]

(1/(2π α^2))^(2/3) (1 + α/2 + 2π α^2) / (1 + α + π α^2) [0.0027371]

The best fit solution at the top of this list is the same as the one given by the "GUT".

What do we conclude from this little exercise?

It is

*really*easy to do exhaustive searches to find best-fit solutions. The above fit works as well as it does because it starts with two different quantities (α) and (2π α^2) (where π is

*not*a rational fraction), and combines them in various ways using lots of rational fractions to tailor the combination, which then leads to a dense set of candidate solutions from which the best-fit solution can then be picked.

Unless you happened to pick the physically correct parametric form to search over (Balmer got lucky with atomic spectra, but that is not to be used as a justification for this approach), then there is

*no physical significance*to solutions that are obtained in this way. If you hedge your bets by searching over a large set of parametric forms, then you will almost certainly find many solutions that have a good fit to the target value, but this doesn't guarantee that any of them is physically significant. Interestingly, a related problem occurs in the context of the Landscape.

The approach used in this "GUT" is numerology, pure and simple. Of course, I only

*suspect*that this is the way that the above expression for mμ/me was "derived"; I can't

*prove*that this is the case.

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