Hello, concerning the analysis or error. What I used to take is v+3*e, and v-3*e, v is the value and e = error. We can assume that the error is like a bell-shaped curve it would normally means that v +/- 3*e is a 99% confidence interval. For example with 1.00137841870 +/- 0.000 000 000 58 (Spacing for readability) would give 1.00137841 or 1.00137842. --> this gives 8 valid decimal digits and maybe 9. But on the other hand some values cannot be placed into a valid interval without loosing a lot of precision as with : Tau-Muon; 16,8183; Phi9 L[10]^(1/11) /L[4]; 16,81830004 That one is right on the nose but the valid interval is 16.8264 and 16.8102 so we have 3 decimal digits for sure and maybe 4. We can hardly make any analysis since we can approach the value as near as we want. Those values with the golden ratio were the only ones that could get near a mass ratio without being too long as an expression. After all that expression is only powers of phi (in disguise). Is there a simpler answer to that ? The expressions I choose to put in the summary table are the ones I picked among the tons of candidates from which some of them were exactly 11 digits in precision. But this exercice is maybe missing the point. All of them with only powers of phi. Maybe (if I may emit this explanation) those values are average values in fact and the mass ratios are just oscillations of values among all those Lucas-Golden ratio values. There are 3 main reasons perhaps to think that way. The first obvious reference is what is found in nature about the golden ratio, as you may know perhaps, 95.6% of all plants grow with a factor of 1.618033. If you take a whole field of daisies and count the petals on each flower you will find 21 or 34 petals. But this is the point, some flowers will have 22 and some others will have 33 but in average it is 21 or 34. So the law of growth remains as being based on phi= 1.618033 but not exactly. The second principle is what is known on the hard hexagon problem, it was found recently that the piling of ice, snow is based on a critical exponent which is phi^5 = 11.0901699... The 3rd reason : the proposed values are (in my opinion) the best possible and simplest answer based on the big tables of constants and what is found in nature. Among all the real numbers, there are ripples of values around powers of phi, lucas numbers and fibonacci numbers and that phenomena is unique because of the 3 main transformations which are near 1. These transforms are the simplest possible. Actually, (I did those search also) with values like (1+sqrt(2))^n and some other simple algebraic values but only the golden ratio Fn and Ln can simplify like that. Since it was the only model having that property, I picked that one as being perhaps significant. I based my assumption also on the fact that [1,sqrt(5), phi^n] can produce a false answer when passed through the LLL algorithm and that is precisely the point, there are so many values and expressions near the point that even LLL will hang on it because the values are near integers. Simon Plouffe