Koide's Formula, recall, is the observation that the sum of the masses of the three charged leptons, divided by the square of the sum of the positive square roots of the charged leptons, is equal to two-thirds. It has been confirmed to five signficant digits which is consistent with experimental evidence, predicts a tau mass to a couple of significant digits more than the currently most precise value, has held up even though it was quite a bit off the most precise values at the time it was formulated several decades ago, and is interestingly exactly at the midpoint of the highest possible value for that ratio (1) and the lowest possible value for that ratio (1/3).
His main points are as follows:
(1) It is much easier to find approximate, but surprisingly close, mathematical coincidences than you would think but manipulating a handful of constants in every conceivable way.
(2) Since the formulation is dimensionless, it is actually a function of two lepton mass ratios, rather than three independent mass values, which makes it somewhat less remarkable.
(3) If the ratio 2/3rd is conceptualized as a 45 degree angle, rather than a ratio, it is not at the midpoint of the possible values, making it less special.
(4) Koide's formula uses the real valued numbers for charged lepton masses, rather than the complex valued charged lepton masses, called "pole masses" that include an adjustment for the decay width of unstable particles (basically, their half lives, converted into mass units), and when Koide's formula is applied to pole masses, the 0.79 ratio that results don't seem as special. Lubos thinks it is unnatural to use something other than pole masses in anything that expresses a fundamental relationship of charge lepton masses.
In the Standard Model, the masses of charged leptons arise from the Yukawa interaction term in the Lagrangian, [which is a simple function of] y . . . a dimensionless (in d=4 and classically) coupling constant; h . . . the real Higgs field; [and] Ψ,Ψ ¯ . . . the Dirac field describing the charged lepton or its complex conjugate, respectively. To preserve the electroweak symmetry – which is needed for a peaceful behavior of the W-bosons and Z-bosons – one can't just add the electron or muon or tau mass by hand. After all, the electroweak symmetry says that the left-handed electron is fundamentally the same particle as the electron neutrino. Instead, we must add the Yukawa cubic vertex – with two fermionic external lines and one Higgs external line – and hope that Mr Higgs or Ms God will break the electroweak symmetry which also means that he will break the symmetry between electrons and their neutrinos. . . . [In turn] In the vacuum, the Higgs field may be written as h=v+Δh. Here, v is a purely numerical (c -number-valued) dimensionful constant whose value 246 GeV was known before we knew that the Higgs boson mass is 125 GeV. The value of v is related to the W-boson and Z-boson masses and other things that were measured a long time ago. The term Δh contains the rest of the dynamical Higgs field (which is operator-valued) but its expectation value is already zero. . . . [And,] m e is just a shortcut for m e =y e v where the Yukawa coupling y e for the electron and the Higgs vev v= 246 GeV are more fundamental than m e. If you write the masses in this way, v will simply cancel and you get the same formula for Q where m is replaced by y everywhere. However, this is not quite accurate because the physical masses are equal to yv up to the leading order (tree level diagrams i.e. classical physics) only. There are (quantum) loop corrections and many other corrections. Moreover, the values of y that produce Q=2/3 are the low-energy values of the Yukawa couplings. Even though the Yukawa couplings are more fundamental than the masses themselves, their low-energy values are less fundamental than some other values, their high-energy values.
In other words, both arguments (4) and (5) are arguments that in the ordinary formulation of the Standard Model, the charged lepton mass inputs into Koide's formula are not fundamental and therefore have no business exhibiting profound and mysterious relationships to each other that have any basis in fundamental physics and hence are probably just numerological coincidences.
I'm not sold on the argument Lubos makes, for a few reasons, that I'll note with Roman numerals to avoid confusion with his reasons:
(I) Koide's formula has come into closer alignment with the experimentally measured values of the charged lepton values as they have been discovered more precisely, while most numerical coincidences (e.g. efforts to describe the electromagnetic coupling constant as a simple integer valued number) fall apart as the experimentally valued number becomes known with more precision. A five significant digit match to a simple, dimensionless, rational number shouldn't be dismissed lightly.
(II) Lots of well motivated Standard Model derived constant predictions (e.g. the W and Z masses, the proton and neutron masses) are not know to any more precision than Koide's formula, so judged by its fit to empirical evidence, Koide's formula is holding its own.
(III) Almost everyone who understands particle physics well enough to be a professional in the field intuitively agrees that the many constants of the Standard Model are not simply random and have deeper interrelationships to each other than we have yet come up with well formulates laws of physics to explicate. Put another way, there is clearly some formula out there that if discovered would derive particle masses, particle decay widths, CKM/PMNS matrix phases, coupling constants of the fundamental forces, and the constants of the running of the fundamental force coupling constants, from a much smaller set of more fundamental constants, and there is no a priori reason that we aren't capable of discovering that relationship.
If you start from the assumption that there is some deeper relationship between these constants, then the question is which are the proposed relationships between these constants has proven most fruitful so far and has tended to become more rather than less accurate as more empirical evidence has become available.
Put another way, if you assume that these constants do have a deeper relationship, then any other empircially relationship between them that is observed necessarily derives in some way from the deep relationship and hints at its nature. The empirical validity of the dimensionless Koide's formula to great precision, at the very least, is proof of a no go theorem for other proposed deeper relationship between charged lepton masses that does not observe that relationship. It fairly tightly constrains the universe of potentially valid deeper theories.
At the very least, Koide's formula poses an unsolved problem in physics akin to the Strong CP problem, i.e. "why is no there observable CP violation in the physics of the strong force?"
In the same vein, the phenomenological and predictive success of the modified gravity theory "MOND" as originally formulated by Milgrom in describing galactic rotation curves with a single numerical constant, doesn't necessarily mean that this phenomena is really caused by the law of gravity being misformulated rather than dark matter. But, it also necessarily implies that any dark matter theory that takes multiple unconstrained numerical constants to produce results that MOND can match with one numerical constant with similar accuracy is missing some very important factors that cause real galaxies to have far more tightly constrained structures than their formulation permits. The fact that a strong phenomenological relationship exists doesn't tell you its cause, but it does generally establish that there is some cause for it.
(IV) Lots of phenomenological relationships in physics that aren't fundamental at the deepest sense and can be derived from mere approximations of theoretical physics formulas which are known to be more accurate are still remarkably accurate and simple in practice.
For example, the phenomenological fact that planets follow orbits around the sun that are ellipses with foci at the planet in question and the sun, turns out to be extremely accurate, and possible to express with high school algebra and derive with elementary first year calculus, even though it ignores all sorts of more accurate physics such as the corrections between general relativity and Newtonian gravity for objects that are in motion, and the fact that planetary orbits are actually determined via supremely difficult to calculate many bodied problems that include the gravitational effects of every little bit of matter in the solar system and beyond, not just a two body problem and a formula in the form F=GmM/r^2. Before Kepler figured out that the orbits were ellipses, Copernicus came up with a simpler approximation of the orbits as spheres around the sun (which are degenerate forms of the equations for ellipses), which while also wrong, was still an immense leap relative to the prior formulations.
Similarly, the classical ideal gas law, PV=NkT, involves physics that aren't fundamental (it can be derived from first principles from statistical mechanics and a few simplifying assumptions, and statistical mechanics, in turn, relies on classical mechanics that have to be derived at a fundamental level in far from obvious way from quantum mechanics). Yet, we still teach high school and lower division physics and chemistry students the ideal gas law because it, and non-ideal gas variants of it that use empirically determined physical constants to fit real gases, turn out to be useful ways to develop quantitative intuition about how gases behave and approximate that behavior with accuracy sufficient for a wealth of applications. The ideal gas law, in turn, was derived from even simpler observations about two variable proportionality or inverse proportionality relationships (e.g. V=cT for a gas of a constant volume) that were observed phenomenologically, long before all of the pieces were put together.
Thus, the fact that Koide's formula doesn't naturally and obviously correspond in form to current physically well motivated electroweak unification models doesn't necessarily count as a strike against it. It may be that the terms in more complex formulations of fundamental sources of charged lepton masses, either cancel out or have insignificant physical values that are swamped by other terms. For example, I suspect that a more exact formulation of Koide's formula for leptons may require the inclusion of all six lepton masses. But, the neutrino masses are so negligible relative to the charged lepton masses that their impact on Koide's formula may be invisible at the currently level of precision with which we know the charged lepton masses.
Odds are that a some level of precision, Koide's formula will cease to hold. But, for example, if the amount by which it is off is at an order of magnitude that could be accounted for via the inclusion of neutrino masses and tweaking the sign of electron neutrino mass term (a Brannen suggested possibility), then Koide's formula starts looking like an incomplete approximation of more exact theory that holds for reasons considerably deeper than coincidence, rather than merely a fluke.
(V) A notion of what is fundamental and what is derived, with a set of constants that are all tightly constrained to be related to each other mathematically, is to some extent a matter of perception. The notion that inferred Yukawa coupling constants, or pole masses of particles must be more fundamental than observed particle rest masses without adjustment for rates of decay, is not at all obvious. There is nothing illogical or irrational about described Yukawa coupling constants and pole masses as derived values, and charged lepton rest masses as fundamental.
My strong suspicion, for example, given the strong patterns that are observed in decay widths, is that the decay width of a particle is a derived constant that is the product in some manner or other of rest masses and some other quantum numbers, rather than having a truly independent value for each particle. Pole mass may be a more useful description of a particle's mass in some equations, just as a wind-chill adjusted temperature may be a more useful description of the ambient temperature in a location for some purposes. But, that doesn't necessarily mean that it is truly more fundamental.
And, reliance upon Standard Model formulations of particle masses as a source of the "true" nature of particle mass is also questionable when one of the deepest problems of the Standard Model is that its formulations can't provide particle masses from first principles for fermions or the Higgs boson (although the photon, W and Z boson rest masses can be derived from it).
(VI) Lubos ignores the relatively productive recent efforts that have been made recently to express other Standard Model particle mases (where the true values are often known to just one or two signficant digits) in Koide-like triples or other Koide-like formulations, an apparent three to one relationship between Koide-like formulations for quarks and for leptons (that fits the three to one relationship betweeen quarks and leptons seen in precision electroweak decay products), possible derivations of fermion mass relationships from CKM/PMNS matrix elements and visa versa (often finding links via the square root of fermion masses to be more natural), the phenomenological observation of quark-lepton complementarity in CKM/PMNS matrix elements, and so on. If there was just one Koide triple in the fermion mass matrix, it might just be a fluke. When there are multiple Koide triples in the fermion mass matrix that all seem to take on some kind of integer value to well within the range of empirically measured masses, dismissing the result as a fluke is problematic.
The implied angle of forty-five degrees from Koide's formula, for example, also comes up in quark-lepton complementarity, which relates to CKM/PMNS matrix element relationships.
(VII) Lubos also puts on blinders to the potential relevance of the square root of fermion mass as a potentially fundamental matter having some relationship to emerging evidence in his own string theoretic field of the similarity between gravity (which is a force that acts on mass-energy) and a squared QCD type gauge group, in which color charge is replaced with kinematic terms.