I was reading the following article that tries to use some equations originally proposed by Pauli in 1951 to reason from one of two reasonably plausible axions that there are tight constraints on the fundamental particle content and mass spectrum of the Standard Model together with BSM extensions in a manner that rules out both the SM as a complete set of particles, and lots of other BSM theories such as supersymmetry, mirror universe models, and two Higgs doublet models. Damian Ejlli, "Beyond the standard model with sum rules" (September 14, 2017). The paper argues that there are three respects in which a weighted sum of terms related to fundamental fermions should equal a weighted set of terms related to fundamental bosons. Each fundamental particle is assigned a "degeneracy factor" that serves as it weight. The formula (1) for the weights, which was formulated in 1951 by Pauli, before second and third generation particles were known to exist, before quarks and gluons were discovered, before the modern graviton was conceived, and before neutrino mass was known to exist, is correct. Each quark counts 12 points. Each charged lepton counts 4 points. A massive Dirac neutrino counts 4 points, while a massive Majorana neutrino or a massless neutrino counts 2 points. The W bosons count 6 points, the Z boson counts 3 points, the Higgs boson counts 1 point, the photon counts 2 points and gluons apparently count 2 points each for 8 flavor variations of gluon. In the Standard Model, the fermion side apparently has 68 more points than the boson side; there is an excess mass on the fermion side equal to about 530 GeV squared in one equations; and there is an excess mass on the fermion side equal to 560 GeV to the fourth power in another equation. Purportedly: (1) The sum of the fermion degeneracy factor for each of the fundamental fermions should be equal to the sum of the boson degeneracy factor for each of the fundamental bosons. (2) The sum of the fermion degeneracy factor times the square of the mass of each of the fundamental fermions should be equal to the sum of the boson degeneracy factor times the square of the mass of each of the fundamental bosons. (3) The sum of the fermion degeneracy factor times the fourth power of the mass of each of the fundamental fermions should be equal to the sum of the boson degeneracy factor times the fourth power of the mass of each of the fundamental bosons. The trouble is that except for some trivial cases that bear no similarity to reality, it appears that this will never be true. Naively, it appears to me that a sum of raw weights, squared masses with same weights, and fourth power masses with the same weights are never going to simultaneously balance, unless all of the fundamental particle masses are identical. In that special case, the sum of the weights for the fermions equals the sum of the weights for the bosons, so if every particle on the fermion side has the same mass as every particle on the boson side, then mass squared on each side will be the same and mass to the fourth power on each side will be the same. But, if the masses are different for each particle, as in real life, it isn't at all obvious that the weighted sum of mass squared can every be equal to the weighted sum of mass to the fourth, because the square of mass squared is not a linear transformation, but linear parity of masses terms must remain. In other words, if you start with the SM set of fundamental particles which have massively unequal masses. It isn't obvious to me that any combination of addition particles can solve these problems, therefore the assumption from which the equations flow must be false or ill defined (or the equations might be misformulated by Pauli with a scheme that wasn't properly generalized). Is my intuition wrong, or does applying the same set of unequal weights for each particle to the raw weights, weights times mass squared, and weights times mass to the fourth, leave the set of equations over constrained with no non-trivial solutions?