I record here the existence of two papers by Ervin Goldfain 1 2 claiming to derive the LC&P sum rule.

His concept seems to be that the effective dimension of space-time varies with energy scale, that the masses of SM particles define special scales, and that the LC&P formula follows from a "closure relation" that must connect these different scales.

Incidentally, he is not just talking about spaces with an integer number of dimensions, as in Kaluza-Klein theories or string theories, where e.g. the number of dimensions may increase from 4 to 10 or 11, at energies above the compactification scale. Instead he talks of there being 4+ε dimensions, reminiscent of dimensional regularization... but the modified concept of dimensionality that he really emphasizes is that of fractals.

Informally, one might say that Goldfain's concept is that space is crinkled or creased in a fractal way, so that e.g. the volume of space inside a cube doesn't simply vary as the third power of the side of the cube. Instead, the exponent describing the change in volume is non-integer, and also varies with the size of the cube (length of its side). If we take a cube and shrink it, we might find that as the side shrinks to one millimeter, volume is proportional to size^3.1, but by the time we have shrunk to one micrometer, volume is proportional to size^3.3. Apparently in the world of fractals, such behavior is called multifractal.

The references to millimeter and micrometer above are purely illustrative. Goldfain seems to believe that the first significant deviations from integer dimensionality (4 space-time dimensions) only begin to occur above the electroweak energy scale, which would correspond to distances less than 10^-18 meters.

Goldfain is an independent investigator who publishes at vixra and in various web "journals", but the concept of multifractal space-time isn't just some whimsy of his, it has seen some mathematical development. The real problem I am having with his work so far, is that I don't understand where the "closure relation" comes from - and that's the crucial step towards obtaining the LC&P formula.

See for example equation 5 in paper "1". The "r"s are the different scales, and the "D" is a fractal dimension. The LC&P formula is a sum of squares, and so if scales were associated with masses, and if D was equal to 2, then we might be able to obtain it from equation 5.

Goldfain has written other papers trying to obtain SM mass ratios from fractal dimensional flow. A skeptical reading might say that all we have here is a conceptual framework in which multiple length scales can assume a special significance, and since masses can be mapped to length scales in physics, this multiscale conceptual framework can be a playground for a physics numerologist trying to explain particle masses.

I am skeptical, but dimensional flow is not a bad thing to think about. I will make a follow-up post if I have anything more concrete to add.