1) Carl Brannen once made a great discovery: the circulant representation of the Koide relation, with parameters a phase of 2/9 radians, and a mass scale of about 313 MeV. Koide acknowledged (section 3.1) the phase relation, but deemed it too difficult to explain for now.
Brannen has now returned to the subject. His overall philosophy may be seen at his site, especially in the "Operator Guide". I do not endorse his framework, but it may help the reader understand his latest paper, and whether it does contain any new progress.
2) There was a PF thread on gravitational vacuum polarization in which the author attributed significance to the sum of the squares of all particle masses. That is the LC&P formula for the square of the Higgs vev.
In the "General physics" category at arxiv (its counterpart of vixra), two papers (1 2) on obtaining the fine-structure constant from consideration of "virtual parapositronium in the vacuum". It could be bogus but it reminds me of various speculations here about the criticality of αem.