It's not precisely 1/137, so looking for 137 of something doesn't make sense.
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Yeah... I sort of suspected as much, I know the decimals continue like a number chorizo.
So flipping the question, then - does the black hairline gap in spectral emissions with up and down electrons represent some sort of irreducible step, and what would that be?
I'm getting to the edge of what I know, but emission and absorbtion lines so represent a irreducible step: an electron moving from one energy state to another inside an atom. That irreducibility is where the "quantum" in quantum mechanics comes from. It's also how physicists figured how how electrons in an atom are grouped into shells and subshells.