Muonic Hydrogen: Vacuum Polarisation a Hundred Times Larger, and the Proton Radius It Revealed

Replace the electron in hydrogen with a muon and the atom shrinks two hundredfold, its orbiting particle spending far more of its time inside the reach of vacuum polarisation and of the proton's own size. The muonic Lamb shift is dominated, unusually, by the Uehling effect of virtual electron pairs, an anchor of about -205 milli-electronvolts in the stated accounting, and its finite-size part varies as 5.20 times the square of the proton radius in femtometres. This paper tabulates that variation across the historically contested radii: moving from 0.88 to 0.841 femtometres changes the level by 0.349 milli-electronvolts, hundreds of times the experimental precision, which is how a vacuum-dominated spectrum became the sharpest ruler for the proton's size and produced the proton radius puzzle. The amplification factor behind the sensitivity, the cube of the reduced-mass ratio at 6.448e+06, is computed in the script. All values derive from stated coefficients; the spectroscopy itself is reported by the cited experiments, not repeated here.