The Counting Rule and the Electron Anomalous Magnetic Moment: A Purchase Smaller than the Muon's by Four and a Half Orders of Magnitude

The counting rule tested against the muon g-2 hadronic dispersion sum in a companion analysis (DOI 10.5281/zenodo.24102027) is run for the electron. Evaluating an explicitly parametrised dispersion integral anchored to the published a_e^{had,LO} = 1.866(11) x 10^-12 gives: 82.87% of the electron's hadronic sum below 1 GeV, 93.66% below 1.8 GeV, 16.82% in the decade 1-10 GeV and 0.315% in 10-100 GeV. The part above the individualisation scale E_ind = 3.02 x 10^9 GeV is bounded by 8.9 x 10^-32, a fraction 4.8 x 10^-20 of the hadronic sum (the same order as the muon, since both tails scale as the squared lepton mass over E_ind^2), but a fraction 7.65 x 10^-29 of the total electron anomaly, against 2.66 x 10^-24 for the muon: the counting rule's purchase on the electron is smaller by a factor 2.9 x 10^-5, four and a half orders of magnitude. The mass-independent QED terms carry all but 3.94 x 10^-9 of a_e; the Schwinger term alone is 100.152% of the total; hadronic light-by-light (0.039 x 10^-12) exceeds the bounded tail by a factor 4.4 x 10^17. All numbers are produced by the deposited compute.py. AI use is disclosed: drafting and computation assistance was provided by Dave, an AI system built and operated under the author's direction. The author, William Alexander Aldred, directed the work and is sole author of record.