 Finite precision tests of the leading electron anomaly

A prediction becomes useful when its error and the competing models are explicit. Consider an original hypothetical readout $a_{\ast }=0.00116$ with stipulated absolute tolerance $2\times 10^{-5}$. Hold the teaching input $\alpha =1/137$ fixed. Compare three models: the Dirac equation baseline $a_e=0$, the leading QED value $a_e=\alpha /(2\pi )$, and a trial coefficient twice as large, $a_e=\alpha /\pi$. Their absolute residuals are respectively $0.00116$, approximately $1.7149\times 10^{-6}$, and approximately $0.00116343$. Only the leading QED value lies inside this stipulated tolerance. The example tests both the existence and the size of a correction; it does not quote an actual observation or assign the tolerance a statistical interpretation.

At tighter precision, including more loop orders and propagating the uncertainty in the fine-structure constant becomes essential. The first omitted term has order ${\alpha }^2$, but the order symbol alone gives no numerical error bound. If $\alpha$ changes by a small $\delta \alpha$, this leading prediction changes by $\delta a_e=\delta \alpha /(2\pi )$. Thus a comparison must state which input determined $\alpha$: extracting it from the very electron magnetic moment being tested would not supply an independent test of that input. Agreement with one leading term at one tolerance does not establish agreement to arbitrarily many digits.