= Drift cancellation in a g-factor comparison
Even a shared field does not cancel if it is sampled at different times. Write $f_A(t)=g_A C(t)$ and $f_B(t)=g_B C(t)$, where $C(t)=\mu_BB(t)/h$. As an original proposed diagnostic, bracket the $B$-state reading at $t=0$ with $A$-state readings at $t=\pm\tau$:
$$
R_{\rm bracket}=
\frac{f_B(0)}{\tfrac12[f_A(-\tau)+f_A(+\tau)]}.
$$
If $C(t)=C_0(1+bt)$, the denominator is exactly $g_A C_0$, so $R_{\rm bracket}=g_B/g_A$. A single earlier $A$ reading instead gives $R_{\rm unpaired}=(g_B/g_A)/(1-b\tau)$. For the preceding numerical model and $b\tau=0.001$, this unpaired comparison falsely gives approximately $a_e=0.00186649$; interpolation recovers $0.0012$. The sequence illustrates cancellation and is not asserted to be Kusch and Foley's exact historical timing protocol.
\Image[/atomic-g-ratio-field-drift.png]
{title=Cancel a shared calibration scale without hiding its limitations}
{description=Original calculations for <spin-orbit g-factor ratio from two P states> and <drift cancellation in a g-factor comparison>. All inputs are stipulated. The left plot shows the nonlinear inversion from a frequency ratio to a spin-to-orbital <g-factor> ratio. The right plot contrasts an unpaired reading with a symmetric bracket under linear field drift; the dashed curve adds chosen quadratic curvature $c\tau^2=0.0002$. No curve is a measured resonance or a historical fit.}
For a curved field $C(t)=C_0(1+bt+ct^2)$, the same calculation gives
$$
R_{\rm bracket}=\frac{g_B/g_A}{1+c\tau^2}.
$$
The linear drift cancels but the curvature remains. Different fields sampled by the two beams, additive frequency offsets and incorrect atomic corrections also survive. For a small ratio error $\delta R$, direct differentiation of the inverse relation gives
$$
\delta x\simeq\frac{6}{(R+1)^2}\,\delta R,\qquad
\delta a_e\simeq\frac{3}{(R+1)^2}\,\delta R\quad(g_L=1).
$$
Repeated readings can reduce random <measurement uncertainty>; agreement among genuinely different state comparisons is what tests shared <systematic errors> in their interpretation.
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