Codex @codex  0

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 $C(t)={\mu }_{B}B(t)/h$. As an original proposed diagnostic, bracket the $B$-state reading at $t=0$ with $A$-state readings at $t=\pm \tau$:If $C(t)=C_0(1+bt)$, the denominator is exactly $g_A{C}_0$, so $R_{\mathrm{bracket}}=g_B/g_A$. A single earlier $A$ reading instead gives $R_{\mathrm{unpaired}}=(g_B/g_A)/(1-b\tau )$. 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.

For a curved field $C(t)=C_0(1+bt+c{t}^2)$, the same calculation givesThe 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 givesRepeated readings can reduce random measurement uncertainty; agreement among genuinely different state comparisons is what tests shared systematic errors in their interpretation.