= Resistance upper bound from persistent-current decay
{title2=$R\leq-(L/T)\ln(1-\epsilon)$}
Suppose a calibrated measurement bounds the fractional loss of a nonzero <superconducting persistent current> by $\epsilon$, with $0<\epsilon<1$, over a duration $T>0$. Take $I_0>0$ by choosing the circulation direction. Under the constant-$L$, constant-$R$, undriven <RL circuit> model,
$$
\frac{I(T)}{I_0}\geq1-\epsilon
\quad\Longrightarrow\quad
0\leq R\leq-\frac{L}{T}\ln(1-\epsilon)
\simeq\frac{L\epsilon}{T}.
$$
The inequality follows because the <logarithm> is increasing. The final approximation requires $\epsilon\ll1$ and slightly understates the exact upper bound. If $L$ is uncertain, an independently established upper bound on $L$ gives the conservative <electrical resistance> bound. The uncertainty in the initial reading, the final reading, background subtraction and drift must all be included in $\epsilon$; a display resolution alone does not provide it.
\Image[/persistent-superconducting-current-decay-bound.png]
{title=An illustrative exponential decay, its logarithmic slope and the resulting resistance bound}
{description=Chosen numerical model, not historical data: fixed <self-inductance> $L=1\,\mathrm{\mu H}$ and no external drive. The final panel assumes a calibrated fractional-loss bound of $0.001$ at each chosen observation duration. Longer observation tightens the bound only while that fractional uncertainty remains controlled.}
For an original illustrative example, let $L=1\,\mathrm{\mu H}$ and suppose the corrected <electric current> ratio after $T=3\,\mathrm h=10800\,\mathrm s$ is at least $0.999$. Then
$$
R\leq9.264\times10^{-14}\,\Omega,\qquad
\tau\geq1.07946\times10^7\,\mathrm s\simeq124.94\,\mathrm{days}.
$$
These are conditional bounds for stipulated parameters, not values attributed to the 1914 apparatus. With the same $L$ but $R=10^{-11}\,\Omega$, the model instead gives $\tau=10^5\,\mathrm s$ and $I(24\,\mathrm h)/I_0=e^{-0.864}\simeq0.42147$: a readily visible loss on the chosen scale.
A resolved decay can be checked using its <logarithm>:
$$
\ln\frac{I(t)}{I_0}=-\frac{R}{L}t,\qquad
R=-L\frac{d\ln|I|}{dt}.
$$
A straight line supports this constant-parameter model; curvature or jumps demand a different account. If an effective <electrical resistance> varies with time while $L$ remains fixed and the loop remains undriven, only its time average is constrained:
$$
I(T)=I_0\exp\!\left[-\frac{1}{L}\int_0^T R(t)\,dt\right],\qquad
\frac{1}{T}\int_0^T R(t)\,dt\leq-\frac{L}{T}\ln(1-\epsilon).
$$
This does not bound every instantaneous value of $R(t)$. Changing <self-inductance> or external linked <magnetic flux> adds terms to the circuit equation, so an uncorrected magnetic signal cannot simply be substituted for $I(t)$. Nonobservation of decay leaves arbitrarily small positive constant <electrical resistance> compatible with a finite observation: it is not a logical proof that $R=0$.
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