= Solution
Let $\delta_1$ and $\delta_2$ be gauge-invariant phase drops across the two identical junctions, oriented between the same terminal phases. Their currents obey the <DC Josephson effect>: $I_1=I_C\sin\delta_1$ and $I_2=I_C\sin\delta_2$. For negligible loop <self-inductance> and no additional phase drops in the arms, <fluxoid> consistency fixes their difference modulo $2\pi$:
$$
\delta_1-\delta_2=2\pi\frac{\Phi}{\Phi_0}.
$$
Define the mean terminal phase $\delta=(\delta_1+\delta_2)/2$ and $\phi=\pi\Phi/\Phi_0$. Then $\delta_{1,2}=\delta\pm\phi$ and the two parallel branch currents add:
$$
\boxed{I=I_C[\sin(\delta+\phi)+\sin(\delta-\phi)]
=2I_C\cos\!\left(\frac{\pi\Phi}{\Phi_0}\right)\sin\delta.}
$$
Consequently the maximum magnitude of the current is $2I_C|\cos\phi|$, not a negative critical current when the cosine changes sign. The phase shifts by $\pi$ between such stable branches. Junction asymmetry or appreciable self-generated flux modifies the simplest interference pattern, so the stated idealization is required before treating $\Phi$ as an externally prescribed flux.
\Image[/past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2015-two-junction-flux-tunable-inductor.png]
{title=Two-junction superconducting interference and its stable inductive response}
{description=Original circuit-level diagram of the stated ideal two-junction model, with a flux-dependent critical-current envelope and stable small-signal inductance. It does not reconstruct the missing source raster components.}
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