  Schmid predicted that a single Josephson junction coupled to a resistive
environment undergoes a quantum phase transition to an insulating phase when
the shunt resistance $R$ exceeds the resistance quantum $h/(4 e^ 2)$. Recent
measurements and theoretical studies have sparked a debate on whether the
location of this transition depends on the ratio between the Josephson and the
charging energies. We employ a combination of multiple innovative analytical
and numerical techniques, never before explicitly applied to this problem, to
decisively demonstrate that the transition line between superconducting and
insulating behavior is indeed independent of this energy ratio. First, we apply
field-theory renormalization group methods and find that the $\beta$ function
vanishes along the critical line up to the third order in the Josephson energy.
We then identify a simple fermionic model that precisely captures the
low-energy physics on the critical line, regardless of the energy ratio. This
conformally invariant fermionic model is verified by comparing the expected
spectrum with exact diagonalization calculations of the resistively shunted
Josephson junction, showing excellent agreement even for moderate system sizes.
Importantly, this identification provides a rigorous non-perturbative proof
that the transition line is maintained at $R=h/(4 e^ 2)$ for all ratios of
Josephson to charging energies. The line is further resilient to other
ultraviolet cutoffs such as the plasma frequency of the resistive environment.
Finally, we implement an adiabatic approach to validate the duality at large
Josephson energy.
