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For $ab\ne 0$, a magnetic field line stays at fixed $R$ and satisfies $d\varphi /dz=B_{\varphi }/(R{B}_z)=a/b$. It is therefore a helix with axial pitch $2\pi b/a$; the sign of $a/b$ fixes its handedness. The axial magnetic flux is $\pi b{R}_0^2$, and the toroidal magnetic field supplies the twist. If $a=0$ the field is untwisted; if $b=0$ it consists of azimuthal circles rather than helical lines carrying axial flux.

A convenient globally regular magnetic vector potential has $A_R=0$ andIndeed, the cylindrical curl gives $B_z=R^{-1}d(R{A}_{\varphi })/dR$ and $B_{\varphi }=-d{A}_z/dR$, with $B_R=0$. The tangential potential components are continuous at $R_0$, so their derivatives introduce no unwanted delta-function magnetic field at the tube boundary. The $C$ term is a constant axial potential.

The most general regular potential is this one plus $\nabla \Lambda (R,\varphi ,z)$ for an arbitrary single-valued gauge function. A putative extra $C_{\varphi }/R$ term in the inner $A_{\varphi }$ is excluded by regularity at the axis; the exterior $1/R$ circulation is already fixed by the enclosed magnetic flux. Since the potential is globally defined across the axis and exterior, any remaining difference is a curl-free vector field, hence a gradient. The constant $C$ can itself be absorbed into a gauge function proportional to $z$.

For the displayed gauge family, inside the tubeand henceFor example, requiring $A_z=0$ outside sets $C=a{R}_0^2/2$ and gives $\boxed{H_m/L=\pi ab{R}_0^4/2}$. The equally regular choice $C=0$ gives zero. More generally, for a length-$L$ segment the gauge transformation contributesIn particular, $\Lambda =\Delta Cz$ shifts $H_m/L$ by $\pi b{R}_0^2\Delta C$. This gauge dependence of helicity in an open magnetic flux tube is possible because field lines penetrate the end caps when $b\ne 0$. There is no contradiction with part (b), whose no-penetration condition fails at those caps. A gauge or end-cap convention is needed to give a unique value per unit length. For $b=0$ the ordinary helicity is zero and this end-cap ambiguity disappears.