Kernel quotient and gauge for discrete curl-curl

Assume Countable Choice for the graph-space uniqueness and $L^2$ conventions of The real H(curl) graph space on a polyhedron. Let $V_h=N_h^1$ be the lowest-order edge space on a finite tetrahedral mesh, as in Lowest-order Whitney and Nedelec de Rham spaces. It is finite dimensional: restriction to the finitely many cells injects it into the finite product of the local spaces, each spanned by at most six Whitney edge fields. Injectivity holds because the cells cover $\mathrm{\Omega }$ up to their measure-zero facets. Use the componentwise real $L^2$ inner product defined in The real H(curl) graph space on a polyhedron. Define its curl kernel and the $L^2$ gauge space by $K_h:=\{v_h\in V_h:\mathrm{curl}{v}_h=0\},Z_h:=V_h\cap K_h^{\perp }\mathrm{.}$ The quotient ${\dot{V}}_h:=V_h\mathrm{/}{K}_h$ has norm $\mathrm{\parallel }[v_h]{\mathrm{\parallel }}_{\mathrm{curl}}:=\mathrm{\parallel }\mathrm{curl}{v}_h{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$; this is well-defined and positive definite because two representatives differ by an element of $K_h$. The bilinear form induced by curl-curl is $a_h([u_h],[v_h])=(\mathrm{curl}{u}_h,\mathrm{curl}{v}_h{)}_{L^2}$. A representative of a quotient solution is fixed by the gauge condition $u_h\in Z_h$. The mixed gauge system uses a multiplier $p_h\in K_h$: $a_h(u_h,v_h)+(p_h,v_h{)}_{L^2}=f(v_h)(v_h\in V_h),(u_h,q_h{)}_{L^2}=0(q_h\in K_h)\mathrm{.}$ The quotient energy norm, rather than the full $H(\mathrm{curl})$ norm, is the norm controlled by this definition.