We construct, for the first time, an off-shell supersymmetric continuous spin gauge theory in 4-dimensional Minkowski spacetime, in both constrained and unconstrained Lagrangian formulations. As an extension to the on-shell description [1], we employ an auxiliary field to close the algebra of supersymmetry transformations off-shell. The 4d N\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal{N} $$\end{document} = 1 massless continuous spin supermultiplet is then denoted by (Φ, H; Ψ), comprised of a dynamical and a non-dynamical complex scalar continuous spin gauge fields Φ and H, as well as a Dirac continuous spin gauge field Ψ. In particular, we demonstrate that the off-shell continuous spin supermultiplet, in a limit, reproduces off-shell supersymmetry transformations of the known scalar supermultiplet (0, 1/2), all integer-spin supermultiplets (s, s + 1/2), s ≥ 1, and all half-integer spin supermultiplets (s − 1/2, s), s ≥ 1.
