$c=1$ full-theory uniqueness source report

Eberhardt–Levine's Theorem 1.1 reports that every unitary two-dimensional $c_L=c_R=1$ CFT with discrete spectrum is the compact free boson, its ${\mathbb{Z}}_2$ reflection orbifold, or one of three exceptional $SU(2{)}_1$ orbifolds by the tetrahedral, octahedral, or icosahedral subgroup of $SO(3)$. Their modular-bootstrap theorem is narrower in logical content than that full-theory statement: under A1–A5 it classifies partition-function densities and includes a spurious average of two distinct Narain densities. Sections 4.1–4.4 then use full CFT data, including unitarity, the operator-state correspondence, current Ward identities, OPE/crossing, and invertible or non-invertible gauging arguments, to exclude the spurious average and identify the theories. This item reports what the sources state; it does not formalize or prove the full-theory uniqueness theorem and is not a prerequisite supplier.

Carpi–Gaudio–Giorgetti's Theorem 5.6 classifies simple discrete-series vertex-operator-algebra extensions of $L(1,0)$ as fixed-point algebras $V_{L_2}^H$ for closed subgroups $H$ of $\mathrm{Aut}(V_{L_2})$ isomorphic to closed subgroups of $SO(3)$. Their Remark 5.7 explicitly says this theorem assumes neither strong rationality nor effective central charge $1$. This is a chiral VOA result, so its scope does not by itself establish the nonchiral full-CFT uniqueness claim above. Their Theorem 5.27 separately assumes strong rationality of $V_{L_2}^{A_5}$; Remark 5.29 reports a separate preprint proof of that hypothesis, but that cited preprint was not inspected here and is not adopted as an input.

Eberhardt–Levine, Theorem 1.1 and §§1, 4.1–4.4

Carpi–Gaudio–Giorgetti, §5.1 and §§5.5