0_9969 Fresh evidence [1,2] suggests that monolayer WTe2 is the long-sought excitonic insulator (EI), a permanent Bose-Einstein condensate of excitons that form in the absence of optical excitation. A surge of experimental claims has recently addressed layered materials, because of reduced Coulomb screening. However, the transition to the putative EI is ubiquitously accompanied by the softening of a phonon inducing a structural change; therefore, it remains unclear whether the observed phase is genuinely excitonic or instead stabilized by electron–phonon interaction. On the contrary, no charge density wave is seen in WTe2, which rules out the role of lattice distortion in this material.Here we present a full theory of the EI phase of WTe2, which is consistent with the key experimental findings at charge neutrality and builds on unbiased calculations from first principles. The ab initio solution of the Bethe-Salpeter equation shows that the exciton binding energy is larger than 100 meV and the radius as small as 4 nm, explaining the observed formation of excitons at high temperature and doping levels [1]. The excitons responsible for the instability experience giant exchange interactions (of the order of 20 meV), which originate from the strong spin-orbit coupling that hybridizes conduction and valence bands. As a consequence, the EI ground state is a spin density wave. The multivalley mean-field calculation of the EI chemical potential as a function of doping is in quantitative agreement with the experiement. Finally, we predict unique features of the EI phase to appear in photoemission and THz absorption spectra. This work is partially funded by MUR PRIN2017 No. 2017BZPKSZ “EXC-INS”.                                  [1] Sun et al., Evidence for equilibrium exciton condensation in monolayer WTe₂. Nature Phys. 18, 94–99, (2022).[2] Jia et al., Evidence for a monolayer excitonic insulator, Nature Phys. 18, 87-93 (2022).