The extremely sharp transition between molecular and ionized gas in the Horsehead nebula

(Abridged) Massive stars can determine the evolution of molecular clouds with their strong ultraviolet (UV) radiation fields. Moreover, UV radiation is relevant in setting the thermal gas pressure in star-forming clouds, whose influence can extend from the rims of molecular clouds to entire star-forming galaxies. Probing the fundamental structure of nearby molecular clouds is therefore crucial to understand how massive stars shape their surrounding medium and how fast molecular clouds are destroyed, specifically at their UV-illuminated edges, where models predict an intermediate zone of neutral atomic gas between the molecular cloud and the surrounding ionized gas whose size is directly related to the exposed physical conditions. We present the highest angular resolution (~$0.5$", corresponding to $207$ au) and velocity-resolved images of the molecular gas emission in the Horsehead nebula, using CO J=3-2 and HCO$^+$ J=4-3 observations with ALMA. We find that CO and HCO$^+$ are present at the edge of the cloud, very close to the ionization (H$^+$/H) and dissociation fronts (H/H$_2$), suggesting a very thin layer of neutral atomic gas (<$650$ au) and a small amount of CO-dark gas ($A_V=0.006-0.26$ mag) for stellar UV illumination conditions typical of molecular clouds in the Milky Way. The new ALMA observations reveal a web of molecular gas filaments with an estimated thermal gas pressure of $P_{\mathrm{th}} = (2.3 - 4.0) \times 10^6$ K cm$^{-3}$, and the presence of a steep density gradient at the cloud edge that can be well explained by stationary isobaric PDR models with pressures consistent with our estimations. However, in the HII region and PDR interface, we find $P_{\mathrm{th,PDR}} > P_{\mathrm{th,HII}}$, suggesting the gas is slightly compressed. Therefore, dynamical effects cannot be completely ruled out and even higher angular observations will be needed to unveil their role.