PAC in DESI. II. Galaxy-halo connection into the $10^{6}{\rm M}_{\odot}$ frontier

Understanding dwarf galaxy formation is crucial for testing dark matter models and reionization physics. However, constructing stellar-mass complete spectroscopic samples at low masses is increasingly difficult, and the potential existence of a local void complicates studies in an average environment. The Photometric object Around Cosmic webs (PAC) method, which combines deep photometric and spectroscopic data to measure the excess surface density $\bar{n}_2w_{\rm{p}}(r_{\rm{p}})$ of photometric objects around spectroscopic tracers, offers a promising path forward. We model 349 $\bar{n}_2w_{\rm{p}}(r_{\rm{p}})$ measurements from DESI Y1 BGS and DECaLS, reaching $M_*=10^{6.4}\,{\rm M}_{\odot}$, using a stellar mass-halo mass relation (SHMR)-based subhalo abundance matching framework applied to two high-resolution $N$-body simulations from the Jiutian suite. The resulting SHMR is constrained down to $M_{\rm h}\simeq10^{8.0}\,h^{-1}{\rm M}_{\odot}$, revealing a clear upturn at $\sim10^{10.0}\,h^{-1}{\rm M}_{\odot}$ toward lower masses, indicating rising star-formation efficiency (SFE) in small haloes. This feature persists under extensions of the model that allow mass-dependent scatter, reionization-induced suppression of the halo occupation fraction, galaxy assembly bias, and alternative cosmologies. Together with the finding from Paper I, we find that central red galaxies dominate the low-mass regime. Our results motivate a hypothesis in which SFE is significantly higher than previously thought prior to reionization, enabling relatively massive galaxies to form in small haloes. These systems are subsequently quenched by the UV background, producing the central red dwarf galaxies observed. Finally, we obtain $3σ$ and $5σ$ upper mass bounds of $10^{8.38}\,h^{-1}{\rm M}_{\odot}$ and $10^{8.71}\,h^{-1}{\rm M}_{\odot}$ on the smallest haloes required to exist by the data.