The modular bootstrap has been a powerful tool for carving out the landscape of allowed two-dimensional conformal field theories (CFTs).  In this colloquium, I describe a complementary approach to standard modular bootstrap bounds: using modern machine learning strategies to actively search for CFT spectra that yield a valid torus partition function.  Using insights from statistical inference and a custom singular-value-based optimizer, I present evidence for an obstruction to finding CFTs with small central charge and large spectral gaps, and I speculate on what this might imply for the structure of the CFT landscape.  Along the way, I reflect on "centaur" approaches to theoretical physics, where human physicists and artificial intelligence collaborate to explore spaces of theories that would be difficult to navigate alone.