Unsupervised Discovery of Intermediate Phase Order in the Frustrated $J_1$-$J_2$ Heisenberg Model via Prometheus Framework

The spin-$1/2$ $J_1$-$J_2$ Heisenberg model on the square lattice exhibits a debated intermediate phase between Néel antiferromagnetic and stripe ordered regimes, with competing theories proposing plaquette valence bond, nematic, and quantum spin liquid ground states. We apply the Prometheus variational autoencoder framework -- previously validated on classical (2D, 3D Ising) and quantum (disordered transverse field Ising) phase transitions -- to systematically explore the $J_1$-$J_2$ phase diagram via unsupervised analysis of exact diagonalization ground states for a $4 \times 4$ lattice. Through dense parameter scans of $J_2/J_1 \in [0.3, 0.7]$ with step size 0.01 and comprehensive latent space analysis, we investigate the nature of the intermediate regime using unsupervised order parameter discovery and critical point detection via multiple independent methods. This work demonstrates the application of rigorously validated machine learning methods to open questions in frustrated quantum magnetism, where traditional order parameter identification is challenged by competing interactions and limited accessible system sizes.