Investigations of many-body physics in an AMO context often employ a static optical lattice to create a periodic potential. Such systems, while capable of exploring, e.g., the Hubbard model, lack the fully emergent crystalline order found in solid state systems whose stiffness is not imposed externally, but arises dynamically. Our multimode cavity QED experiment will introduce a new method of generating fully emergent and compliant optical lattices supporting phonon-like excitations to the ultracold atom toolbox. The system provides new avenues to explore quantum liquid crystalline order. We will report experimental progress, including our recent observation of a "supermode-density-wave-polariton condensate." Future employment of spinful atoms will provide opportunities to explore quantum neural networks and the order of short-range spin glasses.