Universal Frame Potential Hierarchy in Critical Projected Ensembles

Local measurements transform a many body wavefunction into a statistical ensemble of conditional quantum states. In chaotic systems, the higher moments of such ensembles diagnose the emergence of quantum state randomness, but their structure at equilibrium quantum criticality is largely unknown. Here we show that the projected ensemble of a Tomonaga-Luttinger liquid exhibits a universal nonlinear hierarchy of state overlap moments. Remarkably, the leading scaling exponents are independent of the continuously varying Luttinger parameter. Replica boundary conformal field theory reveals a geometric origin: outcome locking combines the active replica swaps into a single collective rotated sector, while intersecting compact replica branes eliminate the leading interaction dependent zero mode contribution. Matrix product state calculations across the interacting XXZ critical phase and direct free fermion calculations independently confirm the interaction independence and nonlinear hierarchy.