Beyond the No‑Go Theorems: Structured Vacuum as a Thermodynamic Resource for Projective Ground‑State Preparation

Recent no-go theorems — by Guryanova, Friis & Huber (2020), Mohammady & Miyadera (2023), Candeloro, Debarba & Binder (2026), and Shahbeigi & Mohammady (2025) establish that ideal projective measurements require divergent thermodynamic resources, a consequence of the third law of thermodynamics. All these theorems assume that the environmental spectral density at the measurement frequency is finite, forcing the measurement pointer to be prepared in a pure state at finite resource cost. We show that when the measurement is performed inside a structured vacuum engineered by a chiral quantum Hall edge, a parity-protected superconducting island, and a complete three-dimensional photonic bandgap: the reverse spectral density at the oscillator frequency vanishes identically. Because the pointer qubit is dispersively coupled to the oscillator and shares the same engineered electromagnetic environment, it is likewise decoupled from the thermal bath at the relevant frequency. This eliminates the thermodynamic bottleneck common to all existing no-go theorems. We prove that under these conditions the effective measurement is classifiable within the Shahbeigi–Mohammady hierarchy as a type-III (fully thermodynamically consistent) operation, despite achieving a perfectly projective outcome on the target system. We further demonstrate that the structured vacuum does not merely shield against noise but acts as an active thermodynamic resource, reclassifying measurement back-action from heat into topologically directed work. This connects projective ground-state preparation to the emerging field of measurement-powered thermal machines (S´anchez et al., 2026; Elouard et al., 2026) and defines a broader class of topological measurement engines.