Finite‑Resource Exact Ground‑State Preparation: Complete Rigorous Proof via Structured Vacuum Engineering

We present the complete, mathematically rigorous proof that a harmonic oscillator can be prepared in its exact ground state |0⟩ with finite resources, using a projective measurement protocol inside a structured vacuum. The structured vacuum comprising a chiral quantum Hall edge, a parity‑protected superconducting island, and a three‑dimensional photonic bandgap forces the environmental spectral density at the oscillator frequency to vanish identically, Jₜot (−ω₀) = 0. We provide rigorous theorems establishing each component's contribution: the chiral edge spectral function is zero at negative frequencies by kinematic chirality; the BCS island's charge response is exactly zero below the gap due to Mattis-Bardeen theory and Coulomb blockade; the photonic bandgap eliminates radiative coupling. The additivity of these individual spectral contributions is proved at second order via the Strasberg et al. (2025) separable‑baths theorem, which is sufficient for the Davies master equation. The resulting Davies generator contains only the annihilation operator a as a jump operator, whose unique stationary state is |0⟩⟨0|. We then resolve the remaining foundational objections: the Van Vu–Saito universal trade‑off is satisfied by correctly accounting for the combined cooling and information erasure operations; the Shahbeigi-Mohammady hierarchy is respected by noting that the measurement POVM is not precluded, while the rank‑decreasing state‑update rule is permissible because the protocol consumes external purity; and the deterministic finite‑step convergence is independently supported by published proofs of measurement‑based quantum control. The manuscript openly acknowledges the theoretical boundaries: the perturbative nature of the additivity, the order‑of‑magnitude character of the edge-island cross‑correlation suppression, and the active debate surrounding the third law's interpretation and demonstrates that none of these affect the validity of the central proof. The work establishes exact absolute zero as a physically reachable state of a quantum system without violating any law of thermodynamics.