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This work presents a constructive thermodynamic protocol that reaches exactly absolute zero using finite resources, within the rigorous framework of the resource theory of thermal operations. The core result is a General Zero‑Work Cooling Theorem: for any heat bath with strictly convex microcanonical entropy that contains a local two‑level subsystem matching the target qubit gap, a cascade of energy‑conserving SWAP operations cools the qubit from a finite initial temperature to any lower temperature in the bath’s accessible range with identically zero work and a finite number of steps. When the microcanonical temperature can be made arbitrarily small, as in an explicit quadratic‑entropy construction; the cascade alone reaches arbitrarily low temperatures work‑free. A single quantum non‑demolition measurement and a conditional π‑pulse then deterministically map the qubit to its exact ground state at a small, constant work cost, achieving exactly absolute zero with finite total work, a finite number of operations, and a finite predetermined time.

The result demonstrates that the unattainability of absolute zero is not a universal law of nature but a contingent restriction to concave‑entropy baths and measurement‑free unitaries. It elevates the escape clause of Masanes-Oppenheim to a generalized unattainability theorem for convex‑entropy systems, with direct connections to black hole thermodynamics, massive gravity, and quantum gravity models. The paper includes a fully explicit quadratic‑entropy cascade, rigorous bounds on all finite‑size corrections, a thermodynamic embedding of the measurement step via Landauer erasure, and a systematic refutation of all known objections.