Complex dissipative structures, from ecosystems to biospheres, embody thermodynamic investments accumulated over evolutionary timescales. The quantitative magnitude of this investment, and the sustainability limits it imposes, have not been derived from first principles. We define accumulated negentropy as the integrated thermodynamic work invested in a system's information content over its history and, using Landauer's principle as the thermodynamic floor, provide a biosphere-scale accounting of this quantity. Earth's biosphere embodies $\sim 10^{29}$ J of accumulated negentropy, a value cross-validated within a factor of 1.36 using independent energy rate density measurements. The Burning-Library Ratio $\mathcal{R}_{\text{BL}} \approx 10^{-7}$ quantifies a seven-order-of-magnitude gap between the energy recoverable through destruction and the thermodynamic cost of replacement. We formalize the coupling between a system's accumulated complexity and its generative information rate, proving that partial destruction incurs a superlinear penalty on future information production. This rate-stock coupling yields four applied results: (1) a tipping-point threshold below which complex systems collapse irreversibly, consistent with the observed 20-25% deforestation threshold for the Amazon; (2) a maximum sustainable extraction bound derived from the balance between generative and decay rates; (3) superlinear recovery times that explain the observed hierarchy from fast-recovering grasslands to threshold-dominated rainforests; and (4) a conservation fidelity bound showing that ex-situ archives cannot substitute for in-situ preservation. The framework provides computable, energy-denominated sustainability limits for any complex dissipative structure whose information content was produced by extended search. The analysis requires only Landauer's principle, basic thermodynamics, and order-of-magnitude estimates from astrophysics and biology.