The paper explores the problem of stabilizer Hamiltonians under local, incoherent Pauli errors by constructing a mapping from the $n$th moment of the decohered ground state density matrix to a classical statistical mechanics (SM) model. Using two approaches—Haah’s polynomial formalism and the CSS-to-homology correspondence—the authors demonstrate that various measures of information capacity, such as quantum relative entropy, coherent information, and entanglement negativity, map to thermodynamic quantities in the SM model. This mapping is used to analyze the 3D toric code and X-Cube model, providing bounds on their optimal decoding thresholds and insights into their information properties under decoherence. The SM mapping is also shown to act as an "ungauging" map, where the classical models describing a given code under decoherence can be gauged to obtain the same code. The paper further discusses correlated errors and non-CSS stabilizer codes, concluding with a discussion on the results and future directions.The paper explores the problem of stabilizer Hamiltonians under local, incoherent Pauli errors by constructing a mapping from the $n$th moment of the decohered ground state density matrix to a classical statistical mechanics (SM) model. Using two approaches—Haah’s polynomial formalism and the CSS-to-homology correspondence—the authors demonstrate that various measures of information capacity, such as quantum relative entropy, coherent information, and entanglement negativity, map to thermodynamic quantities in the SM model. This mapping is used to analyze the 3D toric code and X-Cube model, providing bounds on their optimal decoding thresholds and insights into their information properties under decoherence. The SM mapping is also shown to act as an "ungauging" map, where the classical models describing a given code under decoherence can be gauged to obtain the same code. The paper further discusses correlated errors and non-CSS stabilizer codes, concluding with a discussion on the results and future directions.