Holographic Thermodynamic Computing: Exponential Encoding Advantages from Entanglement Entropy

A holographic system with central charge c encodes exp(c) degrees of freedom in O(c) geometric parameters---an exponential compression dictated by the Ryu-Takayanagi formula. We ask whether this compression yields quantitative advantages for thermodynamic tasks, and find that it does in a precise sense: Holographic structure reduces the resources needed to describe thermodynamic processes, though not the runtime of individual operations. We prove three theorems and state one conjecture. Theorem N1 shows that holographic cooling requires entropy reduction scaling as O(c), exponentially less than the e^c cost for generic states of the same Hilbert-space dimension. Theorem N2 shows that a holographic Maxwell's demon extracts work at efficiency eta = 1 - O(1/c) in the connected Ryu-Takayanagi phase. Theorem N4 shows that holographic states admit tensor network (Multi-scale Entanglement Renormalization Ansatz, MERA) representations with bond dimension polynomial in system size, and provides a constructive algorithm with convergence guarantees. We conjecture (Proposition N3) that holographic error-correcting redundancy reduces erasure costs below the Landauer bound. All results include explicit error bounds, validity conditions, and experimental protocols for near-term quantum hardware.