Keeping a Bit Alive: Maintenance Thermodynamics of Classical Memory

Writing a bit costs at least \(kT\ln 2\), and retaining one passively for a time \(\tau\) requires a barrier \(kT\ln(\tau/\tau_0)\). We derive the third accounting: the power a protocol must draw, in steady state, to keep a physical bit of barrier \(\Delta E\) readable at error rate \(\varepsilon\). For a single KMS bath with Markov jump dynamics and local detailed balance, every protocol class (feedback, autonomous, or error‑correcting) obeys\(\langle P\rangle \ge kT\ln 2\,(1-2\varepsilon)\,e^{-\beta\Delta E}/\tau_0,\)and ideal error correction saturates this floor. Measurement‑free protocols obey a stricter branch, the storage‑side specialization of the Hatano–Sasa housekeeping heat, whose deep‑regime form is\(\langle P\rangle \ge kT\tau_0^{-1}e^{-\beta\Delta E}\ln(1/\varepsilon^{*})\)with local slope \(\mathrm{d}\ln P/\mathrm{d}\ln(1/\varepsilon)=1/\ln(1/\varepsilon)\). These bounds complement, rather than overlap, the thermodynamics of stochastic resetting and of error‑correcting copiers: they price the maintenance of one fixed‑barrier degree of freedom by any protocol, and the optimum is attained, not merely approached. The two branches are experimentally distinguishable: at fixed \(\Delta E\), sweeping \(\varepsilon\) yields the \(1/\ln(1/\varepsilon)\) slope for housekeeping‑dominated maintenance and a flat profile for feedback‑dominated maintenance; at fixed \(\varepsilon\), both give \(\mathrm{d}\ln P/\mathrm{d}\Delta E=-\beta\), requiring no lifetime measurement. Hour‑scale passive memory requires \(\gtrsim 0.75\,\mathrm{eV}\) per bit at 300 K and cannot reside in ionic or hydrogen‑bond states. The brain operates two or more orders of magnitude above the maintenance frontier and does not test it. The decisive measurement belongs to single‑bit, near‑reversible platforms.