The Price of Remembering: A Calibrated Energy Law for Computation

Abstract: Most of the energy a computer spends goes not into computing but into keeping. A bit held in fast storage burns power for every second it stays there, and it costs energy again each time it moves between storage levels. We call the first cost \emph{rent} and the second \emph{fare}, and we state one law: the energy of a computation is at least its operations, plus rent on every live bit for as long as it lives, plus fare on every bit moved. The model beneath the law prices control too: there is no free clock, and any unpriced register would make the theorems false. One lemma carries the results: every use of a value is paid for by rent, by fare, or by recomputing it. Three consequences. Exact attention brings every past token back for every new one, so its energy grows with the \emph{square} of the context length while a fixed-state recurrent model grows linearly; the square is a theorem for machines that never re-read past tokens, and under a stated serving hypothesis it is the fare on every past token, which overtakes the model's own arithmetic near $10^4$ tokens, where long-context serving becomes bandwidth-bound today. Cumulative-memory bounds become joule floors: sorting $n$ items pays rent proportional to $n^2/\log n$ bit-steps on any sequential machine with volatile working storage, on most inputs, and the bound for scrypt makes each password guess cost joules that no parallelism reduces. The law is calibrated: on a synthesized 45\,nm processor whose storage physically moves, the rent constant measured at gate level is a clock term, $0.82$\,pJ per occupied slot per cycle, plus $3.4$\,fJ per net transition, within $4\%$ on every rotating run on operand-isolated hardware. Four fifths of that rent is the clock: keeping is mostly knowing when. Memory hierarchies exist because staging beats flattening, $83\times$ on our instrument. We say how the law can be proven wrong.