A Coding-Theoretic Carnot Cycle: Extremal D-ary Huffman Bounds, the Sagawa–Ueda Identity, and a Closed-Form Efficiency

We construct an exact, closed-form Carnot cycle whose state space is the simplex of probability distributions on a finite source alphabet, and whose working fluid is the optimal D -ary Huffman code attached to that source. The hot reservoir is identified with the uniform distribution, which maximises the average codeword length over all sources on a given alphabet; the cold reservoir is identified with a dominant-symbol quasi-uniform family whose average codeword length approaches its minimum possible value from above as the rare-symbol probability vanishes. Two adiabatic legs, defined by transcodings that preserve the dominant symbol’s unit-length codeword, close the cycle at a work cost that vanishes in the cold-reservoir limit. The resulting efficiency, expressed as the ratio of hot- to cold-reservoir temperatures in units of average codeword length, takes a closed form determined entirely by the alphabet sizes of the hot reservoir, and reproduces standard special cases including a value of eighty-seven and a half per cent for a binary source with two hundred and fifty-six symbols. We then embed the cycle in the Sagawa–Ueda generalised Jarzynski framework, identifying the trajectory-level mutual information with the log-likelihood difference between the source distribution and the codebook’s implicit distribution. Pathwise saturation of the resulting Jensen inequality is shown to require both source uniformity on its support and a matched codebook— a condition strictly stronger than the zero Kullback–Leibler criterion previously invoked in the literature. We derive the leading finite correction to the efficiency and show that it is linear in the rare-symbol weight, whereas the Jensen gap vanishes more slowly, so that pathwise saturation is restored before the Carnot limit is attained. The strict impossibility of reaching average codeword length equal to one for any source with more symbols than codeword letters is reinterpreted as a quantitative, third-law-style unattainability principle. The construction is exact at every step, is internally consistent with the Sagawa–Ueda framework, and closes rigorously the geometric Carnot diagram proposed in earlier work.