The work cost of answering in real time

Many devices must answer each request before the next one arrives, while remembering what came before. This paper determines how much work such a device must spend per episode, one complete run of the process, for every finite interactive quantum process whose outputs carry the same Hamiltonians as its inputs, at every positive finite temperature. Work law for processes. The least work per episode is kT ln 2 times one quantity: an entropy change plus an energy change, both measured against the same adaptive observer, the most demanding one, and minimized over all ways to build the process in the order its outputs are due. For processes that commute with time translation, a strong converse holds for devices with thermal hardware and a sharp battery. What is free. A process costs nothing exactly when it is a limit of devices that run energy-conserving steps, in the required order, on one shared heat bath. Order costs work. A two-qubit process that is free when its outputs may wait costs kT ln 2 [1 - h(p)] per episode when they may not, where p is the bias of a coin the process draws, even when everything is classical. Without time covariance. The same law holds for every finite process when the device's whole controller need only come back approximately, with its actual mean energy and entropy refunded and its startup capital declared. All rates here are optimal asymptotic work rates at vanishing error over the complete interaction, with hardware allowed to grow; where a controller comes back only approximately, its actual mean energy and entropy are also refunded up to sublinear changes. This record archives the manuscript, its TeX source and its machine checks (Python, and Lean with Mathlib). The manuscript is the authority for exact statements and hypotheses; the verification notes list what is checked by machine and what is proved only in the text. How the work was done, including the use of AI models, is described in the manuscript. This paper is one of four on the thermodynamics of time order: Only quantum channels save work by waiting (https://github.com/Apsiape/save-work-by-waiting); The price of keeping time (https://github.com/Apsiape/price-of-keeping-time); The price of time order (https://github.com/Apsiape/price-of-time-order). Scoped licenses: the manuscript, PDF and TeX source are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). The software and machine-readable verification artifacts are additionally available under the MIT License. See RIGHTS.md in the archive. The aggregate Other (Open) category represents this mixed-license package.