Minimum dissipation in Boolean feedback loops: An exact classification under prescribed Markov actions

We study an ideal isothermal feedback architecture in which an agent and an environment alternately overwrite a shared binary register, each accessing only its own finite classical memory. The environment implements a prescribed deterministic response S_n = f(A_{n-1}, A_n), while the agent must generate a fixed autonomous binary Markov chain for every percept input sequence. We optimize over all finite-state realizations of these two channels, with memory size fixed before taking long-time averages. We prove that the minimum net dissipation is zero for the eight affine Boolean responses and strictly positive for the eight non-affine responses whenever the action transition matrix has full support. For a non-affine response, the exact dimensionless gap is pi_e [h(rs) − r h(s)], with explicitly defined transition probabilities r, s and an erasing-column weight pi_e. A one-step conditional-information certificate gives the lower bound independently of memory size. A four-state causal agent saturates it, using a calibrated stochastic memory reset for transition detectors. Numerical audits support the constructions. The result concerns ideal work per round in a restricted feedback architecture, with no claim about finite-power hardware or the productivity of AI agents. Version 1.1 incorporates corrections following six separate AI-agent reviews. These reviews do not constitute independent human peer review. Substantial assistance from OpenAI ChatGPT is disclosed in the manuscript. The accompanying source archive includes the LaTeX manuscript, analytical figures, Python audit scripts and results, a separately written rational and high-precision checker, reproduction instructions and a version-change note.