The Dissipative Cost of Pure Computation: Measuring Cognitive Thermodynamics in an Embodied Half-Cycle

We measure the thermodynamic cost of pure computation in a deterministic physical system (a sliding-puzzle solver running in RAM), and show that this system realizes what we call an embodied half-cycle: only two of the four quarters of a full cognitive-physical cycle (decision and internal action), without external sensing and without self-update. This minimal configuration is chosen because it isolates the cost of pure computation from the additional dissipation channels (sensory filtering, parameter update, maintenance) that a full cycle would introduce. In this regime, Shannon entropy is not needed even as an epistemic tool: the machine has no external world to infer, no uncertainty to filter, and no learning to perform. The only entropy that changes is the Boltzmann entropy of the puzzle state in silicon memory, reduced by irreversible bit overwrites whose cost is bounded below by Landauer's principle. Using a primitive unit of elementary operations (ops), we run 730 experiments across four domains and report: (i) an Ashby scaling of efficiency ηₒps with puzzle size N, dropping by three orders of magnitude from N=3 to N=6; (ii) a non-monotonic optimum in algorithmic sophistication C, with peak at C=4 and a ~2150× collapse toward C=256 (180 runs) ; (iii) a two-dimensional (N, C) cost surface that is non-separable, with a ~7-order-of-magnitude range of ηₒps over a 5×5 grid (250 runs) ; and (iv) a constant ratio of ~6×10¹² between measured energy per operation on modern silicon (1. 7×10⁻⁸ J/op, 50W CPU at 3 GHz) and the Landauer floor (kT ln 2 ≈ 2. 85×10⁻²¹ J/op), identified as a structural property of the substrate rather than of the algorithm. Zero Landauer violations were observed across all 730 runs. Together with the companion papers on absent lexical compression (Lacchè, 2026a), four orthogonal semantic microstates (Lacchè, 2026b), and recursive conceptual divergence (Lacchè, 2026c), these results support the broader claim that the cost of meaning, whether linguistic or computational, obeys the same thermodynamic laws as the cost of any physical configuration.