The Reliability Floor for Switching: Energy versus Error Probability in a Thermal-Noise-Limited Bit

A switch that must distinguish two states against thermal (Johnson) noise errs with a probability set by the energy barrier in units of kT. Using the thermal-activation form p = exp(-E/kT), checked against a Gaussian-noise model of a capacitive node, this paper derives the error probability per operation, inverts it to give the barrier required for one error per 10^15 (34.54 kT, 1.43e-19 J at 300 K), 10^18 (41.45 kT, 1.72e-19 J) and 10^25 operations (57.56 kT, 2.38e-19 J), and scales the result to a machine performing 10^18 operations per second: fewer than one thermal error per year requires 58.71 kT per switch and implies a total floor switching power of 0.243 W, against 21.1 MW measured for the Frontier exascale system at a comparable aggregate rate. Real gates operate roughly a thousand to a million times above the floor, for stated reasons (drive voltage set by the subthreshold slope, noise margins, speed, interconnect, and manufacturing variation). Model calculations are labelled SIMULATED, NOT MEASURED. The deposit includes the paper PDF, the computation script (compute.py) that produces every number in the paper, and its outputs (results.json, results.txt). Computations and drafting were performed with Dave, an AI research system.