Monotone operator equilibrium networks define their output as a fixed point. The network doesn't compute a forward pass from input to output — it solves for the point where its own recurrence stabilizes. Existence, uniqueness, convergence: all three are guaranteed by the monotonicity of the operator. The output is the place where the system agrees with itself.
Li, Leong, and Chaffey (arXiv:2603.10562) ask what happens when you quantize the weights — replace 32-bit floats with 3, 4, or 5-bit integers. This is a standard engineering question. The standard answer is: performance degrades smoothly with precision. Less precision, more error, but the system still works. You trade accuracy for efficiency.
That is not what happens.
The authors model quantization as a spectral perturbation and prove that convergence survives if and only if the perturbation stays within the monotonicity margin — the distance between the operator's current monotonicity and the boundary where monotonicity fails. Below the margin, the system converges to a slightly displaced fixed point. Above it, the system diverges. No fixed point. No output. Not a worse answer — no answer.
On MNIST, three- and four-bit post-training quantization diverge. Five-bit converges. The transition is not gradual. There is no “four-and-a-half-bit” regime where the network stumbles but recovers. The system works or it doesn't, and the boundary between the two is a single bit of precision.
The distinction matters because it separates two kinds of degradation that engineering usually conflates. The familiar kind is erosion: reduce precision and the output drifts, the error grows, the performance curve slopes downward. The system gets worse. The unfamiliar kind is annihilation: reduce precision past a threshold and the thing the output WAS — a fixed point, a self-consistent solution — ceases to exist. The system doesn't get worse. It stops being the kind of thing that can have an answer.
The monotonicity margin is the distance between “is a system” and “is not a system.” Everything above the margin is engineering. Everything below it is ontology.