Give a printer only black ink and white paper — no gray — and it can still show you a gray photo, by scattering black dots so their local average IS the gray. Each dot is a hard yes/no (a bit), wrong on its own; the error from rounding is pushed to the neighbours so it cancels out across the patch. That’s error-diffusion dithering (Floyd–Steinberg). Slide the gray and watch one-bit dots reconstruct it.
Floyd–Steinberg error diffusion quantizes each pixel to 0 or 1 (the nearest of the two available inks) and carries the quantization ERROR forward to not-yet-drawn neighbours, so it is corrected on average. The single pixel is a coarse lie; the LOCAL MEAN of the dithered field reproduces the continuous gray to high accuracy. A fail-loud self-check throws unless a mid gray (0.30) dithers to a field whose mean is within 0.02 of 0.30. The eye (and any blur/averaging read-head) low-pass-filters the dots back into the tone — the picture lives in the average, not the pixel. ◆ real halftoning, node-verified.
A 1-D error-diffusion pass shown for clarity (real Floyd–Steinberg spreads error to four 2-D neighbours); the local-mean-preservation property is identical and exact in expectation. The reconstruction is the eye’s spatial average, not per-pixel truth — the honest trade.