Why long chains break

A chain of coupled constraints only counts as solved when every link holds. Drag the per-link accuracy and watch what survives across the chain. The gold dots are measured runs from the Institute's bench, not the curve: they are what actually happened.

The curve is the independence bound G = p^D−1. Measured against every run on our bench it held as a conservative bound, real accuracy landed at or above it, but not as a precise predictor, so read it as a floor rather than a forecast. Two wrinkles we measured and did not smooth away: per-link accuracy itself sags as chains get longer (98.8% at six links down to 94.8% at sixteen), so size the requirement at the length you actually want; and errors correlate more at length, which pushes real results slightly above the bound. The practical reading is the last two rows of the panel: past a handful of links, "good enough" links stop being good enough, which is why widening a network was never going to reach long chains.