Sensitivity Before Connectivity: Minimal Pairs Reveal How CNNs Learn a Global Visual Rule

A network can see every pixel a visual rule depends on and still not use them. We study this gap with seeded connectivity on near-critical $64\times 64$ grids, where a full-access convolutional network faces zero Bayes risk. Path-conditioned AUC suggests that competence spreads from short to long paths during learning, but this picture is misleading. We introduce minimal pairs for spatial rules: two images that differ in a single pixel outside the query's local window, where closing that pixel disconnects the seed from the query. Size-matched placebo edits, which remove at least as much of the query's component without disconnecting it, separate connectivity from cluster size. Minimal pairs reveal three stages. With $64$k training examples, the model's discrimination does not exceed what the seed position alone provides, and it ignores cuts. With $256$k, it responds to distant cuts at every path length, but mostly in proportion to how many pixels an edit removes. With $1$M, it computes connectivity itself, including on paths that need more certified propagation steps than it has layers. Within individual training runs, the response to cuts appears at all path lengths at once. At first, it depends about equally on how many pixels an edit removes and on whether it disconnects. The weight on disconnection then grows until removal size no longer matters. Path length orders the strength of the response rather than its presence, and response strength is what path-conditioned AUC detects.