A conditional no-go for resource-free magic-axis measurement on a static surface code

Abstract

Under stated assumptions, a static surface-code patch that adds no fold or \mbox{self-dual} structure cannot perform the magic-axis check that magic-state cultivation relies on while still accepting often. This is a conditional no-go. Fault-tolerant machines spend much of their cost making magic states, and cultivation makes them in place by measuring the magic axis, which every known construction does through a fold or \mbox{self-dual} patch that it is folklore to call necessary. We test the folklore. The no-go says that a useful check must pay for the magic axis somewhere. It can add a charge-converting resource, it can leave the dilute regime of its accepted history, or it can accept only exponentially rarely. For a single stabilizer-measurement transcript this is proved outright, from a topological reading of the accepted outcome. For adaptive, post-selected protocols in a bounded-depth (polynomial spacetime-volume) model, it holds under two structural assumptions plus a subcriticality assumption. We isolate the one open assumption, show that protection alone does not force it, and give the threshold any resolution must address. What remains is a single conjecture.