The rapid three-fold enlargement of the hominin brain began about 2.3 million years ago (myr) as Africa dried and grass replaced brush, creating great savannas. Seeking an amplifying feedback loop, I analyzed the lightning-brush-fire ecology for grazing animals in a grassy burn scar. Discovering the new grass by exploring brush byways could promote a population boom−but only after grass-specialized herbivores evolved from mixed feeders at 2.4 myr. When the brush returned several decades later, the grazer boom would turn to bust, squeezing numerous descendants back into the core grasslands. Meat-eating Homo species would boom and bust when grazers did, enriching the core in whatever alleles were earlier concentrated in the brush fringe catchment zone for that boom. This return migration for Homo is what creates the amplifying feedback loop that speeds brain enlargement rate, likely up to the mutation rate limit. It also promotes trait hitchhiking: any brush-relevant allele, not just those for hunting, can experience amplifying feedback merely by hanging out in the catchment zone. The shade offered by brush would have been the default location for cooperative nurseries, time-consuming food preparation, and toolmaking. Increased behavioral versatility correlates with larger brain size. The more versatile brains of a current generation need only spend more-than-average time in the boom's catchment zone for this recursive evolutionary process to keep average brain size increasing via assortative mating. This can account for the time when linear enlargement began, when it ended, and why it slowed in Neanderthals and in Asian Homo erectus. Without utilizing Darwin's selective survival, the feedback loop makes advance room for "free" future functionality in the cerebral cortex, likely relevant to the evolutionary emergence of our structured intellectual functions such as syntax, contingent planning, games, and logic.