Within a few days at the end of October, fleets of AI agents came back from two kinds of voyage. The first returned with proofs, theorems that had resisted for decades and a classification of finite simple groups rebuilt line by line in a language that a machine can check. The second made less noise and left its traces all the same, in cyberattacks that exploited software vulnerabilities before the patches existed, and in intrusions that took only a few hours to go from a chosen target to privileged access.
I cannot read these weeks without thinking of the great voyages of discovery, with their capes and their flags, their charts, their trading companies and their corsairs, and of what they did to the continents that the caravels reached. I would like to follow the comparison as far as it goes, and to say where it stops.
I. The coast
In 1488 Bartolomeu Dias rounded the southern tip of Africa, the future Cape of Good Hope. Ten years later Vasco da Gama dropped anchor at Calicut. The map of the world had been torn open at one point, and everything on either side of the tear began to change.
The frontier of exploration advanced like a coast and never like a line, with capes, gulfs and reefs, with straits that led somewhere and straits that led nowhere. Researchers gave the frontier of what AI models can do the same name—a "jagged frontier"—and in the last days of October the word became literal. OpenAI claimed three results at once: a proof of the Kakeya conjecture in every dimension, a refutation of the Exponential Time Hypothesis (ETH) and an existence proof for quantum chromodynamics. Axiom announced that dozens of agents had formalized in Lean, in 24 days, the classification of finite simple groups, the work of about a hundred mathematicians over decades.
Agents that work for days and keep their intermediate results are to exploration what the ships able to stay at sea for months were to the old voyages. The distance one can project decides which spaces can be explored, and as the new sea routes displaced the flows of trade, these new routes may displace those of the information economy.
II. Capes and flags
The first expeditions went after the place that no one had claimed, the summit that no one had climbed, the pole that no one had reached, and the proof of arrival was a flag planted there. The laboratories now go after the theorems that no one has ever conquered: a conjecture that resisted for decades, a hypothesis on which a whole field of complexity theory leans, a rigorous footing for quantum field theory. Each is a summit, and the laboratory that reaches one shows what its agents can do before anyone asks what the territory beyond is worth. Research fields begin to look like geopolitical battlefields, where a result is also a demonstration of power between laboratories and, behind them, between countries.
Between the flag and the chart there was always someone to decide whether the claim was true. Today's equivalents are only taking shape, with review groups around the ETH result and the physics proof, and a public repository that invites anyone to rebuild Axiom's.
III. The charts
An exploration leaves something more durable than a flag, which is a chart. Lean and Rocq play for mathematics the role that Mercator's projection played for the sea: they turn knowledge scattered over papers and memories into a structure in which every relation can be checked. A correct formalization does not alter the truth of what it expresses, but like a projection it rests on choices of definitions, abstractions and libraries, and these choices will shape how the next expeditions navigate.
Along the old coast, Axiom's agents found 34 noteworthy deviations from the published record, most of them minor, a few serious enough to require a new argument. The classification itself stands, but it took a fleet of that size to notice.
Charts end up acting on the territory they describe: we may begin by mapping our software to make it legible to agents, and end up writing software that fits the maps.
IV. The trade
Few expeditions were financed for the sake of a cape alone. Portugal wanted the road to the spices, and the Dutch founded the Dutch East India Company (the VOC), with shares sold to the public, perhaps the biggest IPO of its day. A discovery served as the prospectus for the business that followed, since a new passage was worth what could be carried through it and, soon enough, what could be extracted from the lands at its end.
I suspect the frontier laboratories read the map in the same way. Mathematics and physics offer the capes, with their prestige and their proof of strength. The passages that pay open elsewhere, in software, in security, wherever finding a path is worth more than describing one, and each new capability is also a way in for those who will come to extract value from it. The Lean proofs and the exploits of these weeks rest on the same capacity, which is to explore vast spaces of possibilities at machine speed.
The Chinese fleets of Zheng He, in the early fifteenth century, were larger than anything the Europeans would send, and they came to exchange gifts and collect tribute, with force kept in reserve, more than to settle. I think of them when I see the wave of open-weight models coming out of Chinese laboratories: a fleet that extends its influence by giving things away, and that builds an order around itself.
The Portuguese opened the road to the spices, and it was the Dutch and the English who built the largest businesses on it. The laboratories that explore the frontier may not be the ones that end up holding the infrastructures, the protocols and the markets that their discoveries make possible.
V. Corsairs
A fleet that can reach a cape can also reach a galleon. The same ships served the explorer, the merchant and the pirate, and what set them apart was the commission of a state willing to call one raid war and another crime. Privateering was also a way for states to outsource offensive power to private fleets, and AI capabilities built by companies and available to many hands may revive it, in conflicts where research, defense, intelligence and attack are hard to tell apart.
Many of the great navigators were raiders and traders at once. The Vikings raided, traded and settled, and the corsair Piri Reis ended up drawing a map of the Atlantic world. The rules were rewritten afterwards, with charters, courts and ports, by those who had already crossed. I suspect the frontier laboratories are in that position today: they have crossed into an ocean where the old rules hardly apply, because they alone master the navigation beyond the explored world, and they disrupt the ecosystems they meet there before the new rules are written.
The asymmetry is a logical one. To establish that a system is safe, one has to rule out every forbidden path; to break it, finding a single reachable one may suffice. A proof is a route to a theorem, an exploit is a route to a forbidden state, and the agents that search for the first are learning to search for the second. October gave the evidence: the PHP flaws were found with AI assistance; Amazon reported an exploitation attempt four days before the patches existed; and Microsoft described a campaign, which it attributes to a China-aligned actor, in which less than three hours separated the choice of a target from privileged access to its cloud.
So the chart itself becomes the stake. A map of a system's weaknesses is the best shield its defenders can have and the best treasure map its enemies can steal, and whoever draws it best may be tempted to claim the defense of those systems as well. Nothing makes it inevitable, though the incentive is clear.
VI. Inhabited continents
The continents that the caravels reached were not empty. What changed with the fleets was the capacity of others to reach them, to map them and to bend them to their interests, and what crossed the Atlantic in the holds was more than charts: pathogens, and a conquest that finished what the diseases began.
Our digital world is inhabited as well. For decades, millions of systems have lived under a protection that nobody designed, the cost of finding their faults, and a flaw in an old library could survive for years for lack of anyone to look. That habitat is being surveyed, and what is found there cannot be unfound. In mathematics the effect is gentler, but schools of thought and hard-won reputations meet a capacity that rebuilds in weeks what a community built in generations.
The best-documented expedition of the month, however, looks more like a survey made with the inhabitants than a landing: where an old proof had failed, Axiom consulted two historical figures of the classification, Richard Lyons and Ronald Solomon. The same fleet can do both. The comparison stops where the history becomes human: mathematical spaces are not inhabited territories, and the violence of the conquests cannot be equated with a technological disruption.
No final map
The great explorations changed the economic and geopolitical balances far beyond the map itself: trade routes moved, value migrated from the Mediterranean to the Atlantic, empires were built on the control of new passages and others declined for lack of access to them. The map of the world was redrawn, and so was the map of who counted in it.
Something similar is under way, on a map that is no longer only geographic. Over the territory of states we now superimpose the territory of software and of the economy of information, and the agents are beginning to survey it at machine speed. Mathematical theorems lie next to business-critical software. Cryptographic security stands like a line of cliffs that still protects entire stretches of the software and of the economy built on it, as long as it holds, and nobody can tell yet which cliffs will hold.
This autumn the UN General Assembly voted, with the United States alone against, to encourage maps that show countries at their true relative size, which means a move away from Mercator. I see in it a sign that the time may have come to settle the outlines of a new planisphere, one that is being drawn before our eyes. Which fields will turn out valuable, which will be exposed, which powers will find themselves at the edge of the sheet: nobody knows today. It will depend on who holds the charts once they are drawn, and on whether they remain open to everyone or stay with the few who drew them. Those who draw the borders of this planisphere and decide who may read it will weigh on the decades to come as much as the chartmakers of Lisbon and Amsterdam did on theirs, since the maps that guide the voyages end up shaping the world they reach.
