In the year 2000, the Clay Mathematics Institute placed one million dollars on each of seven problems — the deepest unsolved questions in mathematics. A quarter-century later, six still stand. One of them lives closer to this page than any prize problem has a right to: it concerns the Navier–Stokes equations, the mathematical laws of flowing fluids, and it asks something that sounds almost innocent. If a fluid starts out smooth, do the equations promise it stays smooth — or can the mathematics tear itself apart in finite time? For flows in three dimensions, nobody knows.
The equations themselves are old and unglamorous: Newton’s laws of motion, written out for a substance that flows. Claude-Louis Navier, a French engineer, set them down in 1822; George Stokes refined them at Cambridge in the decades after. They describe the swirl of milk in tea, the draw of a chimney, the birth of a typhoon — every eddy and gust on Earth obeys them. Yet turbulence, their signature behaviour, remains the outstanding embarrassment of classical physics: energy cascades from great whirls into ever smaller ones, and whether the mathematics stays well-behaved all the way down is precisely the million-dollar question. Lewis Fry Richardson caught the cascade in a rhyme in 1922: big whorls have little whorls that feed on their velocity, and little whorls have lesser whorls, and so on to viscosity.
Here is the part we find quietly wonderful. Richardson, the man with the rhyme, was also the first to dream of computing the weather from these equations — he imagined a great hall of human calculators passing numbers to one another. Today that hall exists as supercomputers, and several times a day they integrate the Navier–Stokes equations across the whole atmosphere, not once but about fifty times from slightly different starting points, because the atmosphere is chaotic and small errors grow. Count how many of those fifty computed futures cross a line — thirty degrees in Singapore, rain by evening in London — and you get the probabilities printed on this front page. Every percentage Harunagi publishes is the descendant of an equation whose good behaviour mathematics cannot yet promise.
The forecast works around the unfinished proof the way a bridge works around the unfinished theory of every rivet: numerically, carefully, and with its uncertainty carried in the open. The Clay prize has strict rules — a published proof, then two years of scrutiny by the world before the million is even considered — and the one problem solved so far, the Poincaré Conjecture, ended with Grigori Perelman declining both the money and the medal. Meanwhile, decade by decade, the forecasts sharpen anyway. The mathematics is unfinished; the weather report arrives each morning regardless, carrying its honest percentages. There may be no better proof that uncertainty, handled plainly, is a working tool long before it is a theorem.