Three of mathematics' hardest unsolved problems.
We found the same structure underneath all of them.
The Clay Institute has offered $1,000,000 for each of seven Millennium Prize Problems since 2000. Only one has been solved. We applied a physics framework to three of them and found they're all asking the same question — where does Pe = 1?
The Péclet number (Pe) is a ratio physicists use to measure when drift overpowers diffusion in a fluid. We showed it applies to information systems — attention, algorithms, turbulence. Pe < 1: the system stays orderly. Pe > 1: it drifts. Pe = 1: the critical boundary.
What we found: three Millennium Prize Problems are asking, in three different mathematical languages, exactly where this boundary sits and what happens at it.
Does turbulence ever blow up to infinity — or does physics keep it safe forever?
Water flowing through a pipe, air over a wing, blood through an artery — all governed by the Navier-Stokes equations since 1845. The question: can a smooth, well-behaved flow ever develop an infinite velocity spike in finite time? Nobody knows. It's been open for 90 years in its modern form.
Drag the slider. At Pe = 1, streamlines are orderly and parallel — laminar flow. Push Pe above 4 and watch them cross, tangle, and break down — turbulence. Blow-up would mean Pe → ∞ in a finite moment.
Where are the zeros of the Riemann zeta function?
The Riemann zeta function encodes the distribution of prime numbers in its zeros. Riemann conjectured in 1859 that every non-trivial zero lies on the line Re(s) = ½. Over 12 trillion zeros have been verified there numerically. Nobody has proved it must be true.
Watch the dots in the visualization. They appear at random positions in the critical strip, then settle on the central gold line — the Pe = 1 boundary. An off-line zero would cost twice as much: the functional equation ζ(s) = χ(s)·ζ(1−s) forces a mirror-image zero to appear on the other side.
Can every problem whose answer is easy to check also be solved quickly?
If you can verify a Sudoku solution in seconds, can you always find one just as fast? P = NP would mean yes — and would break essentially all of modern cryptography overnight. P ≠ NP (what nearly everyone believes) means no — some problems are fundamentally harder to solve than to check.
The visualization shows the information budget H(Y) as a glowing sphere. Gold particles = the NP certificate (the solution that's easy to verify) — they fill the sphere completely. Cyan particles = any polynomial-time algorithm — they approach but are deflected. The budget is full. There's nothing left for the algorithm.