Millennium Prototypes Adjacent to the Clay Problems

Millennium Prototypes Adjacent to the Clay Problems

By C. Rich

In 2000, a group called the Clay Mathematics Institute picked seven of the toughest unsolved problems in math and put a one-million-dollar prize on each one. These aren’t the kind of problems you solve in school; they’re big, foundational questions that have puzzled mathematicians for decades or even centuries. One of them, the Poincaré Conjecture, about the shape of the universe in higher dimensions, was solved a few years later by a Russian mathematician named Grigori Perelman, who turned down the money. The other six are still open.

Over the past couple of years, I’ve been writing a set of short, carefully worded thought experiments, one for each of those six remaining problems. The point isn’t to solve them (I’m not claiming anything that bold). Instead, each thought experiment simply gathers the best-known “near misses”, simpler, fully solved situations that look a lot like the real problem and show why the actual one is so hard.

One thought experiment looks at fluid flow and a famous number-guessing game. The Navier–Stokes equations describe how liquids and gases move; the open question is whether a smooth starting flow in three dimensions can ever suddenly turn chaotic in finite time. The Collatz game says: take any positive integer, if it’s even, divide by 2, if it’s odd, multiply by 3 and add 1, and repeat. Does every number eventually reach 1? The thought experiment builds tiny toy versions: a fluid equation with an extra strong damping term that keeps everything smooth forever (unlike the real case, where blow-up is possible), and a modified Collatz rule with built-in brakes that forces every sequence to settle down. These little models are completely solved, but they highlight exactly what extra ingredient is missing in the real problems.

Another thought experiment is about the quantum theory of the strong nuclear force, the Yang–Mills equations. Physicists use them every day, but mathematicians still don’t have a fully rigorous version in four dimensions that proves the lightest particles have positive mass. The thought experiment collects two classic solvable examples: a two-dimensional version that exactly generates a mass through a quantum anomaly, and a different two-dimensional theory taken to a large-number limit where non-perturbative effects create confinement and a mass gap. Both are textbook successes, sitting right next to the unsolved four-dimensional case.

A third thought experiment tackles the Riemann Hypothesis, the most famous problem about prime numbers. It says that the complicated pattern of the primes is controlled by the zeros of a certain function, all lying on one specific line in the complex plane. The thought experiment brings together three standard pieces of evidence: a complete proof of the exact same statement for “prime numbers” over finite fields (a different kind of arithmetic), the astonishing statistical match between the spacing of high zeros and the eigenvalues of large random matrices, and the known regions where no zeros can hide. None of it proves the original hypothesis, but the pattern looks unbreakable in every direction we can rigorously check.

A fourth thought experiment covers P versus NP, the question at the heart of computational complexity: if you can quickly check a solution to a problem, can you always quickly find a solution? Most experts think the answer is no, but proving it has been brutal. The thought experiment lays out the main roadblocks: artificial “oracle” worlds where P equals NP and others where it doesn’t (showing ordinary diagonalization can’t decide the issue), a barrier theorem showing that most successful proof techniques are blocked by plausible cryptographic assumptions, and strong hardness results in restricted settings like constant-depth circuits. Together, they map why the problem feels so stuck.

A fifth thought experiment is about elliptic curves, smooth cubic equations with surprising ties to deep number theory, and the Birch and Swinnerton-Dyer conjecture. It predicts that the number of rational points on such a curve is encoded in how its associated L-function behaves at a special point. The thought experiment surveys the cases we can prove: when the analytic prediction is small (rank 0 or 1), powerful theorems produce actual rational points and match the ranks perfectly; explicit constructions called Heegner points link derivatives of the L-function to geometric heights; and modern averages show that most elliptic curves have very few rational points, just as the conjecture expects.

The final thought experiment concerns the Hodge conjecture in algebraic geometry: on a complex algebraic variety, certain specially harmonious classes in topology should always come from actual algebraic subvarieties. The thought experiment starts with the completely solved simplest case (classes of type (1,1) come from line bundles), lists families where the full conjecture is known, abelian varieties, many K3 surfaces, some hyperkähler manifolds, and ends with structural results like the unconditional algebraicity of loci where extra classes appear under deformation.

These six short writings are now finished. Each one is deliberately modest: it only collects established results from the literature, states clearly that nothing in it solves the open problem, and cites the original sources. The goal was simply to put the best-understood near-misses in one place, so anyone curious can see exactly what we do know and where the frontier lies.

Behind the project is a loose intuitive picture I sometimes think about: explosive or divergent systems kept in check by some kind of boundary or vacuum constraint. The toy damping terms, the large-N saddle points, the finite-field geometry, the proof barriers, the Heegner points, and the variational constraints all feel like echoes of that idea. Of course, that’s just a personal lens, not a rigorous claim.

The six big problems are still there, unsolved and magnificent. But the ground around each one is full of rigorously explored territory. These thought experiments are just a set of clear signposts pointing out that territory, in the hope that seeing it plainly might someday help someone spot a route across the remaining open space.

A more technical version of these ideas, with references and detailed prototypes for each problem, appears as Pillar 21: Millennium Prototypes Adjacent to the Clay Problems in the Lava-Void Cosmology series. A separate companion note explores a conceptual analogy between Grigori Perelman’s $\mathcal{W}$-entropy for Ricci flow and the entropy structure of Lava-Void Cosmology.

Technical_Prototypes_Adjacent_to_Clay_Problems

Pillar 21 Mathematical Paper

C. Rich