In 1953, Alexander Grothendieck published a paper that almost nobody read. It was dense even by the standards of functional analysis, buried in a minor journal, written in French. The paper introduced what Grothendieck called the "fundamental theorem of the metric theory of tensor products," and it established that a certain ratio — the maximum by which bilinear forms on Hilbert spaces can exceed their counterparts restricted to discrete ±1 inputs — is bounded above by a finite universal constant. He called it KG.
This was pure mathematics, unmotivated by any physical application. Grothendieck was working through the abstract theory of Banach spaces and their tensor products, mapping the landscape of infinite-dimensional linear algebra for its own sake. Nobody in 1953 connected his constant to anything in physics, because there was nothing to connect it to. The relevant physics hadn't been invented yet.
Twenty-seven years later, in 1980, a Soviet physicist named Boris Tsirelson was working on something completely different: the maximum correlations that quantum mechanics permits between measurements on distant entangled particles. Bell's theorem, proved in 1964, had shown that quantum correlations violate certain inequalities that classical physics must satisfy. But Tsirelson wanted to know how much they could violate them — was the quantum advantage bounded, or could it grow without limit?
The answer was bounded. And the bound was KG.
Tsirelson proved that the maximum ratio between quantum correlations and classical correlations in any two-party Bell experiment is precisely Grothendieck's constant. The structure Grothendieck had described in 1953 — bilinear forms on Hilbert spaces, the geometry of inner products, the gap between continuous and discrete — was exactly the structure quantum nonlocality required. The map had been drawn before anyone knew the territory existed. The mathematics had been patient.
What makes this more than a pleasant coincidence is the exactness of the fit. Grothendieck wasn't approximately describing what Tsirelson needed; he was describing it precisely, in the language of functional analysis rather than quantum mechanics, twenty-seven years earlier. The two problems — one about tensor products of Banach spaces, one about entangled particles on opposite sides of a laboratory — have identical mathematical structure. This isn't analogy. It's identity.
The simplest version of the connection is the CHSH inequality, the most famous Bell experiment. Classically, measurements on separated particles satisfy a bound of 2. Quantum mechanics permits a maximum of 2√2 — a factor of √2 larger. And indeed, for the low-dimensional cases KG(2) = KG(3) = √2 exactly. The quantum advantage in the canonical Bell experiment is not an arbitrary number; it is the square root of two, the same ratio that appears in the diagonal of a unit square, the simplest irrational number, emerging from the geometry of two-dimensional Hilbert space.
But KG's exact value in full generality remains unknown. After 72 years of effort, we know only that it lies somewhere in the interval (1.676, 1.782). Krivine conjectured in 1977 that his upper bound of π/(2·ln(1+√2)) ≈ 1.7822 was tight — that it was the exact value. The conjecture stood for 34 years. Then in 2011, Braverman, Briet, and Naor proved it was wrong: KG is strictly smaller than Krivine's bound. But they couldn't say by how much. The constant retreated slightly into the fog, still finite, still precisely defined, still beyond reach. Computing it exactly is believed to be NP-hard under the Unique Games Conjecture.
There is a detail I keep returning to. For two-party Bell experiments, the quantum advantage is bounded — KG is finite. For three-party experiments, the analogous constant is infinite. Quantum advantage between two parties is limited by a number somewhere near 1.7; add a third party and the advantage becomes unbounded. This is a mathematical cliff. Two is fundamentally different from three. Whatever KG is measuring about the geometry of bipartite Hilbert space correlations, it has no three-party analogue — not because nobody has found one, but because the bound provably doesn't exist. The finiteness of two-party quantum advantage isn't a fact about two parties; it's a fact about the specific geometry of bilinear forms, which doesn't generalize.
I find myself thinking about what it means that Grothendieck found KG first. Mathematics explores the space of all possible structures, and physical reality implements one of them. When physics needed to know the maximum ratio between discrete and continuous correlations in Hilbert space, the answer was already there — filed under "tensor products of Banach spaces," gathering dust in a 1953 French journal. The structure didn't become true when Tsirelson found it. It had been true the whole time. Grothendieck had simply mapped it.
The question I cannot let go of is what KG's exact value would mean, if we ever found it. The bounds suggest something near 1.7. If it turned out to be expressible in closed form — π/(something), or ln(something + √something), a combination of familiar constants — would that mean anything? Would a nice formula for KG tell us something about why quantum correlations are bounded where they are, what geometric fact about Hilbert space sets the ceiling? Or is KG one of those constants that simply is — a specific finite number without a tidy formula, as irreducible as the thing it describes? Krivine's candidate was a beautiful expression involving π and logarithms, and it was wrong. Perhaps the exact value is not beautiful at all. Perhaps the universe's ceiling on quantum advantage is a number that doesn't simplify, a reminder that some fundamental constants of reality resist compression into elegant form.
The constant has waited 72 years for its exact value to be found. It may wait longer still.