Connes Rigidity Conjecture

A group is a set of actions that you can combine and undo. Every group has an algebra: sums of actions, with a number in front of each action. To multiply two sums, multiply every pair of terms, and use the group's rule on the actions.

Take the group with two actions: 1, which does nothing, and a switch $s$ , where $s \times s = 1$ . Then

\begin{matrix} (2 + 3 s) (4 + 5 s) & = 8 + 22 s + 15 s^{2} \\ = 23 + 22 s \end{matrix}

Now write each sum $p + q s$ as the pair $(p + q, p - q)$ . The two sums become $(5, - 1)$ and $(9, - 1)$ . Multiply position by position and you get $(45, 1)$ , which is the pair for $23 + 22 s$ . This change of view is a Fourier transform. It shows that the algebra is only pairs of numbers, and the rule $s \times s = 1$ is out of sight.

For an infinite group $G$ , the group von Neumann algebra $L (G)$ also allows suitable infinite sums. Connes's rigidity conjecture says that $L (G)$ remembers $G$ when $G$ is ICC and has property (T). Two such groups with the same algebra must then be the same group, up to renaming.

A group is ICC if it is infinite and every action other than "do nothing" has infinitely many conjugates. A conjugate of $h$ is "undo $g$ , do $h$ , do $g$ ", for some action $g$ . This condition makes $L (G)$ one piece that cannot be split.

Property (T) rules out small motion. Let a group rotate arrows in a space, so that lengths never change. Suppose that for any finite set of actions, some arrow of length 1 moves as little as you like. Property (T) says that some arrow of length 1 then does not move at all.

The whole numbers under addition fail this test. Let "add 1" turn the $k$ th plane by $1 / k$ degrees, for every $k$ . An arrow in a far plane barely moves, but no arrow stays still. Without property (T), the algebra can forget everything: a large family of ICC groups all share one $L (G)$ .

The counterexamples use a gap in what the algebra records. In a group where the order of actions does not matter, the Fourier transform gives lists of numbers, which can involve $i$ , where $i^{2} = - 1$ . The list positions can also be added. The algebra records the positions, each with equal weight, but not how they add.

So one set of positions can have two addition rules, and both give the same algebra. Take the 4 positions $(x, y)$ , where $x$ and $y$ are 0 or 1, and $1 + 1 = 0$ . Here are two addition rules, and the second has a carry, $x x^{'}$ .

\begin{matrix} (x, y) + (x^{'}, y^{'}) & = (x + x^{'}, y + y^{'}) \\ (x, y) + (x^{'}, y^{'}) & = (x + x^{'}, y + y^{'} + x x^{'}) \end{matrix}

The second rule is binary addition if $(x, y)$ stands for the number $x + 2 y$ .

Adding

(1, 0)

to each position, without and with the carry.

OpenAI reports an infinite version, from an internal AI model. A position is now an endless table of such pairs, in a special form. A group $K$ with property (T) shuffles the entries of each table. A shuffle does not depend on the addition rule, so $K$ moves the positions in the same way under both rules.

Reverse the Fourier transform for each rule, and combine each result with $K$ , to get groups $Λ$ and $Γ$ . Their algebras match, because the positions, the weights and the shuffles match. The groups differ: $Γ$ has an action that returns to "do nothing" after exactly 4 repeats, and $Λ$ has none. ICC and property (T) can be read from the algebra, so a check on $Λ$ covers $Γ$ .

The paper extends this to infinitely many different groups with one algebra. Shuoxing Zhou independently reports a different pair, found with help from GPT-5.6 Sol. OpenAI's proof is also written in Lean, a computer program that checks each step of a proof. A public project reports the same for Zhou's proof, apart from one cited theorem. One critique claims that OpenAI's groups fail the ICC test, and that critique is itself disputed.

So property (T) alone does not make the algebra remember the group. Some groups with property (T) are still remembered. A later OpenAI paper reports this for groups such as $S L_{3} (ℤ)$ , a group of $3 \times 3$ tables of whole numbers.