A Stricter Way to Say an AI Is More Than the Sum of Its Parts
In three different corners of AI research, people keep making the same claim in different words. A transformer develops an ability that no single neuron accounts for. A group of agents solves a task that none of them solves alone. A person and a chatbot together reach an answer that neither reaches apart. Each of those results gets written up as emergence, or as synergy, or as the whole being more than the sum of its parts. And none of the three can be weighed against the other two, because nobody has said what the words mean.
That is the complaint behind a paper posted on arXiv on Oct. 20, 2026, by Christoph Riedl, of the Khoury College of Computer Sciences at Northeastern University, and Fernando E. Rosas, of the Department of Informatics at the University of Sussex. Riedl and Rosas are not arguing that the claims are wrong. They are arguing that, as stated, the claims are not the kind of thing that can be wrong: informal assessments unsupported by any framework, which leaves them, in the authors' words, "impossible to compare or even meaningfully dispute." The arXiv record lists the paper as a spotlight paper at NeurIPS 2026. That listing is supplied by the authors and arXiv does not verify it, and the conference meets in December 2026, so there is no proceedings version yet.
Their proposal is to stop arguing about the word and import a measure instead.
Three kinds of information, and only one of them counts
The measure is partial information decomposition, or PID, introduced in 2010 by Williams and Beer for a question that sounds almost too simple: given several parts of a system and some outcome you care about, how is the information about that outcome distributed among the parts?
Three answers are possible, and the whole argument rests on keeping them apart. Some information is unique, carried by one part and no other. Some is redundant, held by several parts at once. And some is synergistic, meaning it is available only from the parts taken together and cannot be recovered from any of them separately.
The authors borrow a viral model failure to make the third kind concrete. Ask a chatbot whether you should walk or drive to a car wash 50 meters away. Distance on its own argues for walking. Washing on its own argues for bringing along the thing to be washed. Only the two clues held together reveal that the thing in question is a car, and that walking defeats the purpose. That last piece of information sits in neither clue alone.
The distinction is not bookkeeping. Redundancy, Riedl and Rosas note, buys robustness, because the same information survives in several places; synergy buys efficiency, because a set of parts can encode far more jointly than separately. The price of that efficiency is fragility. Synergistic information depends on every contributing part at once, so losing one element can destroy it outright, while a redundant code degrades gently. Any system that becomes more capable under tight resources, the authors suggest, is under pressure to trade the first for the second.
A rule that can only say no
Here is the criterion, in the paper's own words: a system's behavior "is emergent only if it carries synergistic information about task-relevant outcomes."
Read that sentence slowly, because its shape matters as much as its content. It says only if. It does not say if and only if. A system that fails the test is not emergent under this definition, and that verdict is firm; a system that passes has cleared a necessary bar and nothing more. The criterion cannot certify that something interesting is happening. It can rule out the claim that something interesting is happening, which is more than the field currently has.
Nor is the criterion new, and the paper does not pretend otherwise. The decomposition is Williams and Beer's. The move of defining emergence through synergy belongs to Rosas and colleagues, who set it out in PLOS Computational Biology in 2020, and the three concrete tests that go with it were laid out by Mediano and colleagues in Philosophical Transactions of the Royal Society A in 2022. What is new here is the decision to point all of it at AI, and to argue that three subfields which barely cite each other have been circling the same quantity.
The machinery behind that argument is real rather than rhetorical. The paper restates the three tests with their formulas: one screens for whether a system can host an emergent variable at all, one checks whether a proposed system-level feature predicts the system's own future better than its parts do, and one is a cheaper practical score for roughly the same question. Nine open-source packages that compute these quantities are listed with their licenses, in Python and R, and the authors publish a quick-start repository at github.com/riedlc/emergence. A reader who wanted to test an emergence claim this afternoon has somewhere to start.
A limit sits directly on top of that promise. The theory fixes none of the choices a measurement needs. "The choice of parts, target, timescale, and redundancy measure is not given by the theory," the authors write, and claims about synergy "are bounded by those modeling choices." Inside a language model a part might be a neuron, an attention head, an expert module or a whole layer, and the answer can shift depending on which you pick. The criterion is falsifiable for a system someone has fully specified. It is not a verdict machine for AI in general.
Three worked examples, and whose work they are
To show the lens working, the paper walks through three settings at wildly different scales.
Inside a single model, it draws on a preprint posted in January 2026 by Pedro Urbina-Rodriguez and seven co-authors, Rosas among them. That study treats attention heads and expert modules as nodes, tracks how information moves between them, and reports a synergistic core in the middle layers of trained models with a redundant periphery early and late, a pattern its authors compare to the organization of the human cortex. It holds across Llama 3, Qwen 3 and DeepSeek V2 Lite. The core is learned rather than built in, since it is absent from randomly initialized networks. Knocking out synergistic heads hurts performance on tasks that need integration far more than knocking out redundant ones, and fine-tuning the synergistic regions with reinforcement learning produces larger gains than fine-tuning redundant components, while supervised fine-tuning shows no such advantage.
Across agents, the source is Riedl's own earlier paper, presented at the ICLR machine learning conference. Ten language-model agents per group each propose an integer, blind to the group's size and to each other's guesses, and are told only whether their total came in too high or too low. Three randomized conditions were compared: a plain one, one assigning each agent a persona, and one instructing agents to think about what the others might do. Both emergence tests came out significantly above zero. A measure of how stable the group's information processing was, sat indistinguishable from zero in the plain and persona conditions and rose sharply under the theory-of-mind instruction. Neither synergy nor redundancy on its own predicted whether a group hit the target; only their interaction did. Those numbers belong to that study, not to this one.
Between a person and a model, the paper leans on a preprint by Riedl and Weidmann, and this is the one case where it does new work of its own. Rather than measuring synergy directly, which the original data were never collected to support, the authors bound the other three kinds of information and show that a positive remainder survives. On items the AI could not solve alone, users still improved over their own solo performance, which rules out the model simply handing over answers. Users switched from a wrong solo answer to a right joint one on 17 to 19% of the easiest questions and on 31 to 38% of the hardest, which is the direction the argument needs. And when a user's measured theory of mind was zero, access to the AI brought no benefit at all. The leftover, the paper argues, was "obtained from evidence never collected with PID in mind."
All three cases come from this pair of researchers or a direct co-author; two of the three are preprints that have not been through review; and the paper itself runs no new experiments. Riedl and Rosas are scrupulous about saying so and name every source. But the picture of three independent fields converging on one quantity is not what is on offer here. It is one research program reading its own results through a single lens, and inviting others to try the lens.
Five predictions, written so they can fail
The most useful part of the paper may be the part with the least evidence behind it. Riedl and Rosas close with five conjectures, each stated with what would confirm it and what would sink it.
Grokking, the jump where a model suddenly generalizes long after it has memorized its training data, should show up as an abrupt swing from redundancy to synergy rather than a smooth drift. Jagged intelligence, the pattern where a model passes the bar exam and fails a children's puzzle, should track what they call synergistic depth, the degree to which an answer requires holding independently informative clues together; the car wash question is their example. Chain-of-thought reasoning, in which a model writes out its steps, should look like synergy unrolled over time, spilling into the written steps what a single pass cannot integrate, with diminishing returns on problems that resist being broken up. The collapse that models suffer when an instruction is split across two turns should turn out to be synergy disrupted rather than information lost. And theory of mind, in a model's input, in a group of agents and in a human collaborator, should act as the same control knob at all three levels.
These are predictions. The authors label them as such and note that only the first has even partial support in existing work. Their own closing line is the right one to end on: "Ultimately, this is an empirical question: the success of these hypotheses will dictate whether this framework is based on a mere analogy or on shared computational reality."
That is an unusual sentence to find at the end of a paper arguing for a framework, and it is the most persuasive thing in it.