In 1887, astronomer W.H.M. Christie wrote an article in Monthly Notices of the Royal Astronomical Society describing a new kind of astronomical device. Tucked inside the courtyard of the Royal Observatory in Greenwich, the Personal Equation Machine was a wooden box containing a small circular opening representing the Sun and Moon and a smattering of pinholes designed to stand in for the stars.
Christie and colleagues sought to record the precise moment at which a star passed a particular position in the sky as the Earth rotated. To establish an accurate position, and accurately record the time at which it occurred, an astronomer might direct a telescope toward the expected path of a star and watch as it approached a fixed reference line in the telescope’s field of view. When the star reached this position, the astronomer dutifully recorded the exact moment of convergence.
Unfortunately, astronomers had different perspectives about when to record a particular movement. Some were too eager to anticipate, while some were too slow. Many naturally had different reaction times that created systematic differences in reporting. This individual timing bias was known as a personal equation.
Such biases left astronomers in a tricky position. They knew that two observers disagreed with each other, but could not determine who was closer to the truth. Any solution needed to provide both the time reported by an astronomer at the point of crossing, and an independently recorded reference time that could be used to determine the margin of error.
The Personal Equation Machine allowed them to view the artificial stars produced by pinholes and press a button when a star passed a specific point, which the apparatus automatically registered at a precise time. Astronomers could independently record the point at which a proxy star strayed into their field of vision, and in doing so received a comparison they could use to identify biases affecting the observations of the real thing.
Put differently: verification in one domain conferred the transfer of knowledge from that area to another. A frontier model trained on complex coding problems may also become better at proposing ideas, surfacing faults, and exploring alternatives. These are not quite the same abilities in every domain, since they operate on different knowledge in each; but they do not need to be learned each time from scratch. Because many of these abilities matter in writing and design too, training on verifiable tasks can improve performance on activities without answer keys.
The verifier’s law?
To understand where this kind of practice fits into modern AI, start with how contemporary models are trained. First, they are exposed to a massive corpus of internet text, and try to predict the next token. Initially, the weights of the neural network are completely randomized, so of course they do a terrible job. As these weights are adjusted through stochastic gradient descent, they come to encode a model of the world, learning ever deeper representations to predict internet text with greater proficiency. But we would like our AI models to do more than just understand the world; we want them to act on our behalf.
Perhaps they can learn to prove theorems and use tools and complete multistep tasks just by this next-token prediction training. In practice, though, we find it easier to elicit these abilities through reinforcement learning, as it rewards models for successfully completing tasks, as opposed to merely predicting what a successful solution looks like. So we take a pretrained base model and place it in an environment rather like a virtual Greenwich courtyard. In fact, we place it in many such environments, letting it try many attempts and nudging it toward whatever earned reward.
One use of this training is to get models to conform more to human preferences through Reinforcement Learning from Human Feedback (RLHF) — having human raters decide, for instance, whether model answers exhibited honesty, clarity, and helpfulness, and then reinforcing those traits. But to make the model more capable, this kind of training is best applied to tasks that are verifiable, such as arithmetic and programming exercises. Reinforcement Learning with Verifiable Rewards (RLVR) rewards reasoning pathways that lead to success, defined via specified mechanical checks like matching an answer or accepting checked proofs. Since there is no need for hand-marking, millions of attempts can be made, so a cheap, consistent check is a very effective teacher.
An early example of this was DeepSeek-R1-Zero, which used mathematical answer checks and programming tests to train a pretrained model. Over the course of this training, the model learned a lot about the particular domains of mathematics and programming (this partly explains the “jaggedness” of frontier model abilities). Although the model was rewarded for little more than correct math and programming answers, it also started to revisit its own working. In one example, it interrupted an algebra problem with “Wait, wait. Wait”, returned to the original equation and worked through the steps again. There was no separate reward for stopping to reconsider.
The prevailing view holds that the ease with which AI can be trained to do a task depends on how easily its success can be verified. Many of those we speak to at the frontier labs draw the further inference that the bottleneck to making progress on a domain is making the domain checkable; writing well, research taste, and other hard-to-check skills must wait.
But the condition used to reward an answer does not fully specify the capability a model develops while learning to satisfy it. The general reasoning abilities models learn in verifiable domains can generalize to domains that are not as verifiable. Consider formal proof checking. Systems like Google DeepMind’s AlphaProof used Lean to check that mathematical proofs followed explicit rules. But verifying that a model correctly solved particular problems is different from establishing what general mathematical abilities it has acquired.
Indeed, someone who thought otherwise would have been surprised when these achievements started extending beyond formal proof search — say, in 2025, when one of DeepMind’s internal systems received a gold-medal-level score (35/42) in the International Mathematical Olympiad, producing natural-language mathematical proofs under competition time limits. And their surprise would have grown further still in September 2026, when OpenAI’s internal model resolved the Navier–Stokes Millennium Prize problem with large-scale RL on a pretrained model. The result was reached in around 88 hours, and the Lean formalization took a further 17 hours. The models did not learn just a knack for satisfying checkers; they are simply good at mathematics.
One must therefore not inflate an observation about where training is cheap into a claim about what training can teach. An easy-to-check mathematical answer may require difficult, inventive reasoning; but no verifier needs to recognize that ingenuity for it to be acquired by the model. The verifier’s law explains why these tasks make effective training grounds, but says less about the abilities models might develop there.
The way of math
One might be relatively unimpressed by non-verifiable skills coming along for the ride in mathematics, since the domain as a whole is verifiable — problems have answer keys and proofs can be checked in Lean. Still, as one commentator recently put it: “I think we can assume that all verifiable domains will eventually go the way of maths. The biggest question in the world is whether non-verifiable domains will follow.” Is there any reason to think they might?
Let’s take writing, which certainly appears to be an unverifiable domain. There is no answer key, no program which can reliably return, “Yes, this is well-written” iff the input was, in actual fact, well-written. The quality of a piece of writing depends on features which are not only in the text: it depends on the expected readership, its purpose, and other contextual factors. And different qualities trade off against each other — clarity against distinctiveness, simplicity against rigor.
Of course, to say that writing is not objectively verifiable is not to say that it is impossible to evaluate. Judgment can be informed without being mechanical: editors are not guessing blind. It’s just that the difficulty lies less in recognizing success than in stating in advance some mechanically checkable criterion for it — some reward function that can be easily hill-climbed to proficiency.
Human judgments can be harnessed to improve model performance in these domains. We have already discussed RLHF, in which people’s comparisons between responses are used to train a model that supplies reward scores. But these scores are imperfect proxies for writing quality, and devising more mechanical checks — say, regarding variation in sentence length or complexity of vocabulary — risks capturing only narrow aspects of good writing. As Orwell famously put it in his rules on good writing: “Break any of these rules sooner than say anything outright barbarous.”
These methods do not offer the cheap, reliable feedback that makes mathematics such a productive training ground. But writing is not the only setting in which abilities relevant to writing are learned; related abilities can instead be acquired through practice on tasks with clearer rewards.
Why should we think such transfer is possible? We have already discussed how pretraining supplies models with a map of the world. But to act effectively in the world, models need to learn how to navigate that map, anticipating how different interventions would change the world, and using those predictions to find a path toward a desired outcome.
These are not completely separable conceptually: of course, text describes people acting and reasoning, and so models might be able to learn enough of the causal structure of the world to be competent agents simply by scaling pretraining. In turn, practice can deepen a model’s understanding of the world. What RL compensates for is the comparative thinness of the data about how solutions are actually found. GitHub contains enormous quantities of code, but relatively little of the process of producing it: the false starts, the failing tests, or the moments a programmer realized what was wrong. Likewise with prose; models mostly just see final results. While scale might help them infer the process from the final results, RL allows models to try different approaches and discover for themselves what works.
Gwern has argued that intelligence is simply search over Turing machines, or programs: there is no “master algorithm”; there is only an enormous number of special-purpose procedures — and of course, the compute required to find them. Navigating the map means running such procedures, each starting from some initial state and using what the model knows to reach the next. On this view, transfer does not require any general faculty. It only requires that procedures found in one domain make related ones easier or cheaper to find in another, once the model has the knowledge that domain requires.
There is some evidence for this picture, though transfer is — at this point on the RL scaling curves — far from universal. Gandhi and colleagues found that a model with some existing inclination to check its work and backtrack was able to improve substantially through RL on a numbers puzzle, while a similar model stalled until primed with examples of these habits — even examples that reached incorrect answers. Hu and colleagues likewise found that RL gains transferred between tasks with similar reasoning patterns, but not to knowledge-intensive domains such as law, medicine, and finance — which is exactly what we would expect if procedures must be learned again for each new body of knowledge.
Finding a mathematical proof, debugging a program, writing a great book, and designing a product all involve proposing possibilities, testing them, and revising them when they fail. If Popper is to be believed, this process is fundamental to how knowledge itself is created. Learning to do these things well — anew in each domain, but more cheaply each time — might be among the most general capabilities a model can acquire.
Particulars to universals
Philosopher of science Karl Popper argued that science advances not by accumulating observations until a correct theory emerges, but by generating conjectures about the world which we test through observation.
This is why, for Popper, the best theories are those that “stick their neck out” in a manner that allows us to gather evidence and attempt to prove them wrong through sustained criticism. This dynamic, between proposing alternatives and exposing them to criticism, informs the direction of scientific investigation without specifying what a subsequent step ought to look like.
Experience enables us to test competing conjectures, eliminating some and prompting new ones. The residue from this trial and error is the source of experiential learning that informs the generation of future conjectures and efforts to refute them. As a result, feedback proceeds by selecting among attempts to find the right answer rather than by offering specific instructions about how to find it.
In verifiable domains, producing knowledge requires both that the model is capable of proposing possible solutions as well as the subsequent criticisms that inform selection and revision. But verification can also occur in fuzzier, non-verifiable domains like writing or design (though the standards by which we determine what counts as successful differ).
The creation of phenomena
In mathematics and programming, such tests are often entangled with the problem — whether the code compiles, the test suite passes, and the answers match the answer key. Elsewhere, we may have to construct the test ourselves. This, after all, was the trick of the Greenwich astronomers: they created an artificial exercise, one in which they only knew the correct answer because they had arranged the conditions themselves. Could something similar work for non-verifiable domains like writing?
In a recent paper, Wang and colleagues propose Reinforcement Learning with Self-Verifiable Rewards (RLSVR). Despite the name, this is not about models marking their own homework; rather, the idea is to turn an activity without an answer key into an exercise that has one, by introducing some fact which is known to the training environment but the model must discover. The authors model the idea on self-supervised learning, the family of methods (including next-token prediction) which create training exercises by hiding information already present in the data, then asking a model to recover it. The hope is to construct artificial exercises whose answers can be checked automatically, and that success on these exercises develops abilities useful for the original, unverifiable activity.
They implement this design in SpyRL, which is based on Who Is the Spy? — a game where one or more players are secretly assigned to be spies, and must figure out the secret word or location before getting caught. Five copies of a model are given some task, say, summarizing a story or writing a story based on a prompt. Four receive the complete input, while the fifth — the designated “spy” — has 20% of it concealed. When all five responses have been produced, each copy inspects the work of the rest, and votes on who they think was working from incomplete information. And obviously since the environment itself chose the spy, it can determine which votes were correct, and use as a reward signal both how effectively the spy was able to disguise itself and how effectively the rest were able to guess the spy.
Notably, this is not a way of “making writing verifiable.” It is a way of isolating a particular criticism and constructing an exercise in which that criticism can be checked. In Popperian terms, each vote can be treated as a conjecture to be tested against the environment’s record. This is redolent of what Ian Hacking termed the creation of phenomena. Experiments are meant to uncover the causal structure of the world, but often, a particular experimental setup must be invented by scientists in order to induce certain phenomena to be investigated; it is not enough to simply observe regularities already handed to us by nature. As with the Greenwich astronomers, the researchers have created an artificial environment in which a judgment relevant to a more complicated activity can be tested.
But we can go further. In this kind of RL, it was the researchers who decided what sort of criticism might be useful for improving writing quality, and constructed an environment in which that criticism could be tested. But could models themselves learn to come up with useful criticisms? A recent paper by Xu and colleagues explores this possibility in programming.
Their method, Test Cases Scaling (TCS), trains models not only to solve programming problems, but to generate tests designed to expose flaws in candidate solutions. First, the model learns to devise valid tests, checking their expected results against a known correct program. Then it is rewarded for finding counterexamples: inputs on which a candidate program fails but the correct one succeeds.
It is one thing to falsify a proposed solution by showing that it fails a test; it is quite another to imagine the circumstances under which it might fail, and devise a test exposing the mistake. This is conjecture and refutation both, integrated within a single system. TCS and SpyRL offer glimpses of a broader possibility: that by constructing verifiable exercises, we can train parts of a much larger process of proposing ideas, criticizing them, and seeking better alternatives. Neither example establishes how far these capabilities will generalize. But we need not make the entire process of knowledge creation verifiable to develop some of the capabilities it requires. AI systems might eventually learn to sustain that cycle themselves, even in domains where no answer key exists.
The boundary of progress will not terminate at verifiable tasks. A verifier determines what earns a reward; it underdetermines what a model learns in trying to earn it. Practice in mathematics, code, and specially constructed exercises can strengthen the machinery for proposing solutions, finding faults, and revising ideas. That machinery may be reused across domains: procedures learned in one setting can make related procedures easier to discover in another. Sure, transfer will be uneven, depending on what knowledge it already has. But as that knowledge accumulates, the same process of conjecture and criticism can improve its performance, even where no answer key exists. If, as we believe, what models learn on verifiable tasks is a way of finding answers rather than a knack for satisfying checkers, non-verifiable domains may soon go the way of their verifiable counterparts: first where models already know what good work looks like, the rest as they acquire that knowledge.
Cosmos Institute is the Academy for Philosopher-Builders, technologists building AI for human flourishing. We run fellowships, fund fast prototypes, and host seminars with institutions like Oxford, Aspen Institute, and Liberty Fund.