Imagine an intelligence for which the right way forward is always apparent. Give it a problem, and it knows how to approach it. The answer might still require a long calculation, but the uncertainty about what to do has disappeared. Could intelligence ever reach that point?

I keep returning to this question because we already know what a small part of that experience feels like. A problem that once left us searching for a beginning can become familiar. We recognize its structure, see which method belongs to it, and get to work. Something from our earlier effort has survived and made the next problem easier.

I will call that reusable guiding knowledge intuition. Through this lens, becoming more intelligent means acquiring knowledge that gives us a useful direction across more situations. A discovery becomes a method, the method becomes teachable, and what once stood at the frontier becomes a routine exercise. The question is how far this process can go.

To examine it, begin with the intuition already acquired. Let $K$ be a finite package containing a system's methods, representations, learned parameters, and remembered answers. Its size, $|K|$, is measured in bits. One package must serve many problems, and it is fixed before the new problem arrives. Even a table of answers is allowed; it simply has to fit. A short algorithm can handle infinitely many inputs, so being finite does not itself impose a limit on the package's reach.

That reach depends on how much work remains when the knowledge is used. For the mathematical discussion, take a family $Q$ of solvable constructions with explicit rules and a checking procedure $V$: $V(x,y)=1$ means that $y$ is a correct answer to problem $x$. A fixed deterministic universal computer runs the code in $K$, with at most polynomial simulation overhead. Each problem starts with $K$ and its own input; reading, reasoning, translating, calculating, writing the answer, and checking all count. Write their total cost as $\tau(K,x)$, with infinite cost for failure.

Now choose an allowance $t(n)$ for a problem described by $n$ bits. The problems the system can finish within that allowance form its routine territory; the rest form its frontier:

The frontier therefore belongs to a system with particular knowledge and resources. A better method can move it. More time can move it too, which is why comparing intuition requires holding the work allowance fixed.

I expect this cycle to continue. Problems at the frontier demand work; some of that work becomes reusable knowledge; a new range of problems becomes routine. An abstraction can travel farther than its examples, covering whole families that were never practiced one by one. Yet even if every fixed problem eventually becomes—and remains—routine, every stage may still leave a different problem beyond the allowance. Progress could continue indefinitely while difficulty keeps reappearing ahead of it.

For humans, that raises a question about time on a much larger scale. Knowledge survives across generations, but each person has to learn enough of it to contribute. As fields deepen, we specialize and collaborate, distributing understanding that no one person can hold. Benjamin Jones's work on the burden of knowledge connects this pressure with longer preparation, narrower expertise, and greater reliance on teams, using patent data and a model of innovation. I see the same question waiting at the frontier of knowledge: if the necessary preparation keeps growing, could some advances eventually require more understanding than one person can acquire in a lifetime? Who will hold the separate intuitions together well enough to see the next connection?

AI changes the scale on which that knowledge can be accumulated. In formal theorem proving, training on checked attempts and generated problems has produced systems that need less search to reach a given solve rate on the evaluated tasks. Earlier computation has become guidance for later computation. In the sense I have defined, we are already computing intuition.

That intuition is also contributing at the research frontier. In August 2026, OpenAI reported ten advances in mathematics and theoretical computer science produced with an internal Astra model across several specialties, with human work on the manuscripts and subsequent formal certificates. I see these developments as the beginning of a cycle in which machines master methods, use them on harder problems, and turn their own discoveries into the material for further learning. I expect systems to carry this process beyond individual human ability and eventually beyond what humanity can accomplish collectively in important forms of intellectual work. Their frontier would no longer be bounded by the examples people had supplied.

This brings the imagined intelligence closer to a concrete question. If we could give a system the best possible intuition at every scale, how much work would still remain? Let $Q_n$ be the nonempty collection of problems of size $n$. With at most $M$ bits available, the best guarantee is

One package has to serve the whole collection: we choose the best package by the work it needs on its hardest problem. A different finite package may be prepared at each size. Allowing more memory can only preserve or improve this optimum, because the old package remains available.

To connect this cost to knowing a direction, consider constructions made through binary choices. A body of rules $\Gamma$ says that certain combinations of values must hold: each rule requires at least one of up to three specified variable values. Together these rules form a 3-CNF formula. A new problem gives some values already fixed, written $\rho$, with the promise that a complete solution exists. Useful guidance chooses a value for the first unset variable while preserving at least one complete solution. If the system can calculate such a choice efficiently for every viable partial assignment, repeated guidance finishes the construction in at most one call per variable. Conversely, a solver that completes any supplied viable assignment can supply its next bit. Here, perfect guidance and completion are efficiently equivalent.

Let $N$ be the encoded size of the rules, including their variable list, and let $Q_\Gamma$ contain every viable partial assignment. Can two fixed polynomial bounds $p$ and $q$ make the following possible?

The system may study the rules for as long as it needs, but it must retain a compact package before the fresh partial assignment arrives. That package must then complete every viable assignment while preserving the supplied values. The rules remain part of the input, so their size and use count; $|K_\Gamma|$ measures the additional prepared knowledge. A polynomial allowance grows at most as a fixed power of the input length, giving us a broad theoretical standard of efficiency.

Under these conventions, $H$ holds exactly when $\mathrm{NP}\subseteq\mathrm{P}/\mathrm{poly}$. NP concerns decision problems whose yes answers have short, efficiently checkable certificates; P concerns decisions computed in polynomial time. P/poly allows a different small Boolean circuit—or, equivalently, a small package of advice—for each input length. This circuit and advice framework fits the permission we gave to prepare knowledge in advance. If preparation itself must also be carried out uniformly in polynomial time, the completion question becomes $\mathrm P=\mathrm{NP}$. Both questions remain open; the familiar conjecture $\mathrm P\ne\mathrm{NP}$ alone is not known to rule out $H$.

I expect some constructions to keep demanding substantial fresh work even after the best compact preparation. Refuting $H$ would establish one strong version of that expectation and would imply $\mathrm P\ne\mathrm{NP}$. We can also pursue narrower answers: for a chosen family and memory budget, prove a concrete lower bound on $T_Q(n,M)$. Such a bound would tell us how much work cannot be prepared away. That work could be a long, correct calculation, so cost alone would not show that failed guesses are necessary.

Representations are already included in this limit question, because their code and use belong to $K$. Any lower bound covering every permitted package would cover the best representation too. But long before we reached such a limit, a better representation could change the reach of a particular system enormously. Seeing a geometric relation as an equation, for example, lets methods developed in algebra become useful in geometry. Problems that looked separate can share the same underlying work.

A useful representation includes that entire passage from one form to another. Let $e$ translate the problem, $a$ solve it in the new language, and $d$ translate the result back. For every problem in the intended collection $C$, these procedures must finish and produce a correct answer:

Their descriptions and running costs count, including the work of recognizing which representation to use. This is what makes the translation useful rather than merely another place to hide the original difficulty. It also explains why brevity alone is insufficient: a tiny program that tries every answer may store very little and still take far too long. The knowledge-compilation literature studies this relationship between compact storage and the operations that the stored form makes efficient.

What I want is the minimum useful representation: the least retained structure that makes a broad collection of problems routine. Using the same definitions, its storage requirement is

The target collection and work allowance give “minimum” its meaning. A smaller package must still cover all the intended problems and remain useful within the allowance; if none does, the value is infinite. To ask for breadth, choose families $C_1,\ldots,C_r$ that differ in their rules, structure, or domain, and require the same package to cover their union. We can also record which entire families it makes routine:

The families must be chosen in advance, because many variations of one familiar problem do not establish the same breadth as a substantial change of structure. This measures what the knowledge can handle. To understand how it handles those tasks, we must ask how much they share inside the representation. That is the connection I want AI to strengthen: learning one thing should make many other things easier to understand.

I expect more useful examples and computation to deepen AI's mastery of particular problems. Yet rewards for solving a task can be earned through increasingly specialized methods. I suspect that success on rewarded tasks can grow faster than the breadth of the intuition behind it. This concern connects to the distinction between demonstrated skill and generalization in François Chollet's account of intelligence.

My thesis is that the human advantage in general intelligence comes from a broader representation. Human intuition seeks simplifications: patterns that remain useful when the subject changes. A geometric idea can illuminate algebra; a relationship learned in one part of life can organize another. This breadth lets us connect fields, see what a problem is really asking, and pose interesting questions that a collection of specialized methods would leave unseen. It is the sense in which I see humans as smarter even when a machine is the stronger solver of a particular problem.

A short human life makes this kind of reuse valuable. We have limited experience and time, so a pattern that explains many things goes further than another method for one narrow case. Machines can afford much finer specialization. I see that difference as one reason their intelligence can be jagged: formidable in one setting, unexpectedly weak in another. Broader representations would connect more of those abilities and make their usefulness carry across situations. The goal is to unite the depth machines can acquire with the simplifying reach I recognize in human thought.

How do we compute that reach? Representation learning is already part of modern AI, so one possibility is that today's methods will discover broader structures as we give them more computation and more varied problems. Another is that a different organizing structure is needed, one our current methods do not reliably find. Defining $M^*(C,t)$ tells us what we want to achieve; finding the knowledge that approaches it is the challenge. I see this as a central problem to work on now: how to organize the world of ideas so that the greatest range of understanding can be carried by the smallest usable body of knowledge.

As that range grows, would its best representation resemble the concepts humans use? Or would a machine organize ideas in a way that makes connections apparent which we cannot yet see? Such an intelligence could reveal new questions through the same structure that helps it answer old ones. Its understanding could also give people a shorter path to the frontier, leaving more of a human life available to explore it.

We are just learning to build intelligence beyond our own. I can't help but wonder how far intelligence itself can go.