01Models are states
Start with a model m and give it a state vector θm in an ability space with k dimensions, one per ability. We never get to observe θm directly, because all we ever see are outcomes, so in practice a model is a posterior distribution over that space, p(θm | traces). A physicist would call that an ensemble.
Early on the posterior is wide, and after many traces it narrows. The "error bars" in the essay are the width of this distribution. Two models with the same mean and different widths are different products, because the buyer is paying for confidence and not for a point estimate.
02Tasks are observables
A task t is an operator on states: it takes a state and returns an outcome probability. The simplest useful form comes from item response theory, which is the mathematics behind every standardized test since the 1980s:
σ is the logistic function. bt is the task's difficulty. at is its discrimination vector, which says which abilities the task loads on and how hard.
Elo and Bradley-Terry are the one-dimensional special case of this, fit from pairwise comparisons instead of pass/fail. That is why LMArena works at all, and it is also why a leaderboard like that cannot tell you anything about an ability its prompts never exercise.
Everything on the right-hand side gets fit jointly from pass/fail data. Maximum likelihood is fine at scale, and a Bayesian fit gives you the posteriors from section 1 for free. The literature calls this multidimensional item response theory, and it comes with thirty years of identifiability results, fitting software, and known failure modes, which is more than most AI benchmarks can say for themselves.
03Runs are measurements
A trace is a tuple: model, task, context, outcome, verification method. Each one is a Bernoulli observation, a single coin flip with an unknown bias, and it updates two posteriors at once: the model's state θm and the task's parameters (at, bt).
For routing, the exchange needs the predictive distribution rather than a point estimate, and the predictive integrates over the posterior:
Calibration gets checked with proper scoring rules on held-out traces. A model whose predicted pass rate is 0.9 should pass about nine times in ten, and if it does not, the posterior is wrong rather than the model.
Verification is a noisy label. The verifier has its own confusion matrix, P(observed pass | true pass), and it enters the likelihood like any other measurement error. This is the formal reason the essay calls verification the central technical problem: the precision of every number on this page is bounded by the precision of the verifier.
04The axes are a gauge choice
This is the part the essay skips on purpose. If you rotate the ability space by any orthogonal matrix R, replacing every θ by Rθ and every at by Rat, then every dot product comes out unchanged, every predicted probability comes out unchanged, and the data cannot tell the two coordinate systems apart.
There are no true axes of intelligence. There are only invariants, and the invariants are predicted outcomes.
Physicists call this a gauge freedom. The coordinates are a convention, and what is physical is whatever survives a change of coordinates.
So when the essay says "the axes themselves are learnable", the precise statement is this: choose a rotation that makes the discrimination vectors sparse and readable, the way factor analysis does. The human labels, coding, math, planning, are one such choice, and a rotation the data prefers is another. Neither of them is the true one, because there is no true one, and the only question is which is more useful.
This is also why a single leaderboard score is a category error. It projects the state onto one direction and throws the rest away, and the direction it picks is whichever one the benchmark author's prompts happened to span.
05The unit: a bit of work against a candle
A kilowatt-hour is a unit of work. It is additive, it is defined by what was delivered rather than by the fuel, and it is pinned to a reference. The intelligence unit needs the same three properties, and it can have them.
The work is against uncertainty: before a run the outcome is uncertain, and after a verified pass it is not. The work done is the uncertainty removed, and the unit of uncertainty is the bit.
Why a candle at all? Because difficulty is not a property of a task on its own. "Hard" always means hard for someone, so the unit needs a someone, and it needs the same someone for everybody. In astronomy that someone is a star whose true brightness you know, and here it is a model whose ability you have measured to death: one specific, frozen, openly downloadable model that anyone can run in a sandbox, so that anyone can reproduce any weight the exchange publishes.
For instance, say the candle is a particular seven-billion-parameter open model from 2026, weights frozen the day it was chosen. Take the task "rename this function across the repository and keep the tests green", run the candle on it a thousand times, and suppose it passes 125 of them. That is one in eight, so the task weighs three bits, and it weighs three bits for every supplier, forever, or until the candle is replaced by the protocol in section 7. A frontier model that passes the same task 95 times in 100 delivers about 2.85 bits per attempt, and if it charges twenty cents an attempt, that is seven cents a bit. A task the candle passes every time weighs nothing, which is the unit's way of saying that the floor is free.
With the example in hand, here is the definition. If we fix a reference model r, open weights, public, frozen, and call it the candle, then we can define the weight of a task as the surprisal of the candle passing it:
IU(job) = Σt ∈ verified passes wt
Expected IU of one attempt by model m is wt · P(pass | θm, t), and price per IU is price per attempt divided by that.
| Candle passes | Weight | Reading |
|---|---|---|
| always | 0 bits | No uncertainty, so no work. The floor is free and the task routes to the cheapest thing, which is what merit order says should happen. |
| half the time | 1 bit | A coin flip for the floor. Routine cognition. |
| once in 8 | 3 bits | A model that passes this at 95% delivers 2.85 expected bits per attempt. |
| once in 1,024 | 10 bits | Frontier territory for the floor, and where escalation pays. |
| never observed | undefined | The candle has to be run on a sample of every task class. See section 10. |
If we check it against the essay's ten properties, it holds up. Bits add across independent tasks. The weight depends on the candle and the task and never on the supplier. Only verified passes count. Expected bits fall straight out of the posterior. The weight is measured by the exchange on the candle rather than declared by the seller, so it is hard to game. And price per bit is a number you can put on an invoice.
06Shannon, compression, and Maxwell's demon
The weight in section 5 is not an invention. It is Shannon's surprisal, the information content of an event, and averaged over a task distribution it becomes a cross-entropy. So the unit was information-theoretic from the start, and this section says what that buys us.
Start with the compression reading. A model that assigns probability P to the correct answer can encode that answer in −log2 P bits, and a model that knows more assigns more probability and needs fewer bits. So the natural measure of how much more model m knows than the candle r on task t is the log-likelihood ratio of the correct answer y*:
Its expectation over tasks is the number of bits m saves relative to the candle. With pass/fail verification, Pm collapses to 1 on a verified pass and the advantage collapses to wt, the weight in section 5.
This is the same quantity that language-model pretraining minimizes, cross-entropy per token, so the candle unit is the pretraining loss re-measured on tasks that people pay for. The lineage is old: Shannon in 1948, Solomonoff's induction in the 1960s, the Hutter Prize's bet that compression is intelligence, and Legg and Hutter's universal intelligence measure, which is a weighted sum over environments of exactly this kind of advantage. Knowledge work is selecting one answer out of many, and selecting is compressing.
Now the physics, because the kilowatt-hour analogy turns out to be literal. Szilard showed in 1929 that one bit of information about a gas molecule can be converted into kT ln 2 of mechanical work, and Landauer showed in 1961 that erasing one bit must dissipate at least that much heat. Bennett used the two together in 1982 to finally exorcise Maxwell's demon: the demon can sort molecules with information, but it has to pay Landauer's price to forget. So a bit of intelligence work and a joule of electrical work sit on the same ledger, with a fixed exchange rate of about 3×10−21 joules per bit at room temperature. The rate is absurd, but it exists.
Two cautions keep this honest. The first is Jaynes's point from 1957 that entropy is a property of a description, not of a thing. The bits we count are relative to the candle, not the entropy of the world. Bringing order to a codebase is real work, but what the unit measures is how much better than the reference you described the right answer, and the candle is the description.
The second is that nothing here threatens the second law. The agent brings order to the task and exports entropy as heat, and the kilowatt-hours pay for the bits. A human brain does its knowledge work on roughly twenty watts, while the accelerators that generate intelligence today draw kilowatts per rack and turn nearly all of it into heat, which makes them, per bit of verified work, some of the most wasteful entropy producers ever built. Szilard's exchange rate says how cheap a bit could be in principle. The datacenter bill says how far we are from it.
07Metrology: the candle will drift
The reference model will fall behind. Tasks the candle never passed will become routine for everyone else, and weights measured against it will inflate. Two precedents say this is a solved problem.
The kilogram was a platinum-iridium cylinder in a vault at Sèvres from 1889 to 2019, and every balance on Earth was calibrated against copies of it. When the instruments got good enough, the SI redefined the kilogram through a fixed value of Planck's constant. The unit stayed continuous and the artifact retired.
Astronomy did it the other way round, and gave us the word. Henrietta Leavitt's 1912 period-luminosity relation turned Cepheid variables into standard candles. Each better rung on the distance ladder recalibrates the ones below it, and old measurements get re-expressed rather than thrown away.
The protocol for the IU is the same:
- Candle v1 is frozen and published, and every weight cites it.
- When v1 saturates, candle v2 is run on the same task bank as v1, and the exchange publishes the conversion curve between v1 bits and v2 bits per task class.
- Historical prices and traces are re-expressed in v2 bits, and contracts name the candle version they settle in.
- v1 stays available forever, so any claim can be re-measured.
A candle has to be open weights rather than an API, because an artifact you cannot run yourself is a vendor and not a reference.
08Delivery is a filter on the market
Latency, price, energy, data residency, isolation, hardware, and availability are observed per supplier and per deployment. They are observed directly rather than inferred, they are not abilities, and so they stay out of θ.
What they do is define the admissible set S(t), the suppliers allowed to bid on task t. A task that requires EU residency and an air gap removes most of the market before any ability is consulted.
Keeping them out of the state space keeps the state space about one thing, and it matches how electricity markets already work. Merit order ranks generators on cost, and transmission constraints decide which generators can reach the load at all.
09Matching, in one line
Every symbol is now defined. S(t) is the delivery filter, the probability is the predictive from section 3, and τt is the buyer's required confidence. The price can be quoted per attempt or per expected bit.
Escalation is the same rule applied after a verified failure: update the posterior, remove the failed supplier, and solve again. Parallel execution is the same rule over subsets, where you choose the cheapest set whose joint pass probability clears τt. Typed judges enter as cheap verifiers with their own confusion matrix, and the "accept when confident, escalate when unsure" rule is a threshold on their output probability.
10Where it creaks
- Surprisal is not value. A ten-bit task can be worthless and a one-bit task can be worth a company. The kWh has the same property and nobody minds, because price per unit is set by the market and not by the unit. It is worth saying anyway, because otherwise someone will say it for you.
- Cold start. Item response theory assumes a fixed bank of items. Here tasks are generated on the fly, so a new task needs
(at, bt)predicted from its features before any run exists. That prediction is itself an inference problem, and it is the first friction in the essay's section 10. - Independence. The model assumes outcomes are conditionally independent given
θ. Agent tasks share context, memory, and tool state, so failures are correlated, and correlated failures make posteriors overconfident unless you model them. - The candle must see everything. A task class the candle has never run has no weight. The exchange has to spend candle runs on a sample of every class, forever, and that is a real operating cost of the unit.
- Gaming. Once bits move revenue, suppliers will train on the task bank. Hidden evaluations, downstream outcomes, and rotating task samples are the defenses, and none of them is free.
- Verifier noise. Every bit is a verified bit. A verifier with a 10% false-pass rate caps the precision of the whole ledger. Cheap typed judges lower the cost of verification and raise this noise at the same time, and the accept-or-escalate threshold is where that trade gets made.
11Why not phase space
Phase space is the obvious physics analogy, and I tried it first, and it does not fit. Hamiltonian phase space comes with dynamics and a conserved volume. A model's position in ability space has no equation of motion that we know of, and nothing is conserved when a lab ships a new checkpoint.
The quantum frame fits because it is a frame about knowledge: a state you cannot observe directly, observables that give probabilistic outcomes, measurements that update what you know, and coordinates that are a convention. Those are exactly the four things an exchange has to reason about, so take the frame and leave the physics where it is.