
I was reading something and boom, this lands in my feed. A Polish theoretical physicist just proved you can recreate all of math's elementary functions from JUST one operation.

$$
E(a, b) = e^{a} - \ln(b)
$$

Every single operation you'd find on a scientific calculator: +, −, ×, ÷, √, trig, log, hyperbolics, inverses, even constants like e, π and i, as you can see below. Extremely mathematically elegant.

My first reaction was, ok cool, another reduction paper, those show up every few years, let me skim it. Then I hit the abstract properly and sat up. The claim wasn't "we reduced 36 operations to 6 primitives", which would be a nice optimisation. The claim was that a single two-input function plus the number 1 is enough to reconstruct every constant, every arithmetic operation, every trig function, every hyperbolic function, every inverse, every radical. Everything. The entire scientific calculator collapses into one button and a seed value.

That sounds like one of those "technically true but useless" results until you realize that continuous mathematics has been looking for exactly this for about a hundred years and nobody had found it.

## the comparison that makes this land

In 1913, Henry Sheffer published a paper showing that a single two-input logical operator, NAND, is enough to build every Boolean function. AND, OR, NOT, XOR, everything. Dump a pile of NAND gates on a table and you can literally construct an entire CPU out of them. This is not a trick, it's how digital hardware actually works. Modern chips are, at the lowest level, seas of identical gates wired into patterns that compute. That one fact is responsible for about half of what makes silicon cheap.

Continuous math never had this. A scientific calculator has thirty-plus keys, and the reason is that every new function the designers wanted to support needed its own button and its own internal routine. Logarithms reduced multiplication to addition back when Napier published in 1614. Euler's formula folded trig into complex exponentials in the 1700s. The exp-log representation formalised that further. And then it stopped. For the last couple of centuries the minimal basis for elementary functions has been "a few primitives", never one. It felt like the bottom of the well.

This paper pulls the bucket out of the well. A single binary operator, plus the constant 1, is sufficient. The gap between this result and everything that came before is roughly the gap between "we use the Sheffer stroke because it's convenient" and "wait, it actually reduces to one thing."

## the operator

Here it is in full. No tricks hidden in the corners.

$$
\mathrm{eml}(x, y) = \exp(x) - \ln(y)
$$

EML stands for Exp-Minus-Log. That's the whole definition.

The reason you need a constant paired with it is that ln(1) = 0, so anywhere you want to drop the log side of the operator, you plug in 1 on the right. That single trick lets you peel the operator apart into its components whenever you need them. For example:

$$
\begin{aligned}
e^{x} &= \mathrm{eml}(x, 1) \\
e &= \mathrm{eml}(1, 1) \\
\ln(z) &= \mathrm{eml}\bigl(1, \mathrm{eml}(\mathrm{eml}(1, z), 1)\bigr)
\end{aligned}
$$

Read that last one carefully, because it's the first time you see what a proper EML expression looks like. The natural logarithm of z takes three nested applications of the same operator and four instances of the constant 1. It's simple in the Kolmogorov sense but very nested in practice. Most things you'd expect to be short turn out to be long.

Before we get into how anyone found this in the first place, play with it.

<EMLTreeExplorer />

Try `ln z` with z = 2.718. The EML tree should spit back 1.0, because ln(e) = 1, and you can watch each sub-node compute its partial value. Then try `e^x` at x = 2 and confirm you get e² ≈ 7.389. The point of the interactive is that you can feel how an elementary function, which you normally think of as a named primitive, is actually just a specific arrangement of a single repeating element. The "function" stops being a label and starts being a tree shape.

## how on earth do you find an operator like this

This is the part that broke my brain a little. You can't just stare at exp and ln and derive their difference by inspection. The paper is very honest about this. It was found by systematic ablation, starting from a list of 36 primitives that together form a standard scientific calculator, which is Table 1 in the paper: 8 constants, 20 unary functions, and 8 binary operations. Elements get iteratively removed from that list, and each remaining set is checked to see if it can still reconstruct the original 36.

The verification itself is where the clever bit lives. Direct symbolic verification is computationally intractable. Kolmogorov-style complexity for these formulas tends to sit around 7 to 9 in Reverse Polish Notation length, which sounds small until you realise the search space at depth 9 is already brutal. So instead of symbolic manipulation, the paper uses a hybrid numeric bootstrap. Free variables x and y get substituted with algebraically independent transcendental constants, specifically the Euler-Mascheroni constant γ ≈ 0.577216 and the Glaisher-Kinkelin constant A ≈ 1.28243, the target expression is evaluated numerically, and then compared against a massive sieve of candidate formulas built from the current operator set.

The reason this works is Schanuel's conjecture. Under Schanuel, a coincidental equality between two algebraically independent transcendentals evaluated at the same point is vanishingly unlikely. So if a candidate formula gives you a numeric match to double-precision on γ and A, it's almost certainly an actual equivalence and not a fluke. This turns formula discovery into a high-precision lookup problem. The double-precision result goes through inverse symbolic calculator software, which returns a candidate closed form, and then you verify it symbolically.

That entire pipeline was originally implemented in Mathematica as a package called VerifyBaseSet. Then GPT Codex 5.3 was used to translate it into Rust, which made it three orders of magnitude faster. Mathematica took hours, Rust takes seconds. That's what made it feasible to run the ablation exhaustively on every element of the 36-primitive list and track how the calculator shrinks.

## the reduction chain

Here is the actual shrink, in order. I'm keeping the paper's naming.

| Name | Constants | Unary | Binary | Total |
|---|---|---|---|---|
| Base 36 | 8 | 20 | 8 | 36 |
| Wolfram | π, e, i | ln | +, ×, ^ | 7 |
| Calc 3 | none | exp, ln, −x, 1/x | + | 6 |
| Calc 2 | none | exp, ln | − | 4 |
| Calc 1 | e or π | none | x^y, log_x y | 4 |
| Calc 0 | none | exp | log_x y | 3 |
| **EML** | **1** | **none** | **eml(x, y)** | **3** |

Each row is a complete scientific calculator on its own. Calc 3 sits near the Wolfram Mathematica primitive set, which has been the optimised minimalist basis for about forty years. Calc 2 is stricter: negation and reciprocal fall out, but you need subtraction as the non-commutative backbone, because without a non-commutative operation you lose the ability to invert. Calc 1 is a radically different attack, a top-down approach built from binary exponentiation and binary logarithm, which needs either e or π as its seed constant. Calc 0 absorbs e into the exp function itself, leaving three primitives total.

Calc 0 was the configuration that strongly suggested a single binary operator could exist. Once you have a system with three primitives and one of them is a unary function, it's natural to ask whether you can absorb the unary into the binary. A month of searching later, eml fell out of the search.

EML is also not unique. It has at least two close cousins.

$$
\begin{aligned}
\mathrm{eml}(x, y) &= \exp(x) - \ln(y) && \text{with constant } 1 \\
\mathrm{edl}(x, y) &= \exp(x) / \ln(y) && \text{with constant } e \\
-\mathrm{eml}(y, x) &= \ln(x) - \exp(y) && \text{with constant } -\infty
\end{aligned}
$$

EDL swaps subtraction for division and trades the constant 1 for e. The minus-eml variant flips operand order and requires the terminal symbol to be −∞, which is weird but valid in extended reals.

The paper then speculates, with empty table rows, that a two-primitive or five-primitive system might exist waiting to be discovered. A ternary operator T(x, y, z) = exp(x) · ln(z) / (ln(x) · exp(y)) already has the property that T(x, x, x) = 1, which would let it act as its own constant. That's a separate paper in preparation for Acta Physica Polonica B.

## the complexity tax

This is the part you can't ignore. Reducing to one operator is elegant, but the cost is that simple things become giant. The paper includes a beautiful table showing how many RPN instructions each primitive needs when compiled to pure EML form.

| Symbol | EML compiler | Direct search |
|---|---|---|
| e | 3 | 3 |
| 1 | 1 | 1 |
| 0 | 7 | 7 |
| −1 | 17 | 15 |
| 2 | 27 | 19 |
| √2 | 165 | >47 |
| i | 131 | >55 |
| **π** | **193** | **>53** |
| e^x | 3 | 3 |
| ln x | 7 | 7 |
| x × y | 41 | 17 |
| x^y | 49 | 25 |
| √x | 139 | 43 |

Computing π in pure EML needs 193 instructions. Not 193 nested evaluations, 193 nodes in the expression tree. That is absurd by any reasonable standard. But it's also deeply interesting, because it tells you something about how much of the "complexity" of elementary functions we have been hiding inside named primitives. We built our scientific calculator to look simple by giving individual names to the expensive things. Strip the names away and the real cost becomes visible. Digital hardware is fine with this, incidentally: a modern CPU has billions of transistors, so 193 of anything is a rounding error.

The uniform structure is what the paper is really selling. Because every EML expression is a binary tree of identical nodes, you get a trivial context-free grammar.

```
S → 1 | eml(S, S)
```

That's the whole language. And that grammar is isomorphic to full binary trees, which means EML expressions are in one-to-one correspondence with Catalan structures. You get a clean combinatorial object to search over. You can enumerate EML trees by leaf count and exhaustively check every small tree. You can compile any elementary function into one. You can even build an analog circuit where each identical component is a physical eml gate, with the same status as a NAND gate in digital hardware or an op-amp in analog electronics. The paper actually draws the schematic symbol.

## the part that surprised me

Then the paper does something I wasn't expecting. It treats the EML tree as a differentiable neural network and tries to recover closed-form formulas from numerical data using gradient descent.

The idea is simple once you see it. Build a full binary EML tree of depth n. At every leaf, instead of hardcoding a 1 or a variable, put a soft mixture of the three possible values weighted by trainable parameters.

```
leafᵢ = αᵢ + βᵢ · x + γᵢ · f
```

The α, β, γ at each leaf are logits pushed through a softmax, which means the tree is parameterised by roughly 5·2ⁿ − 6 continuous weights. A level-2 tree has 14 parameters, a level-3 tree has 34 parameters which simplex reparameterisation reduces to 20. Standard Adam optimiser, PyTorch, complex128 to handle the internal complex arithmetic, clamping to deal with exp overflow, and then a hardening phase that snaps each softmax output to a vertex of the simplex, meaning each leaf commits to being either a 1 or an x or the previous sub-expression.

The result is that for simple target functions, the network recovers the exact symbolic formula. Fit ln(x) using a depth-3 tree, reduce 34 free parameters to 20 via simplex reparameterisation, run plain NMinimize on a black-box basis, and at the end the snapped weights give exactly ln(x). Not approximately. The mean squared error at convergence is on the order of 10^−32, which is machine epsilon squared. When you round the softmax weights to the nearest vertex, you get the actual closed-form formula, and it generalises perfectly outside the training range.

The blind-recovery numbers across 1000+ runs are also honest:

- Depth 2: 100% blind recovery from random initialisation
- Depth 3 to 4: about 25% blind recovery
- Depth 5: below 1% blind recovery
- Depth 6: 0 out of 448 attempts

So it's not magic. Finding the correct EML tree from scratch gets harder fast. But there's an important follow-up experiment. When you take a correct EML tree, perturb its weights with Gaussian noise, and re-optimise, the optimiser converges back to the correct solution in 100% of runs, even at depth 5 and 6. Which means the basins of attraction around correct EML formulas are valid and wide. The hard part is getting close enough to one of them in the first place.

This is the piece that feels different from the rest of the paper. It's saying that because EML expressions are a uniform, differentiable, complete family, you can do symbolic regression by gradient descent on a fixed architecture. No heterogeneous grammars, no handcrafted operator sets, no tree-search bookkeeping. Just an EML tree, a softmax over leaves, and an optimiser. When the generating law is elementary, the weights snap to exact symbolic values. The paper calls this a form of interpretability that conventional neural architectures cannot provide: if your training succeeds, the network is literally a closed-form formula.

## the deep technical dive

If you're the kind of person who wants receipts, here they are.

**Three lines of Mathematica reproduce the discovery:**

```
Import["SymbolicRegression.m"]
EML[x_, y_] := Exp[x] - Log[y]
VerifyBaseSet[{1}, {}, {EML}]
```

That's it. Run this and the package re-generates all 36 elementary operations from Table 1 of the paper within about an hour, depending on your machine. The Rust re-implementation, rust_verify, does it in seconds and also handles arbitrary-precision checks across multiple real transcendentals instead of just two.

**The level-2 master formula:**

```
F(x) = eml(
  α₁ + β₁·x + γ₁·eml(α₃ + β₃·x, α₄ + β₄·x),
  α₂ + β₂·x + γ₂·eml(α₅ + β₅·x, α₆ + β₆·x)
)
```

Fourteen parameters in total. Set α₁ = 0, β₁ = 1, γ₁ = 0 on the left child and α₂ = 1, β₂ = γ₂ = 0 on the right, and you recover exp(x). Set α₁ = α₂ = 1 with everything else zero, you recover e. Set α₁ = β₁ = 0, γ₁ = 1, α₂ = 1, β₂ = γ₂ = 0, α₃ = 0, β₃ = 1, γ₃ = 0, α₄ = 1, γ₄ = 0, you recover exp(exp(x)). The point is that every reachable function at depth ≤ 2 is a specific vertex of a 14-dimensional simplex. Training the EML tree is equivalent to searching that simplex with gradient descent.

**Internals are complex.** EML operates over ℂ using the principal branch. You need complex arithmetic internally to generate things like i and π, because π shows up as −1 × i × ln(−1), which means you have to evaluate ln(−1) = iπ. So every EML evaluator, even for real outputs, runs on complex128 under the hood. Real-valued code using `<math.h>` mostly works but has a subtle i-sign flip on ln(z) for z < 0 that you have to fix with a principal-branch redefinition. The paper also points out that Python and Julia traps on signed zeros, NumPy and PyTorch handle it with overflow signals, and Lean 4's total function convention makes ln(0) return a junk value that breaks formalisation. These are edge cases, not showstoppers, but they exist.

**Reproducibility.** Code lives at github.com/VA00/SymbolicRegressionPackage, archival snapshot at Zenodo DOI 10.5281/zenodo.19183008.

**Caveats the paper is careful about.** The existence of an EML-type operator working purely on the real domain seems impossible. Attempts using pairs of trigonometric or hyperbolic functions found nothing. Complex intermediates are not a bug, they appear to be necessary. Also, the identity function x takes leaf count 9 in its shortest non-trivial EML form (though the compiler can return it directly as leaf count 1). The shortest non-trivial EML expression of anything is, somewhat hilariously, longer than you'd guess.

**Still open.** Whether a truly univariate Sheffer-like function exists, one where even the constant falls out of the operator itself, is open. The ternary operator T(x, y, z) = exp(x) · ln(z) / (ln(x) · exp(y)) with T(x, x, x) = 1 is the current best candidate and is the subject of a follow-up paper. Proving general impossibility is non-trivial because of weird examples like B(x, y) = x − y/2 where B(x, x) = x/2 but B(B(x, x), x) = 0, which the paper uses to motivate why you can't hand-wave the proof.

## what this actually changes

You could argue that EML is just a curiosity. It doesn't let you compute anything new, it just re-shuffles what we already compute. Fair. But I think the framing matters more than people might initially notice.

First, elementary functions now have a canonical normal form. Every closed-form expression you can write with sin, cos, exp, ln, and friends has a unique compiled EML tree, which is a unique element of a context-free grammar. That gives you a clean object to hash, search, and deduplicate. Symbolic regression systems have wanted this kind of uniform structure for years.

Second, this is a genuine interpretability handle for machine learning. The paper quietly makes the point that every standard neural network is, in principle, a special case of an EML tree architecture, because ReLU and similar activations are themselves elementary functions. That means an EML-based symbolic-regression network isn't a toy replacement for a transformer, it's a version of a transformer where the weights, when training succeeds, are exact formulas. The circuit is legible as a closed-form expression. That is not something you can currently say about, say, any large language model, and it's a form of interpretability that no conventional deep learning architecture provides.

Third, the analog-computing angle is intriguing. If you can fabricate a physical eml gate the way we fabricate NAND gates, you get a completely uniform substrate for evaluating elementary functions. No lookup tables, no special circuits for different functions, just a sea of identical elements wired differently. Whether anyone actually builds this is a different question, but the paper provides the theoretical ground floor.

## my honest take

I didn't expect a 2026 paper to close a gap that has been open since Sheffer. I especially didn't expect the proof to come out of a systematic ablation search rather than a deep structural argument. The vibe of the paper is "nobody actually tried this exhaustively and the search space was tractable with a Rust rewrite", and I really like that. It's a reminder that foundational discoveries sometimes happen because someone with the right tools decides to actually run the exhaustive search.

The complexity tax is real. No one is going to compute π as a 193-node EML tree in production. But production was never the point. The point is that elementary functions, for a hundred years treated as an irreducible handful of things, are actually members of a single connected family under one generator. That's a structural fact about mathematics, not a performance optimisation. The paper is clean, the code is reproducible, and the harder variants are left on the table for the next person to chase.

The conclusion ends with, in my paraphrase, "the EML operator may be the tip of an iceberg". I believe it. If you want to read the original and check my numbers, it's on arXiv at 2603.21852v2.

~ Ashish Kumar Verma 🫡


A new arXiv paper shows that a single binary operator, eml(x, y) = exp(x) - ln(y), plus the constant 1, can generate every elementary function. It's the continuous-math analogue of NAND, and it took 112 years to find.

You Can Recreate All Math Functions From JUST One Operation


Give two smart people the same dataset, the same model, the same code. One of them looks at the output and says, "this looks great, ship it." The other looks at the exact same output and says, "hm. What happens in the tails?" I've watched this happen over and over, at university studying mathematics, in finance, now working as a quant developer. And the thing that surprised me is that both people are often equally intelligent. They both understand the math. They can both build the system themselves. But they interpret the same information in completely different ways.

The difference is not intelligence. It is not coding ability. It is definitely not who memorized more formulas. The difference is how they think about the problem.

## you're thinking in dots when you should be thinking in shapes

Ask someone "what's a good salary for a software engineer in Bangalore?" and most people will give you a number. 18 LPA. 25 LPA. 40 LPA. A single dot on a number line. That number feels precise, feels like an answer. But it is not an answer. It is a summary that has thrown away almost everything interesting about the question.

A salary distribution is not a dot. It is a shape. The shape tells you that the median is around 15 LPA but there is a fat right tail where a small number of people earn 80 or 100 LPA, dragging the "average" way above what the typical person actually takes home. It tells you that the 10th percentile earns 6 LPA while the 90th percentile earns 45 LPA. Same job title, wildly different lives. When someone says "the average salary is 25 LPA", they are technically correct, but in a right-skewed distribution, the average is a number that almost nobody actually earns. Most people earn less. A few people earn much more. The average sits somewhere in between, belonging to no one.

This is the first shift that quantitative thinking produces. Single numbers start to feel suspicious. Not wrong exactly, just incomplete, like being told the average depth of a river is 3 feet and deciding it is safe to walk across. Technically accurate. Potentially fatal.

<DistributionExplorer />

Try it. Drag the skew slider to the right. Watch the orange line (the mean) drift away from the green line (the median). Same dataset. Same "average." But the typical outcome and the reported average are now telling completely different stories. This happens everywhere once you start looking. An algorithm with good average-case performance that catastrophically fails on rare inputs. A machine learning model that is 95% accurate overall but dangerous on the specific edge cases that matter. A financial strategy that delivers steady small gains until the one event in the tail distribution wipes it all out.

Two systems can have identical expected values and behave in completely different ways. The average hides the shape. The shape is where the actual information lives. Once you develop this instinct, you never look at a single number the same way again. You always want to know what the rest of the distribution looks like.

## the habit of attacking your own best idea

Once you start thinking in distributions, something uncomfortable happens. You realize your confident opinions are just point estimates wearing a suit.

Say you are deciding whether to leave your stable job and join a promising early-stage startup. You think it through. The founders are smart. The market is growing. You will get meaningful equity. The upside is enormous. Each of those beliefs feels solid. 85% sure about the team. 90% sure about the market. 75% sure the equity means something. 60% sure they survive three years. Each one on its own feels like a strong bet.

But here is what quantitative thinking does to that confidence. It multiplies. If those are genuinely independent assumptions (they are not, but let's start simple), the probability that ALL of them hold simultaneously is 0.85 times 0.90 times 0.75 times 0.60. That is about 34%. Your "pretty confident" position is, in joint probability terms, slightly worse than a coin flip.

This is why quants develop what looks from the outside like pessimism but is actually something different. It is the habit of attacking your own best idea. Not as a performance. Not as devil's advocate at a meeting. Genuinely trying to destroy the argument you just built. The instinct goes like this: you form a view, it feels right, you have good reasons. And then, instead of feeling satisfied, you spend equal effort constructing the strongest possible case against it. You look for the assumption that, if wrong, would make the whole thing collapse. You find the condition that your conclusion is implicitly depending on and ask what happens when that condition changes.

Most people think confidence means certainty. The quant version of confidence is different. It means, "I tried hard to break this and couldn't." That is a fundamentally different kind of confidence, one that is earned through attempted destruction, not through the absence of doubt.

<AssumptionStressTest />

Click through each assumption. Read the counter-argument. Adjust your confidence. Watch the bottom bar. The gap between the blue bar and the orange bar is not pessimism. It is the information that was always there but hidden by the comfortable feeling of each individual assumption being "probably right." The joint probability is the honest answer. The individual confidences are the story you tell yourself.

Ideas that survive this kind of scrutiny are much stronger. By the time they are trusted, they have already been tested against the scenarios that could break them. And the goal is always the same: discover the weaknesses before reality does it for you.

## the feedback loop nobody talks about

There is a popular framing that puts intuition and formal reasoning on opposite sides. Trust your gut versus trust the data. Instinct versus analysis. Fast versus slow. That framing is wrong.

In practice, the people who are best at quantitative reasoning use both, constantly, in a tight loop. It works like this: intuition moves first. You look at a dataset, a model output, a system's behavior, and something feels off. You cannot articulate exactly what. It is just a sense that the result is too clean, or the convergence was too fast, or the distribution looks wrong in a way you cannot immediately name. That feeling is not noise. It is pattern recognition built from hundreds of similar situations, years of seeing systems behave in certain ways. But intuition alone is not enough because it cannot be audited. You cannot explain "it felt wrong" to a colleague. And sometimes it is just bias wearing a lab coat.

So formal reasoning takes over. You check the assumption your gut flagged. You compute the statistical power of the test. You look at the residual distribution. You stress test the edge cases. Sometimes your intuition was exactly right. The result was too clean because there was a data leak. The convergence was too fast because the optimizer was stuck in a local minimum. Your gut caught something your code missed.

Other times your intuition was right for the wrong reasons. The test was underpowered but the effect was also real, just smaller than reported. Your instinct was firing on the variance problem, not the effect size problem. The formal analysis corrected the diagnosis while confirming the symptom. And occasionally your intuition was completely wrong. The thing that felt suspicious was actually fine. You just had not seen that particular pattern before.

What matters in all three cases is that the formal analysis updates the intuition. The next time you see something similar, your pattern recognition is slightly better calibrated. You flag fewer false alarms. You catch real problems faster. Intuition without formalism is superstition. Formalism without intuition is bureaucracy. The loop between them is where actual thinking lives.

Over time, this process builds a deeper understanding than either one could achieve alone. Instead of relying entirely on instinct or entirely on equations, quantitative thinking moves fluidly between the two. And it teaches you that the question is never "should I trust my gut or the data?" The question is "which one should go first right now, and how quickly can I hand it off to the other?"

## the math for the nerds who stayed

If you are still here, let's look at why these instincts have formal backing.

**Why distributions beat point estimates, an information theoretic argument**

A point estimate is a single number. A probability distribution is a function. The information content difference between them is measurable. The Shannon entropy of a continuous distribution with PDF p(x) is:

```
H(X) = -integral of p(x) * log(p(x)) dx
```

For a Gaussian with standard deviation sigma, this works out to:

```
H(X) = 0.5 * log(2 * pi * e * sigma^2)
```

A point estimate has zero entropy. It contains no uncertainty information at all. The distribution encodes exactly how much you know and how much you don't. When you collapse a distribution to a single number, you are literally discarding measurable information. The gap between the entropy of the distribution and zero (the entropy of the point estimate) is the information you threw away by saying "the average is 25 LPA" instead of describing the shape.

**Adversarial questioning as Bayesian updating**

The stress testing instinct has a clean formalism in Bayes' theorem:

```
P(H|E) = P(E|H) * P(H) / P(E)
```

Your initial confidence in an assumption is the prior P(H). The counter-argument is new evidence E. The likelihood P(E|H) measures how compatible the counter-argument is with your assumption being true. When you "stress test" an assumption, you are implicitly evaluating alternative likelihoods. If the counter-evidence is highly likely regardless of whether your assumption holds, it does not update your belief much. But if the counter-evidence is much more likely under the alternative hypothesis (your assumption is wrong), your posterior drops sharply.

The joint probability calculation in the interactive component above is a simplification (it assumes independence), but the core insight holds. Stacking multiple assumptions where each has even a moderate failure probability leads to joint probabilities that feel shockingly low. For n independent assumptions each with probability p:

```
P(all hold) = p^n
```

| Individual confidence | 3 assumptions | 5 assumptions |
|---|---|---|
| 90% | 72.9% | 59.0% |
| 80% | 51.2% | 32.8% |
| 70% | 34.3% | 16.8% |

Even at 90% confidence per assumption, five stacked assumptions leave you below 60%. This is why quants obsess over assumption counting.

**The intuition-formalism loop as approximate Bayesian inference**

There is a growing body of work in computational neuroscience (often called the "Bayesian brain" hypothesis) suggesting that human cognition approximates Bayesian inference. Your prior beliefs about how systems behave get updated by new observations, producing a posterior that becomes your new prior for the next encounter. The loop described in the previous section, intuition generates a hypothesis, formal analysis tests it, the result updates your intuition, is structurally identical to iterative Bayesian updating. Each cycle through the loop refines the posterior. Over time, the prior (your intuition) becomes increasingly well-calibrated to the domain.

This is why experienced quants seem to have almost supernatural instincts. They are not smarter. They have just run more update cycles.

## honest take

I don't think quant thinking makes you smarter. I think it makes you slower to be wrong. Which sounds like the same thing, but it isn't. Smart people are wrong all the time. They just sound confident while being wrong. The quant habit is less glamorous. You spend a lot of time saying "I don't know, but here's my distribution over the possibilities." Nobody has ever looked cool saying that.

But over time, the people who think in distributions instead of dots, who attack their own best ideas, who run the loop between intuition and formalism, they tend to be wrong less often in ways that matter. Not because they avoid mistakes entirely, but because they discover the fragile assumptions before those assumptions become expensive lessons. The world does not get less uncertain when you learn to think quantitatively. It just stops being as scary. Unknown outcomes become part of a structured problem instead of an unsolvable mystery. And numbers stop being answers and start being invitations to understand the system behind them.

That shift, from treating numbers as conclusions to treating them as clues, is the whole thing.


Two people can look at the same dataset and walk away with opposite conclusions. The difference is not intelligence. It is how they think about uncertainty, assumptions, and the structure behind the numbers.

The Quant Mindset Is Not About Math. It's About How You See Numbers.


Every time you type a message and Gmail suggests the next three words, that's a Markov chain. Every time Google decides which result to show you first, that's a Markov chain. Every time ChatGPT generates a sentence, the foundational idea underneath all of that transformer machinery is still, at its core, a Markov chain. And the whole thing exists because two Russian mathematicians got into a fight about free will in 1906. I'm not exaggerating. The algorithm that made modern AI possible was invented out of spite.

## the fight

In 1905, Russia was on fire. Socialist groups rose up against the Tsar, and the country split into two camps: Tsarists on one side, socialists on the other. This division was so deep that it crept into everything, including mathematics.

On the Tsar's side was Pavel Nekrasov, unofficially called the "Tsar of Probability." Nekrasov was deeply religious, politically powerful, and he had this idea that probability theory could be used to prove free will.

His argument went like this. For 200 years, the foundation of probability had been the **law of large numbers**: flip a coin 10 times and you might get 6 heads, but flip it 10,000 times and the ratio approaches 50:50. The average outcome converges to the expected value as you run more trials. But Jacob Bernoulli, who proved this in 1713, only proved it for **independent** events, meaning one coin flip doesn't affect the next.

Nekrasov looked at social statistics: marriage rates in Belgium, crime rates, birth rates. Year after year, these numbers converge to stable averages, following the law of large numbers. And since the law of large numbers requires independence (or so everyone thought), and these statistics come from human decisions, Nekrasov concluded that those decisions must be independent. Therefore, humans have free will. QED.

On the other side of this argument was Andrey Markov, also known as "Andrey the Furious." Atheist, with zero patience for nonsense, he publicly listed Nekrasov's work among the "abuses of mathematics." Markov didn't just think Nekrasov was wrong; he thought the entire logical chain was wrong. He believed the law of large numbers didn't require independence at all, and he was going to prove it.

## a poem and some counting

Markov needed a system where events were clearly dependent on each other, something where you couldn't pretend the outcomes were independent, so he chose text. Specifically, he chose *Eugene Onegin* by Alexander Pushkin, the poem at the heart of Russian literature. He took the first 20,000 letters, stripped out all punctuation and spaces, and pushed them into one long string.

Then he started counting. 43% of the letters were vowels, and 57% were consonants. Next, he broke the string into overlapping pairs and counted the four possible combinations.

| Pair | If independent (predicted) | Actual count |
|---|---|---|
| Vowel → Vowel | 18% | **6%** |
| Vowel → Consonant | 25% | **37%** |
| Consonant → Vowel | 25% | **37%** |
| Consonant → Consonant | 32% | **20%** |

The actual values were wildly different from what independence would predict, with vowel-vowel pairs three times rarer than expected. The letters were clearly dependent. This makes intuitive sense if you think about it: after a consonant like "str", a vowel almost always comes next, while after a vowel, another vowel is unlikely in Russian. The current letter constrains what the next one can be. So Markov had his dependent system, and now he needed to show it still followed the law of large numbers.

## the prediction machine

This is where it gets beautiful. Markov drew two circles, one for "vowel" (V) and one for "consonant" (C), which he called **states**. From each state, you can transition to either state, so he drew arrows, four arrows total.

Now he needed the transition probabilities. He knew 43% of all letters are vowels and that vowel-vowel pairs occur 6% of the time, so he calculated:

```
P(V → V) = 0.06 / 0.43 ≈ 0.13
P(V → C) = 1 - 0.13 = 0.87
```

And for consonants:

```
P(C → C) = 0.20 / 0.57 ≈ 0.67
P(C → V) = 1 - 0.67 = 0.33
```

That's it. Four numbers. And with just these four numbers, he built a **prediction machine**.

Start at a vowel and generate a random number. If it's below 0.13, stay at vowel; otherwise, move to consonant. From consonant, if the random number is below 0.67, stay at consonant; otherwise, move to vowel. Repeat this thousands of times and track the ratio of vowels to consonants.

At first the ratio jumps around wildly, but after enough steps, it converges to exactly 43% vowels and 57% consonants, the same split Markov had counted by hand from the poem. A dependent system, following the law of large numbers.

Try it yourself:

<MarkovChainExplorer />

Play with the transition probabilities: make them extreme, make them equal, make them asymmetric. The ratio always converges, the steady state always exists. The system is dependent, each step depends on the previous one, and the law of large numbers still holds.

Markov had built a chain of dependent events, a literal chain, and he proved mathematically that convergence happens anyway. He ended his paper with one final sentence aimed straight at Nekrasov:

> "Thus, free will is not necessary to do probability."

Andrey the Furious, indeed.

## what makes a Markov chain a Markov chain

Let me be precise about what Markov actually formalized, because this is the part that makes everything else work. A Markov chain has one defining property, the **memoryless property**, also called the **Markov property**:

```
P(next state | current state, all previous states) = P(next state | current state)
```

The future depends on the present, not the past. Only where you are right now matters, not how you got there.

Think of it like this: you're standing at a crossroads, and there are three paths forward. The probability that you take each path depends on which crossroads you're at, but it does not depend on which crossroads you came from, or the one before that, or the one before that. One state, one set of transition probabilities, no memory of history.

This sounds like a limitation, but it's actually a superpower. Because it means you can take enormously complex systems, systems with millions of possible states and long complicated histories, and ignore almost all of that complexity. Just look at where the system is right now and predict what happens next.

That simplification is what makes Markov chains computable, and computability is what makes them useful.

## the bomb

Markov himself didn't care about applications, writing: "I am concerned only with questions of pure analysis. I refer to the question of the applicability with indifference." But applicability came anyway, and it came with a bang. Literally.

In January 1946, a mathematician named Stanislaw Ulam was recovering from encephalitis. He nearly died and spent months in bed, and to pass the time, he played solitaire. After hundreds of games, a question nagged at him: what are the chances that a randomly shuffled game of solitaire can actually be won?

There are 52! possible arrangements of a deck, which is about 8 × 10⁶⁷ possible games. You can't check them all, and you can't solve this analytically, but what if you just play hundreds of random games and count the wins?

This flash of insight followed Ulam back to Los Alamos, where he was working on nuclear weapons. The problem there was understanding how neutrons behave inside a uranium core. A neutron hits a U-235 nucleus, and the nucleus splits, releasing energy and 2-3 more neutrons. If those new neutrons hit more nuclei, you get a chain reaction. But the physics is complicated: neutrons scatter, get absorbed, escape the core, or cause fission, all with probabilities that depend on position, energy, and velocity.

Trillions of neutrons, trillions of possible outcomes, analytically unsolvable. Ulam shared his solitaire insight with John von Neumann: what if we just simulate it? Generate random outcomes, over and over, and use the statistics to approximate the answer.

Von Neumann immediately saw the power, but he also spotted the problem. Solitaire games are independent, but neutrons aren't. A neutron's behavior depends on where it is and what it's done before. You need a Markov chain.

So they built one. A neutron starts traveling through the core, and at each step it can **scatter** off an atom and keep traveling (arrow back to itself), it can **escape** or get absorbed (chain ends), or it can hit U-235 and cause **fission**, releasing 2-3 new neutrons that start their own chains. The transition probabilities depend on the neutron's current state: its position, velocity, energy. Memoryless, because only the present matters.

Run this chain hundreds of times and track how many neutrons are produced per initial neutron. That gives you **k**, the multiplication factor. If k < 1, the reaction dies; if k = 1, it's sustained but stable; if k > 1, you get exponential growth. Bomb.

<MonteCarloNeutron />

Slide the uranium mass up and watch the histogram shift from blue (subcritical) to red (supercritical). That transition is the critical mass, the minimum amount of fissile material needed for a bomb.

Ulam named the method after his gambling uncle's favorite casino: Monte Carlo. The Monte Carlo method, born from solitaire, powered by Markov chains, used to build nuclear weapons. And today it's used in everything from financial modeling to protein folding to climate simulation.

Ulam later wrote: "It is still an unending source of surprise for me to see how a few scribbles on a blackboard could change the course of human affairs."

## the search engine

Fast forward to the 1990s. The internet explodes, thousands of pages appear daily, and nobody can find anything.

Early search engines like Yahoo and Lycos ranked pages by counting how many times your search term appeared on the page, using simple keyword matching. The problem was that it was trivially easy to game: hide keywords in white text on a white background, repeat them a thousand times, and boom, top ranking. There was no notion of quality, only relevance.

Two Stanford PhD students, Sergey Brin and Larry Page, realized that links between pages contained information about quality. If a page has many other pages linking to it, it's probably important, and if those linking pages are themselves important, the endorsement means even more. They modeled the entire web as a Markov chain.

Each webpage is a state, and each link is a transition. If Amy's page links only to Ben, there's a 100% chance of transitioning from Amy to Ben; if Ben links to three pages, each gets a 33% chance. Now imagine a "random surfer" starting on a random page and following links randomly. Over thousands of steps, the fraction of time spent on each page converges to a stable distribution, and pages where the surfer spends more time are ranked higher.

But there's a problem: some parts of the web aren't connected. The surfer can get stuck in a loop, bouncing between a small cluster of pages forever, never reaching the rest. So they added a damping factor: 85% of the time, follow a link; 15% of the time, jump to a completely random page. This ensures the chain is **ergodic**, meaning every state is reachable from every other state, and convergence is guaranteed.

<PageRankDemo />

Try both modes. Power iteration computes the exact mathematical scores, while the random surfer simulates what actually happens. They converge to the same ranking.

Notice how Ben consistently ranks highest, even though every node starts with equal score. That's because Ben has the most incoming links from pages that themselves have importance, which shows that quality links matter more than quantity.

They called the algorithm PageRank ("Page" for web pages, and also because Larry's last name was Page; he snuck that in). Initially they called the search engine BackRub, after the backlinks it analyzed, but then they decided they needed a name as big as their ambition. They thought of the largest number they knew: 10¹⁰⁰, a googol. When registering the domain, they misspelled it.

Google was born. Today, Alphabet (Google's parent company) is worth over $2 trillion, and at the heart of it is a Markov chain that only looks at the current state to predict what happens next.

## from letters to language models

Remember how Markov started all this by predicting vowels and consonants in a poem? In the 1940s, Claude Shannon, the father of information theory, took this idea further. Instead of vowels and consonants, what if you used individual letters as states, and instead of looking at just the last letter, what if you looked at the last two, or three?

With a first-order model (predict based on the previous letter), you get gibberish. With a second-order model, you start seeing recognizable fragments like "th", "the", "ing". With higher orders, the generated text starts looking almost sensible. Then Shannon asked: what if instead of letters, you use **words** as states?

The generated text got much better because sequences of three or four words often made sense. "Attack on an English writer" is grammatically valid. The predictions improved because each step carried more context, and Shannon had discovered something fundamental: **the more context you include, the better your predictions become.**

This is exactly what modern language models do. At the most abstract level, a large language model is answering the same question Markov asked in 1906:

> Given what came before, what comes next?

<TextMarkovPredictor />

That's a simple first-order word-level Markov chain, where each word is predicted based only on the previous word. It produces text that's sometimes grammatical but rarely meaningful.

Modern LLMs do the same thing, but at a completely different scale. Instead of looking at one previous word, they look at thousands of previous **tokens** (words, parts of words, punctuation). Instead of storing transition probabilities in a lookup table, they learn them as billions of neural network parameters. And the key innovation, **attention**, is essentially a mechanism for deciding which of those thousands of previous tokens matter most for the current prediction. In the phrase "the structure of the cell", attention lets the model look back at "blood" and "mitochondria" from earlier in the context to decide that "cell" means biology, not prison.

But strip away the transformers, the attention heads, the billions of parameters, the RLHF, the prompt engineering, and the fundamental question is still:

```
P(next token | all previous tokens)
```

It's Markov's question, extended, scaled up, made deep, but the same question.

## the memoryless property is the point

Weather systems have histories stretching back millions of years, text has context going back entire books, the web has billions of interconnected pages with link structures built over decades, and nuclear reactions involve trillions of particles with complex interaction histories. And for many of these systems, you can ignore almost all of that history. You can just look at the current state and make a meaningful prediction about what happens next.

That's not a hack or an approximation born from laziness; it's a deep structural property of many real-world systems. The memoryless property is what makes these systems computable, because without it, you'd need to track the entire history of every state to make a prediction. With it, you just need to know where you are right now.

As one paper put it: "Problem-solving is often a matter of cooking up an appropriate Markov chain." And it all started because one furious Russian mathematician wanted to prove another one wrong.

## the math for the nerds who stayed

Here's the formal machinery. **Definition.** A (discrete-time, discrete-state) Markov chain is a sequence of random variables X₀, X₁, X₂, ... satisfying:

```
P(Xₙ₊₁ = j | Xₙ = i, Xₙ₋₁ = iₙ₋₁, ..., X₀ = i₀) = P(Xₙ₊₁ = j | Xₙ = i) = Pᵢⱼ
```

The transition matrix **P** has entries Pᵢⱼ where each row sums to 1 (it's a stochastic matrix).

**Stationary distribution.** A probability vector π is stationary if:

```
π = πP
```

Meaning: if the system is in distribution π, one step later it's still in distribution π. The system has "settled."

For the Markov chain we built from Eugene Onegin:

```
P = | 0.13  0.87 |
    | 0.33  0.67 |

Solving πP = π with π₁ + π₂ = 1:

π₁ = 0.33 / (0.87 + 0.33) = 0.275  (hmm, not quite 0.43)
```

Wait. The stationary distribution of the simple two-state chain gives approximately 27.5% vowels, not 43%. That's because the transition matrix encodes conditional probabilities, and the actual letter frequencies in the poem reflect more complex structure than a two-state model captures. Markov's original analysis was more nuanced: he used the transition probabilities to prove the *theorem* (law of large numbers for dependent events), not to perfectly reconstruct the marginal distribution from a simplified model.

The key theorem is:

**Ergodic theorem for Markov chains.** If a Markov chain is irreducible (every state reachable from every other) and aperiodic (doesn't cycle with a fixed period), then:

1. A unique stationary distribution π exists
2. For any starting state, the distribution converges to π
3. The time-average fraction spent in state j converges to πⱼ

This is the generalized law of large numbers for dependent events, Markov's core result.

**PageRank formulation.** For a web graph with N pages, the transition matrix is:

```
M = d × A + (1 - d) × (1/N) × J
```

Where:
- A is the link-based transition matrix (Aᵢⱼ = 1/outlinks(i) if page i links to page j)
- d = 0.85 is the damping factor
- J is the N×N all-ones matrix
- The (1-d) term is the "random jump" that ensures ergodicity

The PageRank vector is the stationary distribution of M:

```
π = πM
```

**Monte Carlo estimation.** For any function f of the state, the Monte Carlo estimate after T steps is:

```
f̂ = (1/T) Σₜ f(Xₜ)
```

By the ergodic theorem, f̂ → E_π[f] as T → ∞. The estimation error decreases as O(1/√T), which means doubling accuracy requires 4x the samples. This is dimension-independent, which is why Monte Carlo works well in high-dimensional spaces where grid-based methods fail exponentially.

**Connection to LLMs.** A language model defines:

```
P(xₜ | x₁, x₂, ..., xₜ₋₁)
```

This is *not* Markov in the strict sense because it conditions on the entire history, not just the current state. But the autoregressive generation process (sample one token, append it, sample the next) is a Markov process *in the space of sequences*. The "state" is the entire sequence so far, and each generation step depends only on the current sequence, not on which random samples led to it. The Markov property holds at the level of the generation trajectory.

The chain is: generate token → update context → generate token → update context, and each step is memoryless with respect to the process of generation. The model doesn't remember which candidates it considered and rejected; only the sequence itself persists.

So even modern LLMs, in a deep structural sense, are running a Markov chain, just one with an astronomically large state space. Markov would probably be indifferent about this, since he was concerned only with pure analysis, after all.

But I think it's worth saying: a fight about free will in Tsarist Russia led to the mathematical foundation under Google, nuclear physics, Monte Carlo simulation, and every large language model on the planet. That's a pretty good chain of events.

~ Ashish Kumar Verma


A Russian mathematician picked a fight over free will, accidentally invented the most important idea in probability, and 120 years later it powers Google, nuclear physics, and every large language model you've ever used. Here's how Markov chains actually work.

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