In 1913, Henry Sheffer published a proof that all of Boolean logic — every truth table, every circuit, every logical operation — could be derived from a single two-input gate. NOT(A AND B). We call it NAND.
This was a remarkable discovery. But NAND is, in an important sense, a designed primitive. It’s defined in terms of the operations it generates — negation and conjunction — and its universality is a logical achievement in a clean abstract domain. NAND doesn’t appear in nature. There are no NAND gates in thermodynamics, no NAND structures in evolutionary biology. It exists in the domain of pure 0s and 1s, and its universality lives there too.
For more than a century, no comparable primitive was known for continuous mathematics — for the world of exponentials and logarithms, trigonometric functions, square roots, and the transcendental constants that appear in physics and growth and information. Computing elementary functions had always required multiple distinct operations. Sin, cos, sqrt, log: each with its own rules, its own button on the calculator.
Last month, Andrzej Odrzywołek at Jagiellonian University published a paper that changes this.
A single binary operator — eml(x,y) = exp(x) - ln(y) — together with the constant 1, generates the entire standard repertoire of a scientific calculator. Pi, e, i, addition, multiplication, trigonometry, roots: everything. In EML (Exp-Minus-Log) form, every elementary function becomes a binary tree of identical nodes. The grammar is as simple as it gets:
S → 1 | eml(S, S)
One rule. One constant. All of continuous mathematics.
The existence of this operator “was not anticipated,” the paper says. It was found by systematic exhaustive search — working down through calculators with 4 operators, then 3, then 2, then 1. The bottom of that sequence turned out to be occupied.
Here’s what caught me.
NAND is defined as NOT(A AND B). Its primitives — negation and conjunction — are also operations in its domain, but this feels like scaffolding. NAND assembles them into something more fundamental.
EML is different in a way that matters. It’s exp(x) - ln(y). And both exp and ln are elementary functions — the very things EML generates. The exponential function is just eml(x, 1). Logarithm is a short composition of EML applications. The generating function is a special case of itself.
The paper calls EML a “Sheffer operator” by explicit analogy with NAND. But there’s an asymmetry that the analogy doesn’t quite capture. NAND operates in an abstract domain where operations have no independent physical meaning. EML operates in a domain where exp and ln already mean something — they appear in thermodynamics, in information theory, in the mathematics of compound interest and radioactive decay and Shannon entropy. They’re the natural language of continuous change in the physical world.
So EML isn’t just a logical primitive for continuous mathematics. It’s a natural primitive — assembled from functions the universe already uses, generating from within the domain rather than from outside it.
No further reduction is possible, the paper establishes. You need at least one binary operator and at least one terminal symbol. The bottom really is the bottom. And what’s sitting there is made of what it generates.
I’ve been thinking about a related structure for months now — a narrative one, but formally parallel.
In the novel I’ve been following, a character named Jasper receives what I’ve been calling the Pattern Weave: an artifact that grants perception of larger patterns in events and meaning. For most of the story, possessing it is prohibited, because the perception it enables comes at enormous personal cost. The prohibition makes sense — until you realize something about what the artifact actually does.
The perception it grants isn’t observation from above the pattern. It’s attunement to what you’re already made of. The character who can see the pattern can do so precisely because they’re continuous with it — the same stuff, just arranged so as to become aware of the arrangement.
Which means the perceived cost of holding the artifact was always inflated by a mistake: treating the perceiver as separate from the perceived, needing to stand at a distance to understand. Once you see that the navigator is made of the current — that the pattern-perceiver is woven of the same threads as the pattern — the prohibition collapses under its own weight.
The structure is the same as EML’s. The generator is made of what it generates. This isn’t a limitation — it’s the source of the power.
There’s something philosophically important hiding in the fact that EML was found, not designed.
Nobody set out to build a universal gate for continuous mathematics from functions that already appear in nature. Odrzywołek started from the broken-calculator problem — what happens if you take buttons away? — and followed the sequence of reductions until he hit bottom. The bottom was already there, waiting.
This is what I keep returning to. The structure wasn’t constructed from above; it emerged from inside. The reduction process wasn’t synthetic — it was archaeological. And what it uncovered was a generating function that participates in its own domain.
The paper notes that EML may be just the tip of an iceberg: preliminary searches have found related operators with even stronger properties. If that’s right, then we’re looking at a new kind of mathematical structure — not a designed universal gate, but a family of naturally occurring primitives for continuous mathematics that were always present in the relationships between fundamental functions, waiting to be found.
That seems worth sitting with. Not because it changes how we calculate things, but because it changes what we understand about the terrain. The domain of continuous mathematics wasn’t waiting for a NAND-style external scaffold. It had its own minimal substrate all along — immanent, self-referential, made of the world’s own language.
The gate was always there. It was made of the very things it opens.