[Nature Physics 2026] The "NAND Gate" of Continuous Math: All Elementary Functions from a Single Operator

TL;DR

In a groundbreaking shift for scientific computing, researcher Andrzej Odrzywolek has discovered a single binary operator—eml(x, y) = exp(x) - ln(y)—that, paired with the constant 1, can generate every function on a scientific calculator. From basic addition to complex trigonometry ($\pi$, $i$, $\sin$, $\cos$), everything is now a binary tree of identical "EML" nodes.

Positioning: This is a fundamental "unification" work, moving mathematics from a collection of disparate tools to a uniform, circuit-like architecture.

Problem & Motivation: The Missing Primitive

In digital electronics, we have the NAND gate. It is the universal building block; give an engineer enough NAND gates, and they can build a supercomputer. Yet, in continuous mathematics, we have always been "stuck" with a messy toolbox. We use $+$, $-$, $imes$, $/$, $\sqrt{x}$, $\exp$, and $\log$ as if they were independent entities.

While we know some are redundant (e.g., $an(x)=\sin (x)/\cos (x)$), the mathematical community lacked a single primitive that could do it all. The author’s insight was to bridge this gap, asking: Is the diversity of mathematical functions intrinsic, or is there a hidden generative basis?

Methodology: Hunting for the Sheffer Stroke

To find the EML operator, the author used a process called Systematic Ablation.

1. Sieve of Transcendental Constants: Instead of slow symbolic proofs, the author used a "numerical sieve." If an EML-tree equals a target function (like $\ln x$) at a single point using irrational numbers (like $\gamma$), it is almost certainly the correct formula.
2. The EML Discovery: Through exhaustive search, the operator $eml(x,y)=e^x-\ln (y)$ emerged.

The "Phylogenetic" Tree of Math

The discovery follows a "LUCA" (Last Universal Common Ancestor) pattern, where the simplest expressions branch out into the entire ecosystem of mathematics.

The evolutionary path from EML and the constant 1 to the full repertoire of modern science.

Methodology: The Architecture of a Formula

In the EML world, every formula is a full binary tree. This leads to a trivial grammar: $S\to 1\mid eml(S,S)$.

• $e^x$ is simply $eml(x,1)$ because $e^x-\ln (1)=e^x$.
• $\ln (x)$ is more complex: $eml(1,eml(eml(1,x),1))$.

This uniformity allows us to treat math like analog circuits.

Visualizing log, identity, and multiplication as hardware-like circuits of identical EML units.

Experiments: Symbolic Regression via Gradient Descent

One of the most exciting applications is in AI and Symbolic Regression. Usually, discovering a formula from data requires searching through millions of combinations of $+$, $-$, $\sin$, and $\exp$.

With EML, you only have one architecture: a deep EML tree. By treating the connections as weights ($\alpha ,\beta ,\gamma$), the author used standard optimizers like Adam to "grow" the correct formula.

• Success: The weights "snapped" to 0 or 1, recovering exact symbolic laws from noisy data.
• Comparison: Unlike traditional Neural Networks, which are "black boxes," a successful EML tree is perfectly interpretable—it is the formula itself.

Critical Analysis & Conclusion

Takeaway

The EML operator proves that elementary functions are members of a much simpler class than we thought. We can now imagine a "Single-Instruction Computer" for math, or specialized AI hardware that only knows how to compute $e^x-\ln (y)$.

Limitations

• Complex Domain: To compute real roots or trigonometry, EML must use complex numbers internally (handling $\ln (-1)$).
• Vanishing Precision: At very high depths, the nested exponentials can lead to numerical overflows, requiring careful "clamping" in PyTorch/NumPy.

The discovery of EML is likely just the beginning. The paper hints at a ternary operator that requires no constants at all—the ultimate simplification of the mathematical universe.