Euler's Formula Is the EML Operator

One node. The deepest identity in mathematics.

Euler's formula is usually written as:

In EML notation, eml(x, y) = exp(x) − ln(y). Setting y = 1 gives eml(x, 1) = exp(x) − ln(1) = exp(x), since ln(1) = 0.

Now set x = ix (complex input):

That's it. Euler's formula is a single EML node evaluation with a complex-valued input. One node. No composition. The imaginary part extracts sin(x); the real part extracts cos(x).

What this means for the depth hierarchy

Over the reals, we proved that sin(x) requires an infinite EML tree (the Infinite Zeros Barrier). Over the complex numbers, sin(x) costs exactly one node — as the imaginary part of a single complex EML evaluation.

This is not a contradiction. It's a domain shift. The same function has different EML complexity depending on whether you allow complex arithmetic. That distinction — real vs. complex, EML-∞ vs. EML-1 — is what the depth hierarchy formalizes.

In the EML classification:

• sin(x) over ℝ: EML-∞ (provably impossible as finite real tree)
• sin(x) over ℂ: EML-1 (Im(eml(ix, 1)))

The bypass works because Euler's formula is available in the complex domain. Whether it's available from the EML grammar starting with terminal {1} — which can only generate real constants at depth 0 — depends on whether i is constructible from {1}. That's the open problem.

Why this is remarkable

Euler's formula is usually described as a mysterious coincidence: why should the exponential function, which comes from calculus and differential equations, equal a combination of trigonometric functions, which come from geometry?

In EML terms, it's not mysterious at all. The EML operator is precisely exp(x) − ln(y). When you set x = ix and y = 1, you're using the EML operator's most basic identity to unify exponential and trigonometric behavior. The "mystery" dissolves into: these functions are the same thing, evaluated with complex vs. real arguments.

The EML framework doesn't explain Euler's formula. But it does make clear that Euler's formula is a consequence of the structure of the EML operator itself, not an accident.

The Euler Gateway theorem

In the monogate theorem catalog, this is listed as T03 (Euler Gateway):

ceml(ix, 1) = exp(ix) − Log(1) = exp(ix) = cos(x) + i·sin(x). The single depth-1 ceml tree with input ix and second argument 1 equals exp(ix).

It's listed as a theorem — a proved fact — not a conjecture. The verification is direct: Log(1) = 0, so ceml(ix, 1) = exp(ix). Euler's formula is standard. QED.

Full theorem catalog: challenge.monogate.dev/theorems · Paper: arXiv:2603.21852