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, sin(x) is not a finite EML tree at any depth. That is proved in Lean by MachLib's periodic barrier (sin_not_in_eml_any_depth_unconditional); the Infinite Zeros Barrier route to it still has open steps. 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: monogate.org/theorems · Paper: arXiv:2603.21852