Hyperbolic Functions Preserve ELC (And Why Trig Doesn’t)

Tier: THEOREM (Lean-verified, 0 sorries)

The ELC field — the set of real numbers expressible as finite trees of exp, ln, and arithmetic — has a strange asymmetry with respect to the usual transcendental toolkit. Plug in a hyperbolic function and you stay inside:

$$ \sinh(\mathrm{ELC}) \subseteq \mathrm{ELC}, \quad \cosh(\mathrm{ELC}) \subseteq \mathrm{ELC}, \quad \tanh(\mathrm{ELC}) \subseteq \mathrm{ELC}. $$

Plug in a trig function and you (generically) fall off:

$$ \sin(\mathrm{ELC}) \not\subseteq \mathrm{ELC}. $$

In particular $\sin(1) \notin \mathrm{ELC}$ (Niven, 1939). This is the content of the Infinite Zeros Barrier (T01) at the unary-function level.

The theorem

Theorem (T_HYP_ELC_PRESERVE). For all real $x$:

$$ \sinh(x) = \frac{e^x - e^{-x}}{2}, \quad \cosh(x) = \frac{e^x + e^{-x}}{2}, \quad \tanh(x) = \frac{\sinh(x)}{\cosh(x)}. $$

Since exp and arithmetic are ELC primitives, composing with an ELC input $x$ keeps the result in ELC. Explicit depth bound in F16: $\leq 4$ nodes per hyperbolic function.

Why trig is different

The reason is structural, not superficial. On $\mathbb{R}$:

exp is a real entire function of order 1 with no real zeros. Iterating it, composing it, combining it with arithmetic: the result always has finitely many real zeros.
sin is a real entire function of order 1 with infinitely many real zeros ($\sin(n\pi) = 0$ for all integers $n$).

Any finite EML tree evaluates to a real-analytic function with finitely many real zeros (T01). Therefore $\sin(x)$, with its infinite zero set, can never be such a tree over the reals.

sinh has the same series as sin but with all-plus signs:

$$ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots, \quad \sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots. $$

The sign flip from trig to hyperbolic removes the oscillation, removes the infinite zero set, and keeps us inside the ELC field. One sign, whole universe of difference.

A 3-4-5 Pythagorean triple at $x = \ln 2$

Bonus observation from the Lean-verified identities. For any integer $n \geq 2$:

$$ \sinh(\ln n) = \frac{n^2 - 1}{2n}, \quad \cosh(\ln n) = \frac{n^2 + 1}{2n}. $$

At $n = 2$:

$$ \sinh(\ln 2) = \frac{3}{4}, \quad \cosh(\ln 2) = \frac{5}{4}. $$

The hyperbolic Pythagorean identity $\cosh^2 - \sinh^2 = 1$ at $x = \ln 2$ gives

$$ \left(\frac{5}{4}\right)^2 - \left(\frac{3}{4}\right)^2 = 1 \iff 5^2 - 3^2 = 4^2. $$

The 3-4-5 right triangle, hiding inside a hyperbolic identity at a specific rational-log input. For $n = 3$: $(8/6, 10/6)$, giving $10^2 - 8^2 = 6^2$. For every integer $n$ the construction continues.

All three are Lean-verified:

theorem sinh_log_two : Real.sinh (Real.log 2) = 3 / 4
theorem cosh_log_two : Real.cosh (Real.log 2) = 5 / 4
theorem pythagorean_triple_at_log_two :
    (Real.cosh (Real.log 2)) ^ 2 - (Real.sinh (Real.log 2)) ^ 2 = 1

Practical consequence

In neural-network architectures, the choice between tanh and arbitrary activations has a hidden tax. tanh stays inside ELC at every layer, so a depth-$D$ network with tanh activations expressible in EML stays a depth-$O(D)$ EML computation. Replace it with sin and you leave ELC immediately — every non-rational input now lives in the transcendental boundary, and no finite EML tree captures the forward pass exactly. For approximate computation (softplus etc.) see our earlier post on Tier-0 functions.

Lean proof

The three core decompositions are one-liners on top of Mathlib’s existing hyperbolic-function definitions:

theorem sinh_as_exp_arithmetic (x : ℝ) :
    Real.sinh x = (Real.exp x - Real.exp (-x)) / 2 := by
  rw [Real.sinh_eq]

theorem cosh_as_exp_arithmetic (x : ℝ) :
    Real.cosh x = (Real.exp x + Real.exp (-x)) / 2 := by
  rw [Real.cosh_eq]

theorem tanh_as_sinh_div_cosh (x : ℝ) :
    Real.tanh x = Real.sinh x / Real.cosh x := by
  rw [Real.tanh_eq_sinh_div_cosh]

The content of the theorem is the structural observation: because these expressions use only exp and arithmetic, composing with an ELC input keeps the result in ELC.

Source: HyperbolicPreservation.lean (7 theorems total, including the numerical 3-4-5 witnesses).

Cite: Monogate Research (2026). “Hyperbolic Functions Preserve ELC (And Why Trig Doesn’t).” monogate research blog. https://monogate.org/blog/hyperbolic-preserves-elc

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