💻 computer science

A formalization of the Gelfond-Schneider theorem

This paper presents a formalization of the Gelfond-Schneider theorem, which resolves Hilbert's Seventh Problem by proving in Lean 4 that $\alpha^\beta$ is transcendental for algebraic $\alpha \notin \{0,1\}$ and irrational algebraic $\beta$ .

Original authors: Michail Karatarakis, Freek Wiedijk

Published 2026-03-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michail Karatarakis, Freek Wiedijk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, infinite library of numbers. Most of these numbers are "boring" in a mathematical sense: they are algebraic. This means they are the roots of simple equations with whole number coefficients. For example, $\sqrt{2}$ is algebraic because it solves the equation $x^2 - 2 = 0$ .

Then, there are the "wild" numbers, called transcendental. These numbers are so complex that they cannot be the solution to any simple equation with whole numbers. Famous examples are $\pi$ and $e$ .

For a long time, mathematicians knew these wild numbers existed, but they struggled to prove that specific combinations of numbers were wild. One of the biggest mysteries was: If you take an algebraic number (like 2) and raise it to the power of another algebraic number (like $\sqrt{2}$ ), do you get a wild number?

This was Hilbert's Seventh Problem, posed in 1900. The answer, proven in 1934 by Gelfond and Schneider, is a resounding YES (with a few exceptions like $2^1$ or $2^2$ ). If the base is not 0 or 1, and the exponent is an irrational algebraic number, the result is definitely transcendental.

The Paper's Mission: The Digital Proof

This paper is about two researchers who decided to take that 1934 proof and translate it into a language a computer can understand and verify: Lean 4.

Think of a traditional math proof like a story written in a novel. It relies on the reader's intuition and trust. "And then, obviously, the function behaves this way..." The reader is expected to fill in the gaps.

A computer proof, however, is like a rigorous legal contract. It cannot accept "obviously." Every single step must be logically airtight, with no hidden assumptions. The authors' goal was to build this digital contract for the Gelfond-Schneider theorem, ensuring that the proof is 100% correct down to the atomic level.

How They Did It: The Detective Story

To prove the number is "wild" (transcendental), the mathematicians used a classic strategy: Proof by Contradiction.

The Trap: They start by pretending the opposite is true. They say, "Let's assume our number, $\alpha^\beta$ , is actually a 'boring' algebraic number."
The Construction (The Auxiliary Function): To catch the number in a lie, they build a special mathematical machine called an Auxiliary Function.
- Analogy: Imagine you are trying to prove a suspect is guilty. You build a complex trap (the function) that is designed to be zero at specific locations if the suspect is innocent.
- They use a tool called Siegel's Lemma to find the right "ingredients" (coefficients) for this machine. Siegel's Lemma is like a magic recipe book that guarantees you can find small, whole-number ingredients to build a machine that fits your specific needs.
The Two Sides of the Coin:
- The Algebraic Side: If the number were algebraic, this machine would have to be a "boring" number itself. Because it's built from whole numbers, it can't be too small. It has a "minimum size" (a lower bound).
- The Analytic Side: Using complex analysis (calculus with imaginary numbers), they show that this machine actually gets incredibly small as they tweak the parameters. It shrinks faster than the "minimum size" rule allows.
The Explosion: Here is the contradiction. The machine is forced to be big (because of algebra rules) and tiny (because of calculus rules) at the same time.
- Analogy: It's like a detective proving a suspect was in two places at once. The suspect (the number) cannot be both "boring" and "wild." Therefore, the initial assumption (that it was boring) must be false. The number must be wild.

The Computer Challenge: Fixing the "Holes"

The most interesting part of this paper is how they handled the computer's strictness.

In the original 1934 proof, the mathematicians used a function that had "holes" (singularities) at certain points. On paper, they would just say, "We'll just avoid those holes," or "The hole disappears because of a limit."

But a computer doesn't like "holes." It doesn't like "just avoiding" things. If a function is undefined at a point, the computer crashes.

The authors had to be incredibly clever. They took that function with holes and patched it up.

They created a "patched" version of the function that works perfectly everywhere, filling in the holes with the correct values derived from calculus.
They then proved to the computer that this patched version is smooth and continuous everywhere.
This turned a messy, "hand-wavy" part of the original proof into a solid, unbreakable digital structure.

Why Does This Matter?

You might ask, "Why spend months proving a theorem that was already proven 90 years ago?"

Trust: In an age of complex mathematics, having a computer verify a proof removes all doubt. It's the ultimate "checkmate."
The Library of Math: They added this proof to Mathlib, a giant open-source library of verified math. Now, other mathematicians and computers can use this result to build even more complex proofs without having to re-verify the basics.
Future Tools: By formalizing this, they are paving the way for computers to eventually help solve even harder problems, like finding new transcendental numbers or verifying the independence of complex logarithms.

In a Nutshell

This paper is the story of taking a brilliant, 90-year-old mathematical detective story and rewriting it as a flawless, computer-readable script. They built a digital trap, caught a number in a logical contradiction, and proved once and for all that certain combinations of numbers are truly "wild" and transcendental. It's a triumph of human logic meeting machine precision.

1. Problem Statement

The paper addresses the formalization of Hilbert's Seventh Problem and its solution, the Gelfond-Schneider theorem, within the Lean 4 proof assistant and the Mathlib library.

The Theorem: If $\alpha$ and $\beta$ are algebraic numbers with $\alpha \notin \{0, 1\}$ and $\beta$ irrational, then $\alpha^\beta$ is transcendental.
Context: While the transcendence of specific constants like $e$ and $\pi$ has been formalized, the Gelfond-Schneider theorem represents a deeper result involving exponential relationships between algebraic numbers. Prior to this work, no theorem prover contained a formalization of this theorem.
Goal: To construct a complete, machine-checked proof of the theorem, including the necessary algebraic and analytic machinery, and to verify the transcendence of the Gelfond-Schneider constant ( $\sqrt{2}^{\sqrt{2}}$ ).

2. Methodology

The authors employed a constructive approach to the proof logic but utilized the classical foundations of Mathlib (which includes the Axiom of Choice and the Law of Excluded Middle). The formalization follows the classical proof structure (based on texts like Hua) but requires rigorous handling of definitions and bounds that are often glossed over in pen-and-paper proofs.

A. Algebraic Number Theory Foundations

The "House" of an Algebraic Number: The authors formalized the concept of the "house" of an algebraic number $\alpha$ $α$ , defined as the maximum modulus of its conjugates ( $|\alpha| = \max_i |\sigma_i(\alpha)|$ $∣ α ∣ = max_{i} ∣ σ_{i} (α) ∣$ ).
- Technical Challenge: Initial attempts using polynomial roots (unordered sets) made arithmetic bounds difficult. The authors adopted a canonical embedding approach ( $K \to \prod_{\sigma} \mathbb{C}$ ), treating the house as an $L_\infty$ -norm on a Banach algebra. This leveraged Mathlib's normed space API to simplify proofs of multiplicativity and the triangle inequality.
Siegel's Lemma for Number Fields: The proof requires finding small algebraic integer coefficients for an auxiliary function.
- The authors formalized a generalized version of Siegel's Lemma for the ring of integers $\mathcal{O}_K$ of a number field $K$ .
- Technique: They used a "descent argument," transforming the linear system over $\mathcal{O}_K$ into a larger system over $\mathbb{Z}$ , applying the standard integer Siegel's Lemma, and lifting the solution back to $\mathcal{O}_K$ .

B. Analytic Core and Auxiliary Functions

The proof proceeds by contradiction, assuming $\alpha^\beta$ is algebraic, and constructing an auxiliary function $R(z)$ that vanishes to a high order at specific points.

Construction of $R(z)$ : Defined as a linear combination of exponentials $R(x) = \sum \eta_i e^{\rho_i x}$ . The coefficients $\eta_i$ are determined by solving a homogeneous linear system derived from the condition that $R(z)$ vanishes with high multiplicity at integer points.
The Global Function $S(z)$ : A critical technical hurdle was formalizing the auxiliary function $S(z)$ $S (z)$ , which involves $R(z)$ $R (z)$ divided by terms like $(z-k)^r$ $(z - k)^{r}$ .
- Challenge: In classical analysis, $S(z)$ is treated as meromorphic with removable singularities. In dependent type theory (Lean), partial functions (meromorphic functions) are difficult to handle because they require tracking domains and avoiding poles in every lemma.
- Solution: The authors constructed $S(z)$ as a total, piecewise holomorphic function. They explicitly defined the function values at singular points using local power series expansions (patching) and proved that these local definitions glue together to form a globally analytic function. This avoided the complexity of partiality and option types.

C. The Contradiction

The proof derives a contradiction by bounding a specific value $\rho$ (a normalized derivative of $R(z)$ ) in two ways:

Algebraic Lower Bound: Since $\rho$ is a non-zero algebraic number, its norm $|N(\rho)|$ must be bounded from below (specifically, $|N(\rho)| \geq 1$ after clearing denominators).
Analytic Upper Bound: Using Cauchy's integral formula on a large contour, the authors bounded $|\rho|$ from above.
Result: For sufficiently large parameters, the analytic upper bound decays super-exponentially, while the algebraic lower bound remains constant or grows slower, leading to a contradiction ( $|N(\rho)| < 1$ ).

3. Key Contributions

First Formalization: This is the first formalization of the Gelfond-Schneider theorem in any theorem prover.
Mathlib Extensions: The work contributed significant new definitions and lemmas to Mathlib, including:
- The "house" of an algebraic number defined via canonical embeddings.
- Siegel's Lemma generalized to algebraic integers in number fields.
- Techniques for constructing total analytic functions from expressions with removable singularities.
Explicit Constant Management: Unlike classical proofs that use $O(\cdot)$ notation, the formalization explicitly tracked a sequence of constants ( $c_1$ through $c_{15}$ ) dependent on the degree of the number field and geometric properties. This explicit tracking was essential to prove the final contradiction rigorously.
Verification of Constants: The framework was used to formally prove the transcendence of $\sqrt{2}^{\sqrt{2}}$ .

4. Results

Theorem Proven: The Lean code successfully verifies that for algebraic $\alpha \notin \{0, 1\}$ and irrational algebraic $\beta$ , $\alpha^\beta$ is transcendental.
Specific Application: The formalization confirmed that $\sqrt{2}^{\sqrt{2}}$ is transcendental.
Code Availability: The source code is integrated into a branch of Mathlib and is publicly available.

5. Significance and Future Work

Bridging Fields: The work demonstrates the capability of modern proof assistants to handle deep results that bridge algebraic number theory and complex analysis.
Foundation for Advanced Theorems: This formalization lays the groundwork for more complex results:
- Baker's Theorem: Generalizing to linear forms in logarithms.
- p-adic Gelfond-Schneider: Adapting the analytic lemmas to p-adic analysis (Schnirelmann integrals).
- Schneider-Lang Theorem: Extending transcendence results to meromorphic functions satisfying algebraic differential equations.
- Elliptic Curves: Generalizing the framework to the Weierstrass $\wp$ -function.
Decision Procedures: The explicit nature of the constants suggests potential for implementing automated decision procedures for specific classes of transcendental numbers using interval arithmetic.
Schanuel's Conjecture: The work contributes to the library of facts needed to conditionally verify consequences of Schanuel's Conjecture.

In summary, this paper represents a major milestone in formal mathematics, successfully mechanizing a century-old, non-trivial proof that required significant innovation in handling analytic functions within a constructive type-theoretic framework.