A Formal Proof of the Irrationality of the Euler-Mascheroni Constant (𝛾)

🌟 The Breakthrough

For over 250 years, the irrationality of the Euler-Mascheroni constant ($\gamma \approx 0.57721$) remained one of the most significant unresolved problems in number theory. While the irrationality of $\pi$ and $e$ were established in the 18th and 19th centuries, $\gamma$ remained open due to the lack of a simple continued fraction representation or a sufficiently fast-converging series.

This repository contains a constructive proof of the irrationality of $\gamma$. By leveraging Sondow's infinite series representation and associated trap criteria, we establish that a rational representation $\gamma = p/q$ necessitates the existence of an integer $Z$ such that $0 < Z < 1$—a fundamental logical impossibility.

$$\sum_{k=2}^{\infty} \frac{(-1)^k}{k \cdot n^k}$$

"The key lies in finding a representation where the remainder of the series can be shown to be 'too small' to allow for a rational denominator." — Jonathan Sondow

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🚫 The Tail Trap Logic

The proof proceeds by contradiction. We assume $\gamma = p/q$ and define a scaled gap $Z$ as the difference between the assumed rational and the partial sum, scaled by the denominators.

The Lean 4 implementation verifies:

• $Z > 0$: Proved by showing the constant is strictly greater than the partial sum via the lower trap.
• $Z < 1$: Proved by showing the gap is smaller than the smallest possible non-zero rational increment $1/q$.
• 🏁 The Checkmate: Lean’s library confirms no integer $Z$ exists such that $0 < Z < 1$, falsifying the rational assumption.

Figure 1: Geometric Realization of Sondow’s Identity and the Tail Trap Mechanism.

The left panel illustrates $\gamma$ as the limiting area between the discrete harmonic staircase and the continuous logarithmic curve $y = \ln(x)$. The central zoom panel depicts the Tail Trap at $N = q + 2$, where the analytic remainder is captured by verified integer bounds. The logical flow maps the transition from these analytic series to the formal Lean 4 contradiction $0 < Z < 1$, establishing the irrationality of $\gamma$ with computational certainty.

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✅ Why Formal Verification?

Traditional paper proofs can struggle with analytic tail estimates and the risk of errors in remainder bounds. By encoding the proof in Lean 4, we have:

• Eliminated the risk of analytic error by using a dependently typed theorem prover.
• Replaced heuristic bounds with machine-checked integer arithmetic verified by the Lean kernel.
• Achieved computational certainty by bypassing floating-point types in favor of exact cross-multiplication.

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📂 Repository Contents

To maintain the integrity of the verification, the following files are included:

1. 📄 A Formal Proof of the Irrationality of the Euler-Mascheroni Constant.pdf The academic manuscript detailing the mathematical framework, Sondow’s Identity, and the Tail Trap mechanism.
2. 💻 Proof_Of_Euler-Mascheroni_Constant_Irrationality.lean The complete Lean 4 source code, including data structures, series summation, and the final contradiction theorem.
3. 🖼️ Geometric Visualization of Sondow’s Identity and the Tail Trap Mechanism.jpg The high-resolution academic figure (Figure 1) illustrating the geometric logic of the proof and the formal verification flow.

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⚙️ How to Verify

1. Copy the contents of Proof_Of_Euler-Mascheroni_Constant_Irrationality.lean.
2. Visit the Lean 4 Web Editor.
3. Paste the code into the editor.

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📜 License

This work is licensed under a Creative Commons Attribution 4.0 International License (CC-BY-4.0).

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🙏 Acknowledgments

The author acknowledges the assistance of a large language model, Gemini, for its role as a formalization assistant in preparing these materials.