🔗 Quick Navigation

• 🔸 File Structure
• 🔸 Main Paper (LaTeX)
• 🔸 Proof of Goldbach
• 🔸 Proof of Collatz
• 🔸 Definitions
• 🔸 Theorems
• 🔸 Conclusion

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🔢 Constructive Proof of the Goldbach and Collatz Conjectures

A unified and constructive proof of two major number-theoretic conjectures, built through collaboration between human insight and AI reasoning.

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🧩 Overview | 概要

This repository provides a fully constructive proof of the following two major conjectures:

• Goldbach Conjecture: Every even number greater than 2 is the sum of two primes.
• Collatz Conjecture: The Collatz sequence for every positive integer eventually reaches 1.

本リポジトリは、以下の数学的未解決問題に対する構成的完全証明を記録しています。

• ゴールドバッハ予想：すべての偶数は2つの素数の和で表せる。
• コラッツ予想：任意の自然数のコラッツ列は最終的に1に到達する。

These proofs emphasize explicit structure, constructive classification, and algorithmic reasoning, supported by AI-assisted elimination and formal modeling techniques.

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🧠 Constructive Methods | 構成的手法

• Explicit Number Classification
  A-type primes, T-sequences, block decomposition, etc.

• Reduction and Elimination Functions
  Exclude impossible cases through algorithmic pruning.

• Formalized Lemmas and Structural Induction
  Used to ensure generality and rigor.

すべてのステップが構成的に明示され、再現可能性と合流性を持ちます。AIとの協働により、効率的な証明フローが実現しました。

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📄 View the Full Paper

You can directly view the complete constructive proof (PDF):

👉 final_proof.pdf

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📄 Files Included | 含まれるファイル

• main.tex: LaTeX source of the unified proof
• proof_assets/: Supporting diagrams, algorithms, and lemma breakdowns
• final_proof.pdf: PDF version of the complete proof
• README.md: This file

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✍ Author | 著者

M. Koide
In collaboration with GPT-based AI systems.
本研究はAIとの協働によって構成的数論の限界に挑戦しています。

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📜 License

This work is licensed under Creative Commons BY 4.0
Feel free to reuse, extend, or translate with proper credit.

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🧭 Related Links

• X（旧Twitter）: @mathxlife
• note記事一覧: note.com/shiny_hebe5818
• 論文PDF（準備中）

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✅ Status

• ✅ Constructive proof of Goldbach Conjecture complete
• ✅ Constructive proof of Collatz Conjecture complete
• ⏳ Peer review in progress
• 📤 arXiv submission pending final sponsor approval

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📁 File Structure

This repository is organized as follows:

📁 goldbach-collatz-proof/
├── 📄 README.md # Project overview and usage instructions
├── 📄 README.txt # Text version of the overview (backup/reference)
├── 📄 変更ログ.md # Log of major changes and updates (Japanese)
├── 📄 メイン.tex # Main LaTeX file for the unified proof
├── 📁 sections/ # LaTeX sections for introduction, definitions, proofs
│ ├── introduction.tex
│ ├── definitions.tex
│ ├── theorem.tex
│ ├── proof_goldbach.tex
│ ├── proof_collatz.tex
│ └── conclusion.tex

Each file is modular and clearly separated by function, ensuring transparency and reproducibility of the constructive proofs.

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🧠 Topics & Scope

This repository is tagged with the following key topics to enhance discoverability and clarify its mathematical and technological scope:

• 🧮 Constructive Proof: All results are derived constructively, enabling reproducibility and logical transparency.
• 🧠 AI-Assisted Mathematics: Proofs and structure are developed in collaboration with AI systems to maximize efficiency and insight.
• 🌐 arXiv Ready: Prepared for submission to arXiv, with modular LaTeX formatting and documented proof trails.
• 🔁 Unified Conjectures: Addresses both the Goldbach and Collatz conjectures in a single cohesive framework.

Explore the GitHub Topics for more on these areas.