## 1 Introduction  
Blockchain technology has emerged as a transformative force, promising decentralized, secure, and transparent systems that can disrupt traditional industries ranging from finance to supply chain management [1]. By enabling peer-to-peer transactions without intermediaries, blockchains have the potential to democratize data ownership [2] and foster trust in digital interactions. However, this revolutionary technology faces a fundamental challenge known as the Blockchain Trilemma [3]. The Blockchain Trilemma posits that a blockchain system cannot achieve optimal decentralization, security, and scalability levels simultaneously [4]. This trilemma forces to prioritize two of these properties at the expense of the third:  
* Decentralization (D) [5]: Ensuring that control and decision making are distributed among a wide network of participants, preventing any single entity from exerting undue influence. For example, Bitcoin achieves decentralization through its distributed network of miners and nodes [6]. * Security (S) [7]: Maintaining resilience against attacks, safeguarding data integrity, and ensuring that the system can reach consensus even in the presence of malicious actors. As an example, the Proof-of-Work [8] consensus mechanisms contribute to security by making it computationally expensive to alter transaction history. Scalability (C) [9]: The ability to handle an increasing number of transactions efficiently, which is essential for widespread adoption and practical usability. However, scalability remains a challenge for Bitcoin and Ethereum, leading to high transaction fees and slow processing times [10]. As illustrated in Figure 1, the trilemma represents the trade-offs between these properties, making it challenging to optimize all three simultaneously. Moving closer to one vertex inherently draws the system away from the other two, forcing practitioners to navigate delicate design choices. The trilemma remains a pivotal obstacle despite numerous innovations attempting to overcome this challenge, such as sharding [11], off-chain transactions [12], and novel consensus algorithms. Sharding partitions the blockchain to improve scalability but may impact security or decentralization. Off-chain transactions, like those enabled by the Lightning Network [13], improve scalability but introduce trade-offs in security and decentralization. Novel consensus algorithms like Proof-of-Stake [14] aim to enhance scalability and energy efficiency but may affect decentralization. This raises a critical question: Can a blockchain system be mathematically proven incapable of simultaneously achieving maximum decentralization, security, and scalability? This paper aims to address this question through a rigorous mathematical proof. Unlike previous studies that have explored empirical or heuristic solutions, we provide a formal impossibility proof. By formalizing the definitions of decentralization, security, and scalability, we demonstrate that achieving all three properties at their maximum levels concurrently leads to inherent contradictions. Our contributions are the following:  
1. Mathematical Formalization: We provide precise mathematical definitions and quantifications for each of the three properties, grounding them in measurable terms using tools from graph theory and probability theory. 2. Impossibility Proof: We prove by contradiction that maximizing any two properties inevitably compromises the third by constructing theoretical models that highlight these trade-offs. 3. Computational Complexity Analysis: To reinforce our theoretical findings, we delve into blockchain systems' communication and processing complexities by analyzing time and space complexity within distributed networks. The paper is organized as follows. Section 1 introduces the problem, laying the foundation for understanding the Blockchain Trilemma. Section 2 reviews the related works, discussing previous efforts to formalize and address the Blockchain Trilemma, focusing on  
Figure 1: The Blockchain Trilemma.