## 7 Conclusions  
The Blockchain Trilemma, which restricts the simultaneous optimization of decentralization, scalability, and security, is a fundamental challenge that continues to shape the evolution of blockchain systems. In this paper, we presented a formal proof demonstrating that this trilemma remains unsolved under current architectural paradigms, even with innovative approaches such as randomized committees and advanced consensus protocols. Our formal analysis confirmed that the inherent limitations of blockchain technology necessitate trade-offs between its three core properties. Despite the promising results of newer protocols like Algorand and Ethereum 2.0, as demonstrated in related studies, none have successfully broken free from these trade-offs. Attempts to achieve scalability through sharding, as well as the introduction of committee-based consensus mechanisms, while helpful in addressing some scalability issues, ultimately compromise either decentralization or security. Our results underscore the need for careful prioritization based on application-specific requirements. Blockchains that aim to serve as a store of value must emphasize security and decentralization, even at the cost of reduced scalability, while others, such as those intended for high-throughput transactional use, must balance scalability against a reduction in network-wide participation. In the future, hybrid approaches that combine Layer-1 and Layer-2 scaling, as well as a deeper exploration of consensus hybrids (e.g., PoW/PoS combinations), may provide a more nuanced path forward. The interplay between on-chain and off-chain scaling solutions presents a promising avenue for addressing the Blockchain Trilemma in more nuanced ways. On-chain approaches, such as sharding, improve scalability by enabling the parallel processing of transactions across smaller network subsets. However, they often reduce decentralization by limiting the number of nodes participating in each shard's consensus and increase security risks, such as shard compromise or inter-shard communication vulnerabilities. Off-chain solutions, such as Layer-2 protocols (e.g., rollups, state channels), shift computational and transactional load to auxiliary layers while periodically anchoring back to the Layer-1 chain for security. These methods enhance scalability without significantly altering the base layer's performance. However, they may introduce decentralization trade-offs, as some off-chain systems depend on trusted intermediaries or centralized operators, and security challenges, such as vulnerabilities in the anchoring process or reliance on coordination mechanisms. By integrating on-chain and off-chain solutions, it is possible to mitigate some of these trade-offs. For example, Layer-1 could focus on providing robust security guarantees and decentralized data anchoring, while Layer-2 systems handle high-throughput transactional needs. This hybrid approach allows the strengths of both scaling strategies to complement each other, optimizing performance for specific use cases while navigating the inherent constraints of the trilemma. In conclusion, the Blockchain Trilemma is not just a theoretical constraint, but a practical reality that developers and researchers must navigate when designing systems for various applications. Our framework provides a foundational tool to evaluate these trade-offs, offering practical implications for real-world blockchain systems. For example, developers working on high-throughput networks, such as payment systems, can use the framework to assess the impact of scaling techniques like sharding or rollups on decentralization and security. Conversely, projects prioritizing decentralization, like Bitcoin,can leverage the framework to understand the scalability sacrifices necessary to maintain security and network-wide participation. Real-world blockchain protocols further illustrate these trade-offs.