## 6 Discussion  
The Blockchain Trilemma, as formalized in this paper, remains an essential framework for understanding the inherent trade-offs among decentralization, scalability, and security in blockchain systems. Our findings provide formal mathematical proof that, under current blockchain architecture paradigms, it is impossible to achieve optimal levels of all three properties simultaneously. In this section, we interpret our results in the context of previous studies, reflect on the validity of our hypotheses, and discuss the implications and possible avenues for future research. The formal proof presented in this work confirms the existence of a fundamental trade-off that restricts blockchain systems from achieving high levels of decentralization, scalability, and security at once. This proof is consistent with the hypotheses previously articulated by [PERSON] (2021) [15], which suggested that the trilemma is intrinsic to blockchain architecture. Our mathematical formalization builds on the conceptual groundwork laid by [PERSON] et al. (2023) [20]. [PERSON] et al. formulated a theoretical representation of the trilemma for PoW blockchains, demonstrating that the product of the three core properties is constrained by a constant value. Our proof goes beyond [PERSON]'s work by generalizing this formulation and demonstrating its applicability across diverse consensus mechanisms beyond PoW, encompassing both PoS and other hybrid consensus protocols. This generalization is crucial, as it broadens the applicability of the trilemma beyond energy-intensive PoW networks, allowing us to assess newer more scalable blockchain technologies. Moreover, our research aligns with and extends the findings of [PERSON] et al. (2024) [21]. Their comparative study of Algorand and Ethereum 2.0 quantitatively evaluated the effectiveness of these platforms in addressing the trilemma challenges. Our work contributes a theoretical underpinning that can help explain their empirical results. For example, Algorand demonstrated relatively higher decentralization metrics but also faced inherent scalability limitations under certain conditions, validating our finding that improving decentralization without a proportional sacrifice in scalability is implausible. The implications of our findings are significant for the design and development of future blockchain networks. First, we suggest that blockchain protocols must prioritize specific attributes based on their intended use cases rather than striving for an unattainable balance of all three. For example, highly secure and decentralized systems like Bitcoin will inevitably sacrifice scalability, making them suitable for value storage rather than high-throughput applications. Our findings also help explain why different blockchain architectures have been unable to achieve the \"ideal\" level of performance that some proponents envisioned. [PERSON] et al. (2020) [19] presented an architecture that claimed to disprove the trilemma, suggesting that scalability could be achieved without compromising decentralization or security by leveraging committee-based consensus. However, our results indicate that any such approach, when fully analyzed under realistic constraints, reveals limitations that fall back into the trilemma framework. Specifically, increasing the number of nodes or the network load without sacrificing performance requires mechanisms that introduce a dependency on select trusted nodes, effectively reducing decentralization. The case of Algorand, as analyzed by [PERSON] et al. (2019) [16], further illustrates the potential security vulnerabilities associated with attempting to optimize all three properties simultaneously. Their study demonstrated that Algorand's Byzantine Agreement mechanism, while innovative, was vulnerable to denial-of-service (DDoS) attacks under specific conditions. This vulnerability suggests that optimizing scalability while maintaining security under real-world network conditions remains a challenging task, as attackers can exploit gaps in validation processes.