The Turing machine, the cornerstone of modern computing, has made great contributions in defining the boundaries of algorithmic computability. However, its inherent linear and deterministic nature shows clear limitations when dealing with the complex, nonlinear real world and in constructing intelligent systems with emergent behaviors. As Gödel’s incompleteness theorem reveals, any sufficiently complex single formal system contains inherent incompleteness. Therefore, to build systems capable of effectively interacting with the unpredictable real world and achieving robust distributed consensus, we must go beyond the constraints of single formalism and embrace new computational paradigms.

The Inspiration of the P/NP Paradigm: Multi-System Interaction and Emergent Intelligence

The P/NP problem framework offers a promising breakthrough. Its core idea is to design computing systems as an interactive collection of different types of distributed formal systems: NP-class “solvers” that handle complex search problems and P-class “verifiers” that provide efficient verification schemes. This non-linear computational difference provides the foundation for the emergence of behaviors, where the overall capabilities of the system exceed the mere sum of its individual components.

The GEB Project: Integrating Three Formal Submodules

Against this background, we focus on the GEB project, which aims to build an adaptive intelligent system. Its core architecture includes three key formal submodules:

1. Formal Blockchain Technology: As the cornerstone of the system, it is responsible for constructing and maintaining a decentralized consensus ledger, ensuring consistency and immutability of transactions and states. This corresponds to the (λ calculus + consensus algorithm) module in the BEVM (λ) concept, aiming to provide a trustworthy execution environment.
2. Formal System of Human–Machine Interaction Theory: GEB emphasizes user-centered design and formalizes the theory of human–machine interaction to construct an account system for the self-mapping of the “Individual.” Here, “Individual” represents an account model directly associated with users or entities, embodying their autonomy and interactive behavior. This echoes the Individual Model in BEVM (λ), aiming to achieve a more intuitive and user-aligned interactive experience. Bitcoin’s UTXO model is an early embodiment of this concept, achieving a 1:1 mapping between users and on-chain assets.
3. Formal System of P/NP Self-Referential Emergent Nonlinear Computational Models: This is the key innovation in GEB’s design. This module aims to utilize the computational complexity of P/NP problems to construct mechanisms capable of perceiving and responding to real-world information. Its core idea is to simulate complex systems in nature through self-reference and emergence, ultimately achieving human–machine symbiosis with adaptive and self-organizing evolution capabilities. This corresponds to the consensus-aware algorithm in BEVM (λ). Bitcoin’s proof-of-work (PoW) mechanism can be seen as an early implementation of this concept—it ensures network security and value by consuming real-world energy, while the longest-chain consensus acts as a distributed, implicit “oracle machine” that connects the computing system with a form of “reality.”

Overcoming the Limitations of Traditional Blockchains: Learning from Bitcoin

Traditional blockchain technologies, such as Ethereum, primarily focus on the formal consensus ledger (corresponding to the first submodule of GEB). However, as the analysis indicates, relying solely on this single formal system can lead to disconnection from the real world and neglect of individual user needs, ultimately risking centralization and closure.

Bitcoin’s success lies in its integration of the Individual model (UTXO) and the P/NP-based consensus-aware mechanism (PoW and longest chain) on top of its consensus ledger. The UTXO model enables more direct human–machine interaction, while PoW strengthens the robustness and value basis of consensus by anchoring it in real-world energy. The longest-chain consensus, as an implicit oracle machine, connects the computing system with the “reality” of computational power input.

GEB’s Vision: Building an Adaptive Future

The GEB project draws from Bitcoin’s successful experience and elevates it to a new theoretical level. By explicitly integrating formal blockchain technology, human–machine interaction theory, and P/NP self-referential emergent nonlinear computational models into three key submodules, GEB aims to build the next generation of distributed systems that can overcome the limitations of traditional blockchains. Its goal is to create an intelligent ecosystem that is more user-aligned, more perceptive of the real world, and equipped with adaptive and self-organizing capabilities. The upcoming new white paper will further elaborate on how the GEB project, while inheriting the core principles of BEVM (λ), will explore practical strategies for realizing these ideas—heralding a future beyond the traditional blockchain paradigm.