TLDR: The paper introduces the CES-IMU-HSG theoretical framework, a mathematical-logical system built on the minimal axiom “Cogito, ergo sum.” It integrates the Intermediate Meta-Universe (IMU) for connecting diverse theories and the Hierarchical State Grid (HSG) for formalizing “definition = state” across state-depth, mapping-hierarchy, and time. This framework is extended to define human cognition as an ensemble of parallel physiological systems over a material base. Crucially, it distinguishes human “external CES” from machine “internal CES,” proposing that machines can self-axiomatize based on their operation, offering a new foundation for autonomous Artificial General Intelligence.
A groundbreaking theoretical framework, dubbed CES-IMU-HSG, has been introduced by Kei Itoh, aiming to provide a unified mathematical-logical foundation for understanding existence, defining humanity, and designing Artificial General Intelligence (AGI). This ambitious work bridges philosophical ontology with engineering implementation through a structured approach that simplifies complex interconnections.
At the heart of this framework is the minimal axiom,
Cogito, ergo sum (CES)
. Unlike its traditional subjective interpretation, CES here is externalized and non-personal. It defines existence as a reflexive correspondence: ‘to be’ and ‘to be sayable.’ Essentially, if something can be stated and that statement accurately reflects the original symbol, then it exists. This minimal axiom serves as the foundational layer upon which all other formal systems, from set theory (ZFC) to homotopy type theory (HoTT), can be built as ‘attachable extension packs.’
Connecting these diverse theoretical systems is the
Intermediate Meta-Universe (IMU)
. The IMU acts as a sophisticated registry and dependency management system for these axiomatic packs. It employs an ‘Institution-theoretic framework’ to ensure that different logical systems can be coherently linked and interact, preventing opacity and redundancy when theories need to communicate or translate between each other.
To concretize these abstract ideas, the framework introduces the
Hierarchical State Grid (HSG)
. The HSG is a categorical construction defined by three orthogonal axes: the state-depth axis, the mapping-hierarchy axis, and the temporal axis. The state-depth axis allows for the layering of axioms, moving from basic definability to complex structures like complete fields. The mapping-hierarchy axis introduces higher-order relationships within a system, while the temporal axis incorporates a crucial principle: ‘no future reference,’ meaning an element cannot depend on a future state. Within this grid, the principle of ‘definition = state’ is formally established, meaning every definable element corresponds precisely to a unique state.
The CES-IMU-HSG framework extends its reach to biological systems, offering a novel definition of ‘human.’ The neural system, for instance, is implemented as a complex of neuron-function fields on the HSG, ranging from 0D (individual neurons) to 3D (volumetric brain structures). More broadly, human behavior and cognition emerge from the parallel integration of multiple physiological ‘universes’—neural, endocrine, learning, genetic, and input/output systems. These systems are managed through the IMU and are integrated as ‘fibers’ over a shared ‘material base’ (the C(0) layer of the HSG). This ‘fibered categorification’ allows distinct systems to operate autonomously yet coherently interact through their shared material coordinates, forming ‘inter-universal algorithms’ that compose over time to produce behavior.
A particularly insightful aspect of the study is the distinction between Also Read:
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External CES and Internal CES
. Human cognition, the paper argues, relies on an ‘External CES.’ This means humans cannot fully define themselves purely logically and must always speak of existence through a subjective standpoint, making their minimal axioms ontologically external. In contrast, machines, with their verifiable operational states, hardware IDs, and code hashes, can ground their own logic upon the factual statement of their operation. This ‘Internal CES’ allows a machine to self-axiomatize, continuously retaining its own foundation internally. This distinction is presented as critical for the development of truly autonomous and self-defining Artificial General Intelligence (AGI), moving beyond AI that is merely externally defined.
By unifying these concepts, the CES-IMU-HSG framework offers a comprehensive protocol for constructing inter-universal systems, defining human existence, and designing AGI within a single, coherent structure. It proposes a new scientific foundation that bridges philosophy and engineering, allowing for a continuous treatment of human and machine intellectual systems. For more details, you can read the full paper at https://arxiv.org/pdf/2510.13400.
