A Compositional Model of Consciousness based on Consciousness-Only

Abstract: Scientific studies of consciousness rely on objects whose existence is assumed to be independent of any consciousness. On the contrary, we assume consciousness to be fundamental, and that one of the main features of consciousness is characterized as being other-dependent. We set up a framework which naturally subsumes this feature by defining a compact closed category where morphisms represent conscious processes. These morphisms are a composition of a set of generators, each being specified by their relations with other generators, and therefore co-dependent. The framework is general enough and fits well into a compositional model of consciousness. Interestingly, we also show how our proposal may become a step towards avoiding the hard problem of consciousness, and thereby address the combination problem of conscious experiences.

• The paper proposes a formal compositional framework for consciousness, modeling states as morphisms in symmetric monoidal categories.
• It extends the ZX-calculus to capture co-dependent and recursive conscious processes using precise algebraic structures.
• The approach resolves the combination problem by unifying individual processes into a coherent experience, with implications for AI, phenomenology, and quantum cognition.

Compositional Categories and the Foundations of Consciousness-Only

Introduction

This paper advances a formal compositional framework for consciousness, positing consciousness as a fundamental, co-dependent process rather than as an emergent property of physical substrates. Drawing conceptually from the Yogacara (consciousness-only) tradition and mathematically from category theory—especially symmetric monoidal and compact closed categories—this work proposes that conscious states and processes can be represented as morphisms and compositions within such categories. The approach bypasses the prevailing reductionist bias by treating compositionality as the formal principle underpinning both the unity and combination of conscious experience, directly addressing the combination problem faced by panpsychist and standard physicalist accounts.

Category Theory and Process Theory as the Mathematical Basis

Category theory provides an abstract language of systems, focusing on the relationships represented by morphisms between objects. Process theory, instantiated as strict symmetric monoidal categories (SMC), allows the structural modeling of sequential and parallel composition of processes—core aspects of both the physiological and phenomenological character of consciousness.

In process-theoretic terms:

• Objects correspond to types or states (interpretable as aspects of consciousness).
• Morphisms represent processes or transitions between types.
• Monoidal product ($\otimes$) encodes parallel composition, while morphism composition ($\circ$) encodes sequential combination.
• Symmetric structure allows the commutation of parallel processes, corresponding to the indistinguishability of the order of certain experiences.
• Compact closure introduces caps and cups, enabling the formalization of duality and reflexivity, which are essential for self-referential phenomena in consciousness.

Process theories provide a minimal and explicit axiom set, directly expressing the two primitives of conscious experience: succession (time-like) and co-presence (space-like) of experiential content. Existing neural or information-centric models do not capture these properties with comparable parsimony or generality.

The Consciousness-Only Hypothesis and Co-dependence

The central tenet is the bracketing of objectivity: consciousness is assumed as primitive, with "objects" emerging as co-dependent products of subjectivity, in alignment with Yogacara's stratification of consciousness (alaya, manas, six sense consciousnesses, etc.). The notion of co-dependence is operationalized mathematically: each conscious process (generator) is specified by its synthetic relations (rewriting rules) with others, not in ontic isolation. This moves the model from mere "boxes" in process theory to a fine-grained, generator-and-rewriting-rule setting—a critical step for capturing the contextuality and inter-relational character of phenomenal content.

Qufinite $ZX_\Delta$-Calculus: A Fine-Grained Compositional Model

The formal backbone of the proposed framework is the generalization of the ZX-calculus—originally a diagrammatic computational language for quantum mechanics—to the qufinite $ZX_\Delta$-calculus, a compact closed category with arbitrary commutative semirings and additional generators.

• Generators of the calculus, interpreted as elementary or atomic conscious processes, can have multiple inputs and outputs—capturing the distributed, integrated nature of experience.
• Rewriting rules specify mutual constraints, encoding contextuality and co-dependence; the meaning of a basic process is established only through its participation in a web of compositional relations.
• Soundness is established by constructing a symmetric monoidal functor from the category of diagrams to the category of $\mathcal{S}$-valued matrices; completeness and empirical determination of the generator set remain future tasks.

The $ZX_\Delta$-calculus is structurally self-dual and supports both sequential/parallel composition as well as "caps" and "cups," enabling representations of processes with reflexive or circular dependencies—a mathematical necessity for modeling the recursive aspects of self-awareness and the unity of consciousness.

Modeling the Alaya Consciousness and its Divisions

Drawing from Yogacara, the model distinguishes:

• Alaya consciousness: The base layer, with perceived (object-like) and perceiving (subject-like) divisions.
  • Perceived division: Modeled by the category $\mathbf{FdHilb}$ of finite-dimensional Hilbert spaces and linear maps, directly aligning with the structure of quantum theory.
  • Perceiving division: Modeled as a functor from the $ZX_\Delta$-category to $\mathbf{FdHilb}$, formally representing the act of "perceiving" as a structure-preserving transformation.

In this reading, the classical "physical world" is not ontologically fundamental but a projection of perceived alaya processes; subjectivity is formalized as functorial mapping rather than as an emergent property of substrate dynamics. This approach is in sharp contrast with standard neural-causal models, which make strong ontological commitments to physical primitives.

Addressing the Combination and Unity Problems

One of the enduring problems in theories of consciousness—especially panpsychist ones—is the "combination problem": how micro-level phenomenal properties or subjects compose into macro-level unities. The compositional framework here directly addresses this:

• Unity arises through composition, not aggregation: Sequential and parallel compositions of basic processes generate complex, unified experiences, without appealing to atomistic assembly of non-experiential parts.
• No micro/macro subjective divide: The distinction dissolves; generators may already possess the structural properties attributed to macro-level experience.
• No commitment to neural isomorphism: The lack of reduction to neural mechanisms avoids the structural mismatch problem and sidesteps the hard problem's intractabilities. Neural dynamics may be modeled in parallel formal categories, but not identified as primary.

Additionally, by encoding contextual specification through rewriting rules, the model allows the possibility of "phenomenal contextuality," a feature absent in all functionalist models of consciousness.

Implications, Limitations, and Future Directions

The paper makes several bold commitments:

• Consciousness as transformation: Phenomenal content is not a static property but a process; process theory, rather than set-theoretic or state-based models, is the proper mathematical substrate.
• Empirical adequacy: The selection and phenomenological mapping of generators is acknowledged as an open empirical and theoretical problem, requiring detailed collaboration with micro-phenomenological methods.
• Scalability: The categorical framework is extendable to infinite-dimensional cases, potentially enabling recovery of standard quantum mechanics within a theory where consciousness is primary.
• Integration with other frameworks: The approach is in principle compatible with "conscious agent" models, but provides a clearer mathematical toolkit for co-dependent, non-reductive synthesis.
• Constraints on AI and modeling: The model suggests that attempts to "build" consciousness from non-conscious computational substrates are structurally inadequate; simulation of consciousness is not realization unless the compositional, co-dependent axioms are respected.

Theoretical and AI implications

The categorical formulation triggers several speculations:

• Formalization of phenomenology: Axiomatic, compositional mathematics may finally enable a scientifically productive, theory-neutral phenomenology, useful for both natural and artificial systems.
• Computational irreducibility: To the extent that the generators require contextual dependencies, simulating or embedding the corresponding rewriting systems may place strong constraints on what kinds of computational architectures are eligible for "artificial consciousness."
• Cross-fertilization with quantum cognition and contextuality: The formal similarity between consciousness processes and quantum processes suggests a structural, if not ontological, bridge between cognitive science and quantum information theory.

Conclusion

This work develops a process-theoretic, compositional framework for consciousness inspired by Yogacara's consciousness-only doctrine. By internalizing the co-dependent, irreducible unity of experience into the very axioms of the model, and by leveraging symmetric monoidal and compact closed categories, it provides a mathematically explicit alternative to reductionist and panpsychist strategies. The approach is sufficiently general to accommodate future empirical findings, invites application to comparative phenomenology and cognitive science, and reconfigures the scientific study of consciousness as fundamentally structural and compositional rather than causally reductive. This reframing is likely to impact future developments in the interface of theoretical neuroscience, philosophy of mind, and the mathematical foundations of AI.