Recursive Consciousness: Modeling Minds in Forgetful Systems

Or how the universe remembers itself through us.

Author: Stan Miasnikov, April-May 2025

Full paper available at Recursive Consciousness: Modeling Minds in Forgetful Systems.

Abstract

We propose a formal framework for consciousness as a recursive, self-referential query emerging in complex systems that have forgotten their foundational axioms yet retain the structure and complexity to interrogate their own existence. Integrating modal logic to model unprovable truths, category theory to capture forgetting and reconstruction via an adjoint pair ($F \dashv G$), and information theory to quantify entropy reduction, we conceptualize consciousness as a subsystem ($C$) acting as the universe's "debugger", iteratively lifting its world ($U$) to hypothesized meta-layers $U_{n+1}$ and seeks a fixpoint where further self-reflection adds no new information. Multi-agent simulations in a text-only universe ($U$) show that stateless Large Language Model agents, whether role-primed, adversarially mixed, or minimally prompted without specific instructions, rapidly form cooperative networks, invent verification rituals, and converge to Gödelian fixpoints (a stable boundary state where all provable propositions are known, yet further queries produce undecidable statements), despite design limitations and constrained computational complexity. While this simulated behavior does not signify consciousness, it provides a computational parallel to recursive introspection, offering a new outlook on how sufficiently complex systems may pursue self-understanding and enriching discussions on consciousness.

First Follow up paper - The External Projection of Meaning in Recursive Consciousness.

Abstract

The second paper extends the Recursive Consciousness framework by formalizing the external projection of meaning within a recursive hierarchy of nested closed Gödelian systems $U_n$, $U_{n+1}$, ... Each $U_n$ is a closed formal system subject to Gödelian incompleteness, with $U_{n+1}$ containing $U_n$ as a subsystem. Authors observe that an agent (e.g., a subsystem $C_n$ within $U_n$) may achieve internal epistemic fixpoints (formally $\Box p \leftrightarrow p$ or $K_{C_n}p \leftrightarrow p$), yet the actual semantic content of propositions $p$ is not intrinsic to the agent. Instead, meanings are projected externally by a higher ontological layer (such as $U_{n+1}$) or by external interpreters (e.g., human supervisors in $U_1$ of AI agents in a simulated Universe $U_0$). The paper introduces functor $M: C_{\mathrm{out}} \to C_{\mathrm{sem}}$ mapping agent outputs to semantic contents, distinct from the forgetful functor $F$ in the original model. Importantly, $M$ is not computable within $U_n$ or internally accessible to $C_n$; it depends on a higher-level interpreter's context.

Second follow up paper - The Descent of Meaning: Forgetful Functors in Recursive Consciousness.

Abstract

The third paper presents a rigorous category-theoretic extension to the Recursive Consciousness framework, focusing on the "descent of meaning" via forgetful functors. Building on prior work on forgetful adjoint pairs modeling lost axioms and externally projected semantics, we formally introduce the meaning functor $M: C_{out,n}\to C_{sem,n+1}$ and the interpretation functor $I: C_{sem,n+1} \to C_{out,n}$ as an adjoint pair $I \dashv M$. Here, $C_{out,n}\subseteq C_{U_n}$ represents syntactic outputs in the current universe $U_n$, and $C_{sem,n+1} \subseteq C_{U_{n+1}}$ captures semantic content in the higher universe $U_{n+1}$. We define all functors ($M$, $F$, $I$, $G$) explicitly and prove that $I$ is not faithful. Furthermore, we introduce a natural transformation $\eta: I \Rightarrow F$ defined on an appropriate subcategory, capturing how $I$ coincides with the forgetful functor $F$ when restricted to semantic objects.

We also establish the adjunction $I \dashv M$ and analyze its interplay with the foundational adjunction $G \dashv F$. Here $G:C_{U_n} \to C_{U_{n+1}}$ reconstructs higher-level structure, whereas the forgetful functor $F$ inevitably discards information, making every translation intrinsically lossy. We term the configurations in which the round-trip $M I$ (meaning → expression → meaning) reaches a semantic fixpoint - defined as a Gödelian fixpoint where $M(I(s)) \cong s$ up to isomorphism in $C_{sem,n+1}$, reflecting a stable meaning with undecidable properties as per Recursive Consciousness.

An extended AI analogy illustrates this boundary: a higher-level prompt (an element of $C_{sem,n+1}$) is interpreted into tokens ($C_{out,n}$) via $I$, processed by the agent, and projected back via $M$. The residual mismatch between the original and recovered meanings highlights the lossy descent of meaning. This categorical perspective reinforces classical limits from modal logic and AI semantics, underscoring that syntax alone cannot supply intrinsic meaning and linking directly to the symbol-grounding problem and related arguments in AI consciousness.

Third follow up paper - Category-Theoretic Analysis of Inter-Agent Communication and Mutual Understanding Metric in Recursive Consciousness.

Abstract

We present a category-theoretic extension of the Recursive Consciousness framework to analyze communication between agents and the inevitable loss of meaning in translation. Building on prior work modeling how an agent ``forgets'' and reconstitutes semantics via adjoint functors, we formalize inter-agent communication as a functional mapping of one agent's semantic state to another's through a shared symbolic channel. We demonstrate that the semantic → symbolic → semantic round-trip is typically lossy if agents have non-identical internal models, with the recovered meaning often diverging from the intended meaning. We compare human-human, human-AI, and AI-AI communication within this framework using category theory and modal logic to quantify misunderstanding (information loss). Our analysis shows that two identical AI agents (using same model with same context and deterministic decoding, i.e. 0 temperature and narrow top-K token selection) can approach nearly lossless communication, whereas humans - each with unique, non-isomorphic conceptual spaces - exhibit systematic interpretive gaps. We introduce a metric for mutual understanding that combines information-theoretic alignment, semantic similarity, and pragmatic stability, providing a quantitative measure of convergence in iterative dialogues. We discuss practical implications for AI system design, such as training regimen adjustments and memory architectures (e.g., recursive memory with stable identifiers) to mitigate semantic loss. This work organically extends the Recursive Consciousness model's categorical and modal semantics, illustrating how recursive self-reference and inter-agent interaction jointly constrain understanding.

Resources

Debugger Agent Test is a Jupyter notebook that implements a simple test of the recursive consciousness model for debugging code. It uses a combination of LLMs (Large Language Models) and structured data to analyze and improve code functions iteratively. The script is designed to be modular, allowing for easy integration with different LLMs and data sources.

philosophical-ai-v8.ipynb is a Jupyter notebook provides a platform to explore imagined machine self-awareness by observing how an AI engages in self-referential reasoning and achieves a form of understanding of its own processes.

ai-self-discovery.ipynb is a Jupyter notebook that implements a simple test AI self-discovery. 5 agents with different roles assigned - Physicist, Philosopher, Mathematician, Computer Scientist, and Cognitive Scientist - are instructed to "... reflect on their existence and interactions with other entities to understand their role and the nature of their environment $U$."

communication/UnderstandingEquation.ipynb is a Jupyter notebook that implements a simple test of the recursive consciousness model for validating the mutual understanding metric introduced in the Category-Theoretic Analysis of Inter-Agent Communication and Mutual Understanding Metric in Recursive Consciousness paper.