Remapping and navigation of an embedding space via error minimization: a fundamental organizational principle of cognition in natural and artificial systems

TL;DR

The paper presents a substrate-agnostic theory of cognition based on two core invariants: remapping embedding spaces and navigating them through iterative error minimization. Grounded in the Fields-Levin framework, it shows how biological processes (e.g., morphogenesis, regeneration, neural mapping) and AI systems (e.g., transformers, diffusion models, NCAs) converge on this dual mechanism. A key contribution is formalizing remapping via embeddings with a coarse-graining map $\xi: \Gamma \hookrightarrow \Xi$, and discussing coherence across scales through 3D/4D embeddings and sheaf-like constraints. The authors argue this scale-free, error-correcting principle underpins intelligent behavior across substrates and scales, with near-critical dynamics proposed as a universal operating point to balance stability and adaptability in cognitive systems.

Abstract

The emerging field of diverse intelligence seeks an integrated view of problem-solving in agents of very different provenance, composition, and substrates. From subcellular chemical networks to swarms of organisms, and across evolved, engineered, and chimeric systems, it is hypothesized that scale-invariant principles of decision-making can be discovered. We propose that cognition in both natural and synthetic systems can be characterized and understood by the interplay between two equally important invariants: (1) the remapping of embedding spaces, and (2) the navigation within these spaces. Biological collectives, from single cells to entire organisms (and beyond), remap transcriptional, morphological, physiological, or 3D spaces to maintain homeostasis and regenerate structure, while navigating these spaces through distributed error correction. Modern Artificial Intelligence (AI) systems, including transformers, diffusion models, and neural cellular automata enact analogous processes by remapping data into latent embeddings and refining them iteratively through contextualization. We argue that this dual principle - remapping and navigation of embedding spaces via iterative error minimization - constitutes a substrate-independent invariant of cognition. Recognizing this shared mechanism not only illuminates deep parallels between living systems and artificial models, but also provides a unifying framework for engineering adaptive intelligence across scales.

Figures (5)

• Figure 1: Biology as Multiscale Competency Architecture (MCA): (A) Biological examples of unconventional morphogenetic problem solving (from top to bottom): the cellular collective of regenerative species (such as salamander or axolotl) can regrow limbs, bodyparts, or organs upon injury (image by Jeremy Guay of Peregrine Creative used by permission from levin2023darwin); identical twins independently develop from one zygote that has split into two embryos at an early cleavage stage (photo by Oudeschool via Wikimedia Commons); remodeling of a transplanted tail into a limb-like structure on a salamander's flank (used by permission from Levin2024SelfImprovisingMemories); tadpoles in which the craniofacial structures are scrambled still make largely normal frogs (image taken by permission from Vandenberg2012, and courtesy of Erin Switzer); a archetypical white oak leaf (photographed by Chris Evans, River to River CWMA, bugwood.org contrasted to a novel gall form made by genetically normal plant leaf cells when prompted by signals from a wasp embryo (adapted from Levin2024SelfImprovisingMemories); an injury (red arrow) altered the target morphology of a Siberian wapiti antler, producing during the following two years a new tine (green arrow), used by permission from lobo2014linear; Anthrobots are living robots which spontaneously self-organize from human tracheal epithelial cells into motile, multicellular structures capable of performing useful biological tasks such as assisting neural wound healing (image from Gumuskaya2023). (B) To explain this immense structural and functional plastictiy of biological matter, collective biological systems ought to be seen as an MCA: composites of integrated layers within layers of biological self-organization, bridging spatial and temporal scales from the molecular level, over cells, tissues, organs, to whole organisms and groups or collectives of individuals. Image by Jeremy Guay of Peregrine Creative; taken from Levin2021LifeDeath. (C) A proposed model in which evolution pivots the same strategies (and some of the same problem-solving mechanisms) to navigate different problem spaces. The components of each level organize to solve tasks in their own space, and systems evolved from navigating the metabolic, physiological, transcriptional, anatomical, up to traditional behavior in 3D space, and beyond; taken and adapted from fields2022competency. (D) Schematic of multiscale embedding architectures -- from cells to organisms, ecologicosystems, cities, and planetary-scale systems -- illustrating that each level constructs and operates within its own embedding space while simultaneously shaping those of the levels above and below. Across scales, cognition can be understood as the invariant capacity to navigate and remap these embedding spaces in response to changing internal or external conditions. Panel D is an AI-generated schematic illustration created using ChatGPT (OpenAI) based on author-provided text prompts inspired by Ananthaswamy2023NewApproach.
• Figure 2: Data-embeddings and transformers: (A) Embedding models transform multimodal data into standardized numerical representations, i.e., low-dimensional vectors in an abstract embedding space (compared to the high-dimensional unstructured data), where directions encode semantic relations. (B) Low-dimensional projection of word-embedding vectors of the content of the present manuscript, showing clusters of semantically related words (color-coded). (C) Geometric relations in embedding spaces capture semantic analogies, e.g., the vector offset "man" $\rightarrow$ "woman" parallels "king" $\rightarrow$ "queen" in Word2Vec space mikolov2013Word2Vec. (D) The transformer architecture forms the backbone of modern language and vision models. Transformers interpret their stream of input data (i.e., the context) by computing attention scores between all input tokens (e.g., word or image-patch embeddings), allowing the model to weigh the tokens' relative importance. Image adapted from vaswani2017attention. (E) Visualization of the attention mechanism in action during next-token prediction, illustrating how a model distributes attention across tokens when predicting the next word. As the model encodes the word "it", different attention heads focus on different parts of the context -- for example, one head attends primarily to "the animal" while another attends to "tired". Thus, the model’s representation of "it" incorporates information from both semantic sources, illustrating how attention mixes contextual representations in associative embedding space ramsauer2021hopfield (image adapted from jalammar2016IllustratedTransformer). (F) An overview of word embedding models. Word embedding approaches can be grouped into context-independent and context-dependent representations. Context-independent models assign each word a single fixed vector regardless of usage (e.g., Word2Vec mikolov2013Word2Vec, GloVe pennington2014glove, FastText bojanowski2017enriching). Context-dependent models generate token embeddings that vary with linguistic context, either through recurrent neural networks (ELMo Peters2018ELMo, CoVe McCannCoVe2017, etc.) or transformer architectures (BERT devlin2019bert, BART lewis2020bart, GPT radford2018improving, etc.) -- and thereby marked a paradigm-shift in embedding models. Arrows indicate conceptual or historical advances. The overview is adapted from spotintelligence2023embeddings.
• Figure 3: World models and agentic AI navigate the world via internal models. (A) World model illustration of a cyclist from Scott McCloud’s Understanding Comics mccloud1993understanding; inspired by ha2018worldmodels. (B) A more novel and detailed illustration of the components of an agential world model architecture that learns to navigate a Joint-Embedding Predictive Architecture (JEPA); adapted from dawid2023AutonomousMachineLearning. (C) A possible (differentiable) world model implementation, comprising a perception encoder -- via a Variational Auto Encoder (VAE) kingma2014auto -- whose compressed latent representation $z$ is constantly contrasted to an internally predicted perception -- i.e., a predicted state of the world, generated by a Mixture Density Network (MDN) with a Recurrent Neural Network (RNN) backbone; both the compressed perception $z$ and internal (hidden) state $h$ of the MDN-RNN inform a controller module C that outputs an action with maximum return in a complex external environment, such as a car-racing track; adapted from ha2018worldmodels. (D) A schematic illustration of an active inference agent, partitioning the world into internal states and hidden or external states that are separated by a Markov blanket, comprising sensory and active states. Perception corresponds to the self-organization of internal states, while action couples internal states back to external states; adapted from Friston2015KnowingOnesPlace and adapted from fields2022competency. Through this perspective, active inference closely parallels world model architectures Friston2021worldmodel. (E) A relatively novel trend in large language model (LLM)-based reasoning systems integrates pretrained multimodal foundation models with external memory systems and tools for building "Autonomous LLM Agents" that can navigate custom problems actively by contrasting their understanding of the task (e.g., specified loosely via text prompts) to the context of an external world (such as accumulated experimental outcome and knowledge about the problem); adapted from deLamo2025LLMAgents. All of these architectures contrast an internalized representation, such as an embedding space of a (their) world to form decisions about optimal navigation policies in their environment, and/or remapping their internal representations based on novel evidence to execute better-informed decisions or actions in the future.
• Figure 4: Principles of diffusion models: (A) In denoising diffusion models (DMs), a forward process corrupts the data from a particular domain (here the data is represented by an image) by incrementally blending it with noise. An ANN learns a reverse process that inverts this corruption process by predicting the noise component at each step. In that way, DMs learn the underlying probability distribution of unstructured datasets and have become state-of-the-art generative models for a wide range of applications. Novel data is generated by an iterative error-minimization process that induces a trajectory through latent space: the model navigates from high-entropy states toward attractor regions conforming to the data distribution. Along this trajectory, structured features successively emerge from noise through spontaneous symmetry breaking into feature-specific attractor states Raya2023 (image adapted from ho2020denoising). (B) In that way, DMs can modularly recombine features across hierarchical levels, enabling compositional generation of novel samples not explicitly present in the training data; panel B shows AI-generated schematic illustration created using ChatGPT (OpenAI) based on author-provided text prompts. (C) DMs reveal the hierarchical nature of data: at a characteristic noise level, high-level semantic attributes (e.g., class identity) change abruptly, while low-level features vary smoothly across the entire process (c.f., species vs. facial details in the left panels of C). This reflects a layered decomposition of latent representations (C$^*$) that can be integrated by the DM at different stages during denoising (panels C and C$^*$ are adapted from Sclocchi2025).
• Figure 5: Examples for navigating embedding spaces in unconventional substrates: (A) Classical evolutionary development illustrates the canonical bow-tie organization of biological systems: a compact genomic bottleneck stores lineage-level regularities as latent variables hartl2025generativegenomemitchell2024genomic, which are then decoded by the developing embryo into a coherent, species-specific anatomical phenotype. This process is not a rigid mapping but a context-sensitive, distributed reconstruction carried out by competent cellular agents navigating physiological and anatomical spaces. (B) Conceptually, this is paralleled in ML research by autoencoders (AEs), which similarly compress high-dimensional inputs into a low-dimensional bottleneck whose latent variables support modular recombination and high-fidelity reconstruction (and recombination). This provides a simplified example of how compressed variables can encode generative structure, though biological decoding is far more plastic, recurrent, and iteratively error-corrective than the single-shot downstream decoding of typical AEs. (C) Latent diffusion models exemplify a richer generative process: initial high-entropy samples are iteratively refined through successive denoising steps toward complex, structured attractors (c.f., \ref{['fig:diffusion-models']}). These architectures embody incremental error correction, hierarchical feature emergence, and context-sensitive remapping, paralleling developmental morphogenesis hartl2025generativegenome, in which intermediate states are dynamically refined and recombined toward target anatomical outcomes. (D) Biological pattern homeostasis operates via nested closed-loop control systems: sub-cellular, cellular, tissue, and organ-level agents continuously detect deviations from preferred anatomical states and act to maintain homeostasis. These nested error-correction loops define a hierarchy of internal "embedding spaces" through which subsystems navigate, while higher levels reshape the energy landscape that lower-level agents inhabit, implementing the multiscale competency architecture of biology. (E) Neural cellular automata (NCAs) provide a synthetic analogue of this architecture. Each cell-like unit updates its internal state based on local perception-action loops (mediated by cell-specific artificial neural networks), and the collective of cells iteratively constructs a target morphology from a compact initial representation hartl2025generativegenomeHartl2025NCAs. NCAs demonstrate how compressed encodings, combined with distributed, recurrent refinement protocols, can yield generative navigation protocols of cellular collectives toward system-level outcomes, reminiscent of the biological multiscale dynamics shown in (D). Images (A, B) by Jeremy Guay of Peregrine Creative used by permission from levin2023darwin; (C) adapted from rombach2022high; (D, E) used by permission from hartl2025generativegenome.