Author:  Richard E Laitinen
Affiliation: Educational and Developmental Therapies, Inc.
Date: 05/10/2025

Abstract

This paper proposes a systems-level model that integrates Calabi–Yau topologies with principles of Relational Frame Theory (RFT), percolation theory, and neurocognitive modeling. It frames derived relational framing as a multi-dimensional, network-driven process involving phase transitions and combinatorial density. Practical examples and developmental implications are offered, and logical consistency is reviewed. Alternative topological models are also discussed for future expansion of the framework.

Calabi–Yau Manifolds as Higher-Dimensional Topologies of Relational Hubs in RFT

Relational Frame Theory (RFT) seeks to account for the generativity, flexibility, and complexity of human language by modeling cognition as a network of derived relational frames. As language behavior becomes increasingly abstract and multidimensional, the field has faced conceptual and quantitative challenges in representing the full extent of relational complexity, especially as repertoires develop combinatorially and exhibit emergent properties. This paper introduces the Calabi–Yau manifold as a useful topological and geometric metaphor for representing these symbolic structures, offering a formally rich model for encoding the curvature, compactification, and entanglement of relational systems.

Calabi–Yau manifolds are well-known in theoretical physics for supporting the compactification of additional dimensions in string theory (Candelas et al., 1985). They preserve internal consistency, allow multidimensional folding, and maintain symmetry-preserving transformations. These mathematical features have strong metaphorical and structural parallels with advanced relational framing—where learners integrate multiple relational types across various contexts into a coherent symbolic system. Just as Calabi–Yau manifolds provide a substrate for vibrational modes in higher-dimensional strings, they can also serve as a model for symbolic propagation across embedded relational domains, both taught and derived.

This proposal is motivated by the increasing need in RFT and applied behavior analysis to represent not only the accumulation of relational frames but also their dynamic interaction, density, and transformation across learning histories. The structure of Calabi–Yau spaces mirrors the development of relational hubs (Ming, 2020, 2021), in which multiple relational types converge, and fluency emerges through repeated engagement. Furthermore, their curvature and holonomy properties parallel transformational changes observed in derived relational responding, offering a potential bridge between abstract topological modeling and neuro-symbolic learning mechanisms.

In integrating this manifold into a developmental and applied framework, we aim to model how symbolic knowledge transitions from discrete, directly taught elements into a high-dimensional, compact, and efficient symbolic space. The use of Calabi–Yau manifolds thus becomes a reasonable and powerful framework for modeling the emergence and transformation of complex language and cognition across individual development, instructional contexts, and symbolic domains.

Relational Frame Theory (RFT) conceptualizes cognition and language as emerging from networks of relational framings and relational networks (e.g., coordination, opposition, hierarchy, comparison) governed by mutual entailment, combinatorial entailment, and transformation of function.

Relational hubs are network nodes (stimuli or concepts) with high connectivity across multiple relational frames, facilitating symbolic generalization and combinatorial density (Ming, 2020).

Calabi-Yau manifolds are complex, compact, Ricci-flat spaces from string theory. Here, they are theorized to support entangled configurations that can be metaphorically linked to directly taught and derived relational edges in RFT, and we hypothesize that they offer a topological metaphor and formal model for the multi-dimensional relational networks generated by advanced human cognition as conceptualized in RFT. Specifically:

Relational hubs, when extended beyond conventional semantic-spatial networks, begin to fold into higher-order relational geometry, which may resemble Calabi-Yau topologies.

These manifolds can encode overlapping, nested, and entangled relational systems (e.g., simultaneous equivalence, opposition, and hierarchy) without contradiction, similar to how Calabi-Yau spaces support stable entangled structures, analogous to relational dependencies taught through multiple exemplar instruction or derived through mutual and combinatorial entailment.

In RFT, multi-hub convergence zones (e.g., nodes linking across multiple relational frames and domains) may form topological folds that compactify relational complexity into cognitively accessible forms—mirroring how extra dimensions in Calabi-Yau spaces are "rolled up" yet essential.

Implications for Higher-Dimensional Relational Networks: Dimensional Extension of Hubs

Traditional relational hubs operate in 2D/3D graph space.

When combinatorial entailment scales nonlinearly, the relational structure may require representation in higher-dimensional curved space.

C-Y manifolds offer complex curvature, non-Euclidean connectivity, and symmetry-preserving compactification, modeling how massive symbolic systems (e.g., language, metaphor) can remain tractable to a finite human mind.

Transformation of Function as Topological Twist:

Just as C-Y manifolds preserve holonomy (transport around closed loops leads to rotation), transformation of function in RFT can be conceptualized as a loop-induced property transformation.

This suggests functional shifts in derived relations are not simply edge reassignments but topological movements through a folded relational network.

Entangled Symbolic Systems:

Concepts like metaphor, irony, and abstraction involve non-linear relational embeddings—these may map onto non-trivial cycles or holes in Calabi-Yau-like topologies, where conceptual movement cannot be reduced to linear graph traversal.

Such embeddings require multi-dimensional representations, consistent with the multi-cycle Calabi-Yau structure.

Formal Model Sketch (Conceptual)

Let G = (V, E, F), where G is a relational network in RFT.

Define:

A local relational patch Ui ⊂ G centered on a relational hub hi ∈ V.  

The set of all such patches forms a relational atlas over GG.

Explanation of Symbols:

G = (V,E,F):

This denotes a relational network where V is the set of vertices or nodes (e.g., stimuli or concepts), E is the set of edges (e.g., relational links between stimuli), and F is a function or set of labels describing the type of relation (e.g., coordination, comparison).

Ui ⊂ G

This represents a subnetwork or subset of the full graph GG, called a local relational patch. It's a small, localized portion of the full network, centered on a key node.

Hi ∈ V

This means that the relational hub (Hi ∈ V) is a specific node in the larger set of vertices (V). A hub is typically a highly connected or central concept in the network.

Interpretation

This expression defines a localized cognitive or symbolic neighborhood within the broader relational network. It centers on a relational hub (a concept like “dog” or “bigger”) and includes all the directly and closely connected relational frames involving that hub. In the model, these local patches can be mapped onto higher-dimensional structures like Calabi-Yau manifolds for analysis of symbolic complexity and transformation.

See Appendix B for explanation of symbols and logical relations

Assume each patch admits a local coordinate chart mapping its internal relational frames to a higher-dimensional manifold M, such that:

Φi : Ui → M, M ≈ Calabi-Yau

The gluing functions 

ϕij = ϕi ∘ ϕj

preserve relational consistency (mutual entailment, etc.), forming a sheaf-like structure.

If the global relational manifold is a Calabi-Yau fiber bundle over the space of symbolic functions, then the functional shifts (ToF) and network phase transitions (e.g., derived generalization) can be modeled as morphisms between sections of this bundle. 

In this context, a morphism refers to a structure-preserving transformation between objects—in this case, between sections of the Calabi–Yau fiber bundle that represent local symbolic networks governed by specific contextual cues. In category theory, morphisms maintain the internal consistency of relational structures while enabling their reconfiguration across different contexts or functional demands. This is particularly important in Relational Frame Theory, where transformation of function (ToF) entails the contextual reassignment of stimulus properties without disrupting the integrity of the relational system.

When we model ToF and derived generalization as morphisms, we are treating symbolic transformation as continuous, coherent shifts rather than disconnected or arbitrary changes. Morphisms thus allow symbolic systems to retain global consistency while locally adapting to new contingencies or frames. For instance, a stimulus that once evoked avoidance may, through recontextualization, evoke approach behavior—yet the relational structure (e.g., “more than,” “opposite of”) surrounding it remains intact.

Crucially, morphisms also have topological consequences. As relational systems undergo functional shifts, the connectivity, density, or curvature of the symbolic manifold may change. These changes reflect not just the addition of new nodes or edges, but reorganization of existing pathways, leading to increased symbolic efficiency or new emergent generalizations. For example, when a new relational hub is formed (e.g., “justice” integrating legal, moral, and social domains), morphisms allow multiple symbolic structures to fold into a compact, integrated representational space.

This perspective helps explain how individuals acquire increasingly flexible, abstract repertoires over time—by undergoing topological transformations that preserve coherence while enhancing generalization. It also aligns with neurodynamic evidence (Bassett & Sporns, 2017; Bullmore & Sporns, 2009; Park & Friston, 2013; Sporns, 2011) that large-scale brain networks reorganize their topology (e.g., modularity, path length) in response to learning and task demands (Sporns, 2011), further supporting the relevance of morphism-based modeling in understanding symbolic behavior (see “Notes” section following References).

Practical Examples Using Core Relational Frames:

Coordination: A learner is taught that "cat" = "feline" and "feline" = "pet." Within a Calabi-Yau-like topology, these equivalence frames form a folded loop that allows generalization to new contexts (e.g., "pet" = "cat") via topological proximity. As more coordination frames accumulate across hubs, percolation thresholds are crossed and a dense network of synonyms emerges.

Distinction: A child learns "red ≠ blue" and "circle ≠ square." When these are linked within a Calabi-Yau framework, the multidimensional space allows embedding distinctions across stimulus classes (e.g., "colors" ≠ "shapes"), leading to emergent generalization that is topologically supported by curvature separation and relational divergence.

Comparison: Teaching "a car is faster than a bike" and "a plane is faster than a car" creates an ordinal chain. Within the framework, comparison frames add directional structure across relational hubs. When mapped onto the manifold, the curvature encodes directionality and magnitude, helping the learner derive novel comparative relations like "a plane is faster than a bike."

Theoretical and Applied Consequences: Developmental Trajectories and Lifespan Implications

This framework provides a topological metaphor for understanding how relational repertoires evolve across the lifespan. In early development, relational framing begins with isolated and concrete associations (e.g., simple coordination). As direct instruction and environmental exposure accumulate, learners acquire increasingly complex relational structures that interconnect across domains.

The Calabi–Yau structure metaphorically represents how initially discrete and low-dimensional relations become compactified into multi-dimensional symbolic systems without overwhelming the learner’s cognitive resources. Just as extra dimensions in Calabi–Yau manifolds are folded into stable forms, symbolic functions are increasingly compressed into fluently accessible patterns via derived relational responding.

Percolation theory adds a critical dynamic: developmental progress occurs not as a linear accumulation, but through threshold-dependent transformations. Once a certain combinatorial density is achieved, relational hubs interconnect explosively, giving rise to qualitative shifts in symbolic understanding (e.g., analogical reasoning, metaphor, abstraction). These developmental transitions are characterized not just by more relations, but by new configurations of connectivity—topological phase shifts.

This view aligns with neurodevelopmental evidence: cortical maturation and synaptic pruning result in denser, more efficient networks that mirror the relational compaction and restructuring described here. Thus, the Calabi–Yau model does not merely describe symbolic structures—it offers a generative map of how those structures may emerge, stabilize, and transform across developmental time.

Neurobiological Plausibility: This framework aligns with emerging understandings of neurobiological processes involved in higher-order cognition. Neural networks—especially in the cortex—exhibit properties analogous to relational networks, including bidirectional synaptic connectivity, recurrent loops, and modular architectures. Hebbian plasticity ("cells that fire together wire together") supports the development of functionally entangled connections, paralleling the directly taught and derived relational edges conceptualized in this framework. Additionally, small-world and scale-free properties in cortical connectivity mirror the high clustering and hub centrality of relational frames. The Calabi-Yau analogy offers a representational metaphor for the multi-dimensional embedding of symbolic relations, with curvature and holonomy corresponding to feedback loops, representational binding, and contextual transformation in neural dynamics.

By integrating Percolation Theory into this Calabi–Yau relational topology, the framework may promote Combinatorial Density through the probabilistic emergence of connected relational clusters. Percolation thresholds—critical points at which sparse relational edges suddenly form dense networks—can model the non-linear growth of relational framing capacities. In this context, Calabi–Yau manifolds act as the latent geometric substrate over which relational connections propagate, and percolation theory describes the tipping point at which derived relational responses saturate across domains.

This overlay enables the system to model both the topological constraints and stochastic expansions of relational learning, supporting more precise forecasts of when generalized derived relational framing will emerge.

By integrating Percolation Theory into this Calabi–Yau relational topology, the framework may promote Combinatorial Density through the probabilistic emergence of connected relational clusters. Percolation thresholds—critical points at which sparse relational edges suddenly form dense networks—can model the non-linear growth of relational framing capacities. In this context, Calabi–Yau manifolds act as the latent geometric substrate over which relational connections propagate, and percolation theory describes the tipping point at which derived relational responses saturate across domains. This overlay enables the system to model both the topological constraints and stochastic expansions of relational learning, supporting more precise forecasts of when generalized derived relational framing will emerge.

This model permits formalization of RFT phase transitions as topological bifurcations, within a geometric interpretation of relational density and fluency, with higher-dimensional embedding correlating to increased generalization potential. As such, the model suggests that the "cognitive compactification" of vast relational networks (e.g., human language) may follow structural rules similar to Calabi-Yau compactification, where only directly taught or derived relational edges remain accessible on the cognitive surface, while opening a bridge between quantum cognition models and RFT via shared geometry.

The model is grounded in and extends the applied work of Ming (2020, 2021), who operationalized the development of relational framing skills through structured curriculum, measurement of fluency, and the architecture of relational hubs. These hubs—defined as nodes of high relational connectivity across frame types—are represented in this framework as topologically significant structures that fold symbolic relations into cognitively efficient forms. Their high degree of connectedness and density mirrors the compact, curvature-based properties of Calabi–Yau spaces, especially where symbolic flexibility and generalization emerge.

In earlier sections, the framework was used to map percolation thresholds (Developmental Trajectories), network phase transitions (Implications for Relational Networks), and the formal structuring of frame transformation through sheaf-like gluing functions (Formal Model Sketch). The use of topological analogies provides both a formal basis for describing symbolic learning and a platform for quantifying how complex relational repertoires evolve—conceptually, developmentally, and instructionally.

This integration supports multiple levels of analysis:

• From the instructional level, it formalizes how curriculum density and fluency-building procedures (Ming, 2021) relate to symbolic compaction.

• From the developmental perspective, it models relational acquisition as undergoing nonlinear growth through topological bifurcations (Barnes-Holmes et al., 2004).

• From the neuroscientific angle, it parallels known organizational principles of brain development, such as modularity, small-world connectivity, and dynamic reconfiguration (Sporns, 2011).

Ultimately, the Calabi–Yau framework synthesizes these findings to propose that symbolic cognition does not simply accumulate but transforms: folding into high-dimensional structures whose surface behaviors manifest as flexible, fluent, and generative language. This geometrically motivated model provides RFT with a systems-level language and visual-metaphorical precision capable of advancing both theoretical coherence and applied instructional design.

Developmental Trajectories and Lifespan Implications

Relational Frame Theory (RFT) posits that the core of complex language and cognition emerges through relational framing—a repertoire that becomes more abstract, generalized, and flexible across development (Hayes, Barnes-Holmes, & Roche, 2001). The proposed Calabi–Yau-based framework provides a geometric and topological metaphor for understanding this transformation over time. Early in life, children acquire simple directly taught relations—such as basic coordination (e.g., 'cat' = 'kitty') and distinction (e.g., 'cat' ≠ 'dog'). These can be represented as isolated nodes or minimally connected relational subgraphs.

As instruction and environmental contact increase, relational frames multiply and begin to form combinatorial networks. The Calabi–Yau manifold metaphor captures this evolution by illustrating how symbolic relations do not expand linearly but compactify into increasingly multidimensional, internally consistent spaces. Much like extra spatial dimensions in Calabi–Yau geometry are 'folded' yet integral, symbolic relations in the developing mind become internally reorganized into fluently accessible hubs. These relational hubs act as convergence points, representing domains of high combinatorial density and cross-frame interaction.

During sensitive periods or instructional breakthroughs, relational expansion undergoes percolation—a network phase transition where previously sparse or isolated relations suddenly form densely connected webs. This shift mirrors observations in applied developmental research, such as the spontaneous emergence of analogical reasoning or hierarchical categorization once a critical mass of training is reached (Barnes-Holmes et al., 2004). These developmental leaps are not merely additive but represent topological reorganizations—transforming the learner’s symbolic landscape.

This model aligns with neurodevelopmental literature describing the brain’s evolving architecture. Research in developmental cognitive neuroscience reveals that the cortex organizes itself into increasingly efficient modular and hierarchical structures (Sporns, 2011; Johnson, 2011). These brain networks exhibit small-world and scale-free properties—mathematical patterns that parallel the structure of dense relational hubs. Moreover, the consolidation of symbolic knowledge resembles synaptic pruning and re-weighting processes observed in the maturing brain.

This topological view also supports lifespan applications. In adolescence and adulthood, as abstraction increases and metacognition strengthens, relational frames often become deeply embedded within hierarchically nested structures. These may correspond to higher-dimensional layers in the manifold metaphor. Conversely, in cognitive aging or developmental disorders, degradation or disorganization of relational hubs may explain declines in symbolic flexibility or generalization.

In summary, the Calabi–Yau topological framework provides a novel, geometrically inspired model of relational development that accommodates:

• Initial acquisition of discrete frames,

• The nonlinear emergence of combinatorial density,

• Critical transitions marked by percolation thresholds,

• Neurobiological maturation of symbolic function,

• And potential regression or plateauing due to cognitive or experiential constraints.

This conceptualization not only aligns with empirical RFT research but extends it by embedding relational development within a dynamic system and topological formalism that is well-suited for modeling cognitive change across the lifespan.

Summary and Integration with Applied RFT

The integration of Calabi–Yau topological models with Relational Frame Theory (RFT) provides a multidimensional framework for understanding symbolic complexity, relational fluency, and emergent generalization. This framework complements and extends the applied work of Siri Ming (2020, 2021), who operationalized the construction and fluency of relational framing skills through systematic instruction and performance-based tracking.

Ming’s approach emphasizes the progressive training of relational repertoires—particularly coordination, distinction, comparison, temporal, and deictic frames—within structured curricula designed for learners with autism and language delays. Central to her model is the concept of relational hubs, which serve as central nodes linking across multiple relational domains. These hubs, when viewed through a Calabi–Yau lens, resemble high-curvature zones within a compact manifold—points at which symbolic density, flexibility, and derived responding converge.

This document's framework models how such hubs dynamically transform across learning contexts, with Calabi–Yau compactification representing the fluency-driven internalization of complex relational structures. Where Ming's (2020) work applies fluency measurement and direct teaching to track and construct relational abilities, the Calabi–Yau formalism accounts for the emergence, stabilization, and phase-shift generalization of those abilities across increasingly multidimensional symbolic environments.

Furthermore, the incorporation of percolation theory into the model mirrors Ming’s emphasis on combinatorial growth: as relational elements are introduced and practiced, learners reach a threshold at which spontaneous generalization and cross-frame transformation occur. This suggests a precise theoretical mechanism for the kind of relational explosion observed clinically when instructional density is sufficient.

Together, this synthesis of topological modeling and applied instructional science proposes a novel systems-level understanding of how relational framing skills evolve—both behaviorally and structurally. It bridges advanced mathematics with practical curriculum design, supporting the vision that RFT can be taught, tracked, and understood in a formally coherent, developmentally grounded, and computationally rigorous manner.

"The combinatorially dense, multi-hub architecture of advanced symbolic systems, as conceptualized in RFT, may require a topological framework capable of non-linear embedding, holonomy-preserving transformation, and dimensional compactification. Calabi-Yau manifolds—compact, curved spaces supporting entangled relational edges, both directly taught and derived—offer a compelling analogy and possible formal structure for modeling the emergence, stability, and transformation of complex relational networks in human cognition."

References

Barnes-Holmes, Y., Barnes-Holmes, D., Smeets, P. M., Strand, P. S., & Friman, P. (2004). Establishing relational responding in accordance with same and opposite as generalized operant behavior in young children. The Psychological Record, 54(1), 71–88.

Bassett, D. S., & Sporns, O. (2017). Network neuroscience. Nature Neuroscience, 20(3), 353–364. https://doi.org/10.1038/nn.4502

Bullmore, E., & Sporns, O. (2009). Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10(3), 186–198. https://doi.org/10.1038/nrn2575

Hayes, S. C., Barnes-Holmes, D., & Roche, B. (2001). Relational frame theory: A post-Skinnerian account of human language and cognition. Springer Science & Business Media.

Johnson, M. H. (2011). Interactive specialization: A domain-general framework for human functional brain development? Developmental Cognitive Neuroscience, 1(1), 7–21.

Ming, S. (2020). Building a relational foundation: Systematically teaching relational framing repertoires to learners with autism and other developmental disabilities. International Journal of Behavior Analysis & Autism Spectrum Disorders, 6(2), 23–42.

Ming, S. (2021). Relational hubs and curriculum architecture: A practical application of relational frame theory to language instruction. Behavioral Development Bulletin, 26(1), 59–73.

Park, H. J., & Friston, K. (2013). Structural and functional brain networks: from connections to cognition. Science, 342(6158), 1238411. https://doi.org/10.1126/science.1238411

Sporns, O. (2011). Networks of the brain. MIT Press.

Notes:

Bassett, D. S., & Sporns, O. (2017) - Discusses how the human brain dynamically reconfigures its network topology across development, task contexts, and cognitive demands.

Bullmore, E., & Sporns, O. (2009) - Provides evidence of small-world and scale-free topologies in brain networks and their relevance to cognitive function.

Park, H. J., & Friston, K. (2013) - Highlights how changes in brain network topology support cognitive reorganization, consistent with symbolic transformation models.

Sporns, O. (2011) - Offers foundational insight into dynamic reorganization, modularity, and efficiency in brain networks.

Appendices

Appendix A: Simpler Topological Alternatives to Calabi–Yau Models

While Calabi–Yau manifolds offer a rich, high-dimensional metaphor for modeling relational complexity, simpler topological structures may serve as effective and more accessible alternatives:

Torus Manifolds (n-dimensional Tori)

       Benefits: Retain compactness and periodicity, model closed relational loops and modular structures.

Application: Coordination and cyclical entailment structures can be modeled through phase-wrapped topological structures.

Hyperbolic Space (e.g., Poincaré Disk)

Benefits: Ideal for modeling hierarchical or exponential growth, matching combinatorial entailment.

Application: Enables representation of symbolic elaboration and generalization through radial expansion from core hubs.

Simplicial Complexes / CW-Complexes

Benefits: Discrete, additive, and geometrically simple, allowing explicit tracking of relational additions.

Application: Each new derived relation forms a higher-dimensional simplex, supporting homological analysis of relational coherence.

Fiber Bundles over Riemannian Manifolds

Benefits: Generalize the concept of layered relational systems without requiring Ricci-flatness.

Application: Supports contextual flexibility by assigning symbolic interpretations to fiber sections over meaningful base domains.

Comparative Summary

These alternatives preserve core features such as relational embedding, combinatorial scalability, and symbolic transformation, while increasing computational and empirical tractability. Future iterations of the framework could adopt or hybridize these simpler models to bridge theoretical sophistication with applied feasibility.

Appendix B: Explanation of Symbols and Logical Relations

G = (V,E,F): Graph of a relational network with:

V: nodes (stimuli/concepts)

E: edges (relational links)

F: frame type (e.g., coordination)

Ui ⊂ G: Local patch of a relational network.

Hi ∈ V: A relational hub (central, high-connectivity node).

Φi : Ui → M: Chart mapping local relational frames to manifold M.

Φij = ϕi ∘ ϕj − 1: Gluing function ensuring consistency between overlapping symbolic domains.

Calabi–Yau Fiber Bundle: A structure where base = symbolic context; fibers = local relational networks.

Sheaf-like Structure: Ensures consistency across overlapping local relational representations.

Percolation Threshold: Critical point where loosely linked symbolic clusters form large-scale generalization networks.