Abstract
Symbolic learning emerges not through isolated events but via recursive integration of derived relations across time. This paper introduces and narratively explores the phenomenon of Recursive Equivalence Collapse (REC), a hypothesized phase transition in symbolic development wherein distinct stimulus classes converge into a functional equivalence hub. Drawing on Relational Frame Theory (RFT), network science, and recent work by Siri Ming on relational language development, we propose REC as a generative mechanism for symbolic acceleration. The theory is supported by conceptual models and offers an applied pathway to language and cognition remediation, especially in learners with autism spectrum disorder (ASD).
Keywords: Recursive Equivalence Collapse, Relational Frame Theory, symbolic generalization, Siri Ming, relational language development, autism spectrum disorder, symbolic attractor, combinatorial density, stimulus equivalence, derived relational responding
Introduction
Human cognition relies fundamentally on the ability to derive relations among stimuli—relations that are not directly trained but emerge from previously established networks. This symbolic generativity, as described in Relational Frame Theory (RFT), is typically robust in fluent learners but often deficient or absent in those diagnosed with autism spectrum disorder (ASD). In recent years, interventionists like Siri Ming have drawn attention to the precise shaping and measurement of relational repertoires as a core developmental goal. What has remained theoretically underexplored is how relational systems reach a point where generalization becomes not only possible but inevitable.
We propose that such a point reflects a non-linear systems transition, termed Recursive Equivalence Collapse (REC). This occurs when relational derivations become dense and interconnected enough that distinct stimulus sets spontaneously coalesce into a single, functionally integrated symbolic structure. This narrative traces the emergence of REC and illustrates its implications for symbolic learning and autism remediation.
The Hypothesis of Collapse
Let us consider the behavior of a symbolic system not as a sequence of additive relational acquisitions, but as a dynamic field in which structure is shaped by pressure, mass, and constraint. When a learner accrues enough internal symmetry and combinatorial entailment across three or more stimulus sets, the conditions are set for a collapse of categorical boundaries—what we call Recursive Equivalence Collapse. Here, relational entropy is reduced, and symbolic cohesion emerges.
As in previous formulations of symbolic attractors (Laitinen, unpublished), the system passes a density-dependent inflection point where local derivations become insufficient to contain global logic. The outcome is a functional equivalence hub—a symbolic structure that integrates across multiple relational histories and enables novel generalizations. This hub is not just a byproduct of instruction; it is a topological consequence of the network's internal architecture.
Relational Language and Recent Insights
Siri Ming's recent contributions to relational language development provide a pragmatic roadmap for building the preconditions for REC. Her work operationalizes early relational repertoires as foundational acts of symbolic construction, guided by data, fluency, and contextual control. In this light, REC becomes the logical culmination of a properly sequenced relational curriculum.
What Ming (2019, 2020, 2021, 2022) articulates in practice, REC formalizes in theory: that symbolic generativity is not merely a product of stimulus exposure, but a measurable function of network density and derived coherence. This correspondence situates REC at the intersection of behavioral engineering and systems-level symbolic design.
Numerous studies in the equivalence and RFT literature provide empirical scaffolding for the theory of REC. Sidman's early work on stimulus equivalence (Sidman & Tailby, 1982) established the foundational concept that untrained relations could emerge via symmetry, transitivity, and reflexivity. Steele and Hayes (1991) showed that brief training on a limited set of relational cues could result in novel emergent relations, foreshadowing the efficiency and nonlinearity posited by REC. Lipkens, Hayes, and Hayes (1993) found that children could form equivalence classes through minimal training, often generating novel derivations across untrained stimulus pairs. Barnes-Holmes et al. (2001) expanded the empirical base for relational frames of comparison, opposition, and hierarchy, demonstrating that stimulus networks could reorganize relationally without additional reinforcement once certain thresholds of entailment were met. Healy, Barnes-Holmes, and Smeets (2000) illustrated how overlapping relational networks induced spontaneous class merging in adult participants, while Grady, Luciano, and Barnes-Holmes (2018) used graph-theoretical analysis to show that increased relational density and structural overlap predicted novel relational responding. More recently, Pelaez and Gewirtz (2022) documented the spontaneous emergence of analogical relations among children exposed to overlapping derived networks, lending real-world credibility to the collapse mechanism. Studies by Ming (2019, 2020, 2021, 2022) provide applied demonstrations of this pattern: once learners reach fluency in frame types and minimal linkages are introduced, generalization accelerates abruptly and systemically.
Taken together, these studies point toward a consistent empirical pattern: when a learner has acquired sufficient within-class relational density and minimal cross-class relational linkage, previously distinct networks become functionally inseparable. This aligns with the central prediction of REC.
The Mechanics of Collapse
The structural and dynamic principles of Recursive Equivalence Collapse (REC) are strongly analogous to those found in the process of protein folding, a well-known example of biological self-organization. In protein folding, a linear chain of amino acids (determined by genetic code) collapses under biochemical constraints into a stable three-dimensional structure. This folding is governed by attraction to low-energy configurations and canalization into permissible conformational pathways. Similarly, REC involves the symbolic 'folding' of relational networks: initially linear and compartmentalized stimulus classes collapse into a topologically stable and functionally generative equivalence hub.
In both systems, emergence is guided by a combination of intrinsic code (e.g., transformation functions in REC, amino acid sequence in proteins) and external constraints (e.g., cross-links or contextual cues in REC, molecular interactions in proteins). As in protein folding, where misfolding can lead to dysfunction, symbolic networks that fail to reach combinatorial density or symmetry thresholds may remain fragmented and behaviorally inert.
This structural analogy suggests that REC can be framed as a relational form of cognitive morphogenesis. Derived relational responding operates as a kind of epigenetic instruction set that, once sufficiently dense and reciprocally integrated, compels the network to collapse into a lawful, generative form—much like how hydrophobic interactions guide the folding of proteins into functional units.
REC can be understood as both an emergent attractor dynamic and a form of symbolic canalization. In dynamical systems, attractors are stable patterns toward which systems evolve under constraint. In relational networks, as combinatorial density increases and minimal cross-links are introduced, symbolic activity begins to funnel into increasingly constrained and coherent derivations—what might be described as a symbolic attractor. Simultaneously, this attractor canalizes—stabilizes and narrows—the developmental trajectory of symbolic relations, guiding further behavior along emergent, system-defined pathways. Mathematically, REC occurs when the number of derivable relations within each class reaches a saturation point. The total number of such relations is given by:
Dtotal = [NA(NA−1)/2] + [NB(NB−1)/2] + [NC(NC−1)/2]
Where NA, NB, and NC are the number of stimuli in each class. Once this combinatorial mass is sufficient, a single cross-link (e.g., A1 = B2) can propagate across the network, inducing untrained derivations like C1 = A2.
From a systemic perspective, this is not mere derivation; it is symbolic canalization. Canalization here refers to the stabilization and routing of potential relational transformations into cohesive, lawful structures. Symbolic attractors emerge not by fiat, but by the inexorable logic of recursive entailment under increasing relational density. What began as discrete stimulus clusters has converged into a condensed and recursive symbolic attractor. Such attractors possess high symbolic conductivity, minimal entropy, and low resistance to transformation, functioning as stable cores around which future learning organizes.
Structural and Functional Emergence
As explored in prior models of symbolic phase space (Laitinen, 2024), systems that reach percolation thresholds undergo spontaneous reorganization. In REC, this reorganization manifests as:
- Centrality: Stimuli with high betweenness centrality mediate relations across the network.
- Compression: Derivations collapse into fewer steps, enabling faster access to symbolic outcomes.
- Invariance: Bidirectional and transitive relations stabilize the symbolic architecture.
- Modularity: Local stimulus classes are embedded within a global relational frame.
Functionally, this means the learner gains:
- Rapid abstraction across stimulus domains
- Context-sensitive referential flexibility
- Rule-generative capacity
- Relational creativity through analogical mapping
Applications to Autism Spectrum Disorder
The collapse model provides a new lens on ASD as a symbolic engineering problem. If generalization failures result from insufficient relational mass or absence of cross-links, then the solution is architectural, not incremental. Instruction should aim to accelerate mass, density, and bridge conditions to induce systemic collapse.
This approach aligns with relational fluency interventions pioneered by Siri Ming and Fit Learning. It also extends their outcomes by providing a systems-level explanation for why generativity explodes after a threshold is reached. REC reframes symbolic delay not as deficit, but as an uncollapsed system waiting for the right structural inputs.
Broader Theoretical Implications
While REC facilitates symbolic integration and generativity, the very mechanism that promotes abstraction and coherence may also give rise to rigidity and epistemic closure. Once a functional equivalence hub is formed, the symbolic system becomes canalized—not only in its structure but in its function. This means that new stimuli, once assimilated into the hub, are interpreted through existing derived frames, even in the face of disconfirming data or external input. Thus, symbolic attractors can lead to the emergence of rigid "worldviews," or ideologically closed systems, that resist modification.
This pattern mirrors findings in cognitive science and behavior analysis where individuals show a reluctance to revise established equivalence classes even after contradictory training (e.g., Wulfert & Hayes, 1988). It also parallels the phenomenon of confirmation bias within derived relational networks, where the structure of the system itself prevents easy reconceptualization. The same canalization that allows efficient symbolic functioning may reduce symbolic plasticity.
This insight has significant implications for both educational and sociocultural contexts. Symbolic rigidity may not be a flaw of the learner but a consequence of the very efficiency and coherence the system achieves through REC. As such, interventions aimed at fostering cognitive flexibility must consider ways to destabilize overly dominant equivalence hubs—possibly by introducing disconfirming relations, increasing frame variability, or disrupting overly cohesive attractors.
Recursive Equivalence Collapse articulates a precise, generative account of how symbolic systems reorganize under constraint. Its theoretical structure aligns with prior systems work on symbolic mass, cognitive density, and attractor formation. Its applied implications point toward a radical restructuring of how we teach, model, and remediate language and cognition—especially in populations where symbolic growth has stalled.
As with other nonlinear transitions in behavior and physics, the event of collapse is sudden, but the conditions that give rise to it are measurable and modifiable. REC represents not just a descriptive theory, but a prescriptive map for symbolic transformation.