Abstract

AToM proposes that coherence—defined as the maintenance of smooth curvature, stable dimensionality, persistent topological structure, and cross-scale coupling—is the structural invariant underlying embodied, embedded, enactive, and extended cognition. Trauma is formalized as a collapse in coherence geometry; entrainment is defined as the class of dynamics that restores it across scales. Drawing from predictive processing (Friston, Clark), enactivism (Varela, Thompson, Di Paolo), interpersonal neurobiology (Siegel, Porges), and information geometry (Amari, Chentsov), AToM integrates 4E cognition within a single cross-scale geometric framework. This paper clarifies novelty, addresses potential objections, and situates AToM as a mathematically grounded extension—not replacement—of 4E theory.

1. Introduction: Why 4E Needs a Fifth Structure

The embodied, embedded, enactive, and extended (“4E”) paradigm has reshaped cognitive science by shifting the explanatory center of gravity away from internal symbol manipulation and toward organism–environment dynamics (Varela, Thompson & Rosch, 1991; Clark, 1997; Gallagher, 2005; Thompson, 2007). Its core corrective insight—that cognition is not located exclusively in the brain but arises from sensorimotor coupling, worldly scaffolding, and relational engagement—has become a foundational assumption across philosophy of mind, cognitive neuroscience, developmental psychology, and human–computer interaction. Yet for all its conceptual breadth, 4E cognition remains under-specified at the structural level. It provides a powerful ontology of where cognition happens (in bodies, worlds, and networks), how it emerges (through interaction, coupling, and activity), and why it is fundamentally relational, but it offers no generalizable mathematical or topological invariant that explains when cognition holds together, why it sometimes collapses, or how it propagates across scales from neurons to dyads to institutions.

In other words, 4E provides a distributed map of cognitive processes, but not the geometry that determines the stability or fragility of those processes.

This absence becomes most apparent in domains where cognition becomes disordered, unstable, or path-dependent. 4E provides no unified account of trauma-induced collapse in sense-making; of why interpersonal coordination sometimes produces mutual understanding and sometimes produces fragmentation or panic; of why institutions drift into incoherence despite rich informational coupling; or of how cultural narratives stabilize collective identity over generations. In each of these cases, what fails or succeeds is coherence—the capacity of a system to maintain stable, low-entropy organization under shifting constraints. Yet coherence, though implicitly assumed across 4E literatures, is never formalized as a structural property. It is treated metaphorically (e.g., “smooth coupling,” “stable attunement,” “integrated activity”) rather than as a measurable quantity grounded in established mathematical frameworks.

A Theory of Meaning (AToM) fills this explanatory gap by proposing coherence as the missing structural invariant of 4E cognition. Coherence here is not a vague descriptor but a formally tractable, cross-scale geometric construct. Drawing on information geometry (Amari, 1987; Amari & Nagaoka, 2000), dynamical systems theory (Kelso, 1995), topological data analysis (Carlsson, 2009), and empirical work on neural synchrony and interpersonal attunement (Feldman, 2012; Porges, 2011), AToM defines coherence as a tuple of measurable properties: curvature smoothness, dimensional stability, topological persistence, and cross-frequency coupling ratios. These invariants quantify how well a system maintains integrated trajectories across fluctuations, perturbations, and transitions.

This structural move allows AToM to articulate what 4E leaves implicit: that cognition requires not only embodiment, embedding, enaction, and extension, but also maintenance of coherent organization across these domains. Without a unifying structure, 4E becomes descriptively rich but explanatorily diffuse—capable of characterizing local processes but not of unifying them across levels of analysis.

The absence of a structural invariant also hampers 4E’s ability to model failure modes. Trauma, for instance, disrupts the system’s capacity to remain stably embodied, attuned, extended, or enactive. Yet 4E has no formal vocabulary for collapse—no way to describe how the manifold of possible cognitive trajectories narrows, how curvature spikes produce hypervigilance, or how dysregulation propagates across neurophysiological, behavioral, and relational levels. AToM resolves this by modeling trauma as dimensional reduction and curvature concentration within the system’s coherence geometry—a collapse detectable via measurable shifts in manifold structure, persistent homology, autonomic synchrony, and linguistic curvature.

Similarly, 4E lacks a principled account of cross-scale propagation: how micro-scale phenomena (neural firing patterns, autonomic regulation) influence macro-scale structures (interpersonal attunement, institutional trust, cultural stability). Enactivists speak of “participatory sense-making” (De Jaegher & Di Paolo, 2007), but this framework remains at the level of phenomenology and interactional description. It does not specify why certain interactions stabilize and others destabilize, nor does it provide invariants that generalize across different scales of coordination. AToM introduces entrainment—in a new, formalized, cross-scale sense—as the dynamical class of processes that preserve geodesic coherence across nested constraint manifolds. This definition integrates neural oscillation, autonomic co-regulation, conversational pacing, organizational alignment, and cultural synchronization into a unified dynamical vocabulary.

By incorporating coherence and entrainment as formal constructs, AToM not only complements 4E but completes its structural architecture. The original 4Es describe the locations and modalities of cognition; AToM adds the geometry that governs their coordinated operation. In this view:

• Embodiment provides the material substrate.
• Embedding provides the ecological scaffold.
• Enaction provides the dynamical process.
• Extension provides the distributed architecture.
• Coherence provides the structural invariant that determines whether these processes remain stable, collapse, or reorganize.

This fifth structural dimension transforms 4E cognition from an interpretive paradigm into an analytically unified framework capable of supporting falsifiable predictions, cross-domain generalization, and computational implementation.

Indeed, once coherence is formalized, a series of long-standing puzzles become tractable. For example:

• Why do certain joint actions yield fluid coordination while others devolve into conflict?
• Why do some therapeutic processes produce sudden insight (“phase transitions”) while others stagnate?
• Why do some cultural narratives stabilize collective identity while others accelerate fragmentation?
• Why do institutions sometimes behave like coherent agents and sometimes like chaotic multi-attractor systems?

Each of these can be described as coherence dynamics under constraint—a framework unavailable to traditional 4E formulations.

The contribution of AToM is therefore not the introduction of a new “E” in a branding sense, but the articulation of a structural variable that 4E theory implicitly relies upon but does not formalize. Without coherence, 4E cognition lacks a criterion for evaluating when embodied–embedded–enactive–extended systems are functioning effectively; without entrainment, it lacks an account of how these systems maintain stability across perturbations or scale their organization from neurons to cultures.

AToM thus positions coherence not as a metaphor or supplement but as the structural completion of the 4E paradigm—the invariant that allows 4E cognition to become a unified, mathematically grounded, empirically measurable science of mind.

2. Precisely What AToM Means by “Entrainment”

The term “entrainment” has a long and distinguished history across the sciences. In circadian biology, entrainment describes how endogenous oscillators align with environmental cycles such as light and temperature (Pittendrigh, 1981). In nonlinear dynamics, it refers to the synchronization of coupled oscillators through phase-locking (Winfree, 1980; Strogatz, 2000). In developmental psychology and interpersonal neurobiology, entrainment captures the rhythmic alignment of movement, affect, gaze, and autonomic states between interacting individuals (Feldman, 2012). In enactive cognitive science, particularly in participatory sense-making (De Jaegher & Di Paolo, 2007), entrainment is invoked to describe how coordinated interaction patterns sustain joint meaning-making. Across these domains, the term reliably points to processes where coupled systems stabilize one another through mutual constraint. Yet the uses remain domain-specific, methodologically heterogeneous, and conceptually unintegrated. Each tradition describes a form of synchronization, but none provides a general account of what makes entrainment a unifying principle or how it scales from physiological oscillators to interpersonal dynamics, institutions, and cultures.

AToM takes these existing insights not as competing definitions but as empirical evidence that entrainment is a cross-domain regularity awaiting formalization. What AToM contributes is a structural, topological, and information-geometric articulation of entrainment that explains why these diverse phenomena share a common profile and how they instantiate a deeper invariant. In AToM, entrainment is defined as the class of dynamics that preserves geodesic coherence across nested constraint manifolds. This definition moves beyond synchrony or coordination in the narrow sense. It identifies the specific mathematical and topological conditions under which systems—biological, cognitive, relational, institutional, or cultural—maintain low-entropy organization under perturbation. It provides the missing geometry that links oscillation, sense-making, and social coordination within a single explanatory scaffold.

The core of this definition is the notion of geodesic coherence: the extent to which a system’s trajectory remains smooth, stable, and integrable relative to the curvature of its underlying manifold. In information geometry, systems traverse statistical manifolds shaped by their generative models, priors, and likelihood functions. In dynamical systems theory, trajectories unfold on state-spaces whose topology constrains possible behavior. In relational and cultural systems, trajectories unfold on constraint landscapes shaped by norms, expectations, roles, incentives, and symbolic structures. Across these levels, coherence corresponds to motion along low-curvature paths that preserve internal consistency and minimize divergence. Entrainment, in the AToM sense, is the process that keeps these trajectories aligned despite local fluctuations, conflicting signals, or environmental perturbations.

To make this precise, AToM grounds entrainment in three measurable constructs: multi-frequency phase alignment, curvature flow, and persistent homology. Multi-frequency phase alignment captures synchronization across heterogeneous oscillatory processes—neural rhythms, autonomic cycles, turn-taking patterns, institutional feedback loops—allowing systems of different temporal scales to couple without collapsing into uniformity. Curvature flow describes how the geometry of a manifold evolves under constraints and how local deformations either destabilize or reinforce coherence. Persistent homology measures the lifespan of structural features—clusters, cycles, bottlenecks—in high-dimensional data, providing a way to track how patterns of interaction or meaning persist or disintegrate across scales.

Bringing these tools together allows AToM to characterize entrainment as a cross-scale coherence-preserving dynamic. In physiology, entrainment occurs when breathing rhythms, heart rate variability, endocrine cycles, and circadian oscillations stabilize into integrated patterns. In cognition, entrainment appears when perceptual, interoceptive, affective, and attentional processes align sufficiently to support stable action and interpretation. In interpersonal contexts, entrainment manifests in dyadic interactions where autonomic rhythms, vocal prosody, gesture, timing, and attentional arcs become mutually constraining, producing a shared basin of sense-making. In institutions, entrainment stabilizes communication loops, role structures, incentive architectures, and procedural norms. In culture, entrainment occurs through ritual, shared narrative compression, collective memory, and symbolic scaffolding that synchronize expectations across large populations.

The novelty of AToM’s formulation lies in showing that these apparently disparate phenomena all satisfy the same geometric criterion: preservation of coherence across nested constraint manifolds. Biological rhythms constrain neural processing; neural processing constrains affective appraisal; affect constrains interpersonal coordination; interpersonal coordination constrains institutional dynamics; institutions constrain cultural trajectories. The stability or breakdown of any given scale depends on how entrainment propagates through these nested structures. When entrainment holds, coherence is preserved and meaning-making remains stable. When entrainment weakens, coherence collapses and systems drift into disorganization, fragmentation, burnout, or conflict.

AToM’s definition also distinguishes entrainment from mere synchrony. Synchrony is alignment in time or phase; it can occur even in rigid or pathological systems. Entrainment, in the AToM sense, requires not only temporal alignment but maintenance of curvature smoothness and dimensional stability. A pair of coupled oscillators may synchronize while becoming increasingly fragile; a social group may fall into rigid lockstep while losing interpretive flexibility. These cases do not count as entrainment in AToM, because they fail to preserve geodesic coherence. Similarly, a culture may become synchronized through propaganda or coercion, but such forced uniformity produces topological collapse rather than stable coherence. True entrainment is stabilizing, not constraining; it preserves the dimensional richness of the system rather than collapsing it.

This distinction allows AToM to explain why certain interpersonal dynamics are nourishing while others are suffocating; why some institutions generate creativity while others generate stagnation; why some cultural formations foster resilience while others spiral into extremism or fragmentation. In each case, the question is whether the system’s coordination dynamics preserve or degrade its coherence geometry. A dyad that aligns breathing, gaze, and vocal rhythm without constraining emotional or cognitive space exhibits entrainment; a dyad that enforces mimicry or compliance exhibits synchrony without coherence. A community whose rituals maintain topological stability across generations exhibits entrainment; a community whose rituals suppress variability collapses dimensionality.

By providing a formal, cross-scale definition, AToM elevates entrainment from an evocative metaphor to a rigorous explanatory principle. It shows why entrainment appears in so many scientific domains: not because different fields happened upon the same metaphor, but because living, cognitive, and social systems must preserve coherence under constraint in order to remain viable. Entrainment is the mechanism by which this preservation occurs. It is the glue that allows 4E cognition to scale beyond local interactions, extending embodiment, embedding, enaction, and extension into multi-level systems where meaning persists across time, space, and complexity.

In this sense, entrainment is not an optional supplement to 4E cognition. It is the structural condition that allows 4E processes to remain coherent at all. It provides the integrative geometry through which organism–environment dynamics become stable enough to support memory, identity, communication, culture, and collective life. AToM articulates entrainment in its full generality: not as synchrony, not as coordination, not as resonance alone, but as the preservation of coherence across the nested manifolds that constitute human experience.

3. What 4E Explains — And What It Cannot Yet Explain

The 4E paradigm—embodied, embedded, enactive, and extended cognition—has transformed contemporary understandings of mind by breaking the hegemony of internalist, representational theories and insisting that cognition is inseparable from the body, the environment, social interaction, and material scaffolds. Yet despite its theoretical reach and philosophical influence, 4E cognition still lacks something essential: a structural invariant that explains not only what cognition is, but what keeps it together, what breaks it apart, and how it propagates across scales of organization. Without such an invariant, 4E accounts remain compelling but incomplete—descriptively rich, but structurally thin.

To see this clearly, it helps to examine each of the four Es in turn and identify both the explanatory power they bring and the methodological gaps they leave behind. These gaps are precisely where AToM intervenes: not by contradicting 4E cognition, but by supplying the geometric and topological structure that allows 4E principles to become measurable, scalable, and predictive.

3.1 Embodied Cognition: Grounded, but Geometrically Unspecified

Embodied cognition begins with the claim that the body is not a peripheral support structure but the primary medium of meaning-making. Perception is shaped by sensorimotor contingencies; action is guided by the lived body rather than abstract symbolic computation. This insight has reshaped fields from robotics to phenomenology.

However, embodied cognition lacks a structural account of how bodily processes maintain coherence. We know that interoception, autonomic regulation, proprioceptive feedback, and metabolic rhythms shape experience, but 4E theory does not specify what holds these processes together or how their integration can be quantified. Embodiment is richly described but structurally underdetermined.

AToM addresses this by introducing coherence as a measurable property of the embodied system. In AToM, embodied coherence consists of:

• curvature smoothness in interoceptive and sensorimotor manifolds,
• dimensional stability across autonomic subsystems,
• and cross-frequency coupling among physiological oscillators.

Rather than treating the body as implicitly integrated, AToM shows how the geometry of coherence—smooth curvature, persistent topological features, and stable coupling—constitutes the embodied ground from which cognition emerges. Embodiment becomes not merely the locus of cognition, but one of the primary generators and constraints of coherence.

3.2 Embedded Cognition: Contextual, but Missing Failure Modes

Embedded cognition emphasizes that organisms inhabit structured environments that shape, constrain, and enable cognitive processes. Cognition is not separable from ecological affordances, physical environments, or socio-cultural norms. This ecological sensitivity is one of 4E’s greatest strengths. Yet it lacks a formal account of what happens when embedding fails.

Traditional 4E discussions speak of “misalignment,” “breakdowns,” or “disrupted affordances,” but these are metaphorical descriptions. They do not specify the geometry of breakdown, nor do they identify what marks the transition from stable embedding to disintegration. Without a structural account, embedded cognition cannot explain why some systems remain resilient under perturbation while others collapse.

AToM fills this gap by modeling failure modes as coherence collapse within an embedded manifold. When environmental constraints exceed the system’s capacity for integration:

• curvature spikes, producing hypersensitivity and hypervigilance;
• dimensional reduction occurs, reducing available behavioral degrees of freedom;
• and topological bottlenecks emerge, constraining transitions between states.

Thus, AToM provides a principled, geometric description of embedded breakdowns. It explains why some environments promote flourishing—by preserving coherence—and why others generate trauma—by collapsing it. Embedded cognition remains essential, but AToM shows what makes embedding stable or unstable across scales.

3.3 Enactive Cognition: Dynamic, but Topologically Abstract

Enactive cognition argues that meaning arises through embodied action in the world. Sense-making is not internal representation but the ongoing negotiation of relevance, possibility, and constraint. This framework captures the lived, dynamic, adaptive nature of cognition better than any previous paradigm.

Yet enactivism does not specify the topology of sense-making. It describes interactional dynamics but leaves unclear the structure of the space in which those dynamics unfold. Without a manifold, a metric, or an invariant, enactive theory cannot distinguish stable trajectories from unstable ones, or productive interaction patterns from collapses into dissociation or noise.

AToM resolves this by defining coherence manifolds on which enactive trajectories unfold. These manifolds encode:

• the curvature of predictive and sensorimotor models,
• the dimensionality of available action-perception cycles,
• and the persistent structures that support stable sense-making.

On this foundation, enaction becomes navigable: trajectories can be modeled, stability can be quantified, and breakdowns can be formally described as distortions or collapses within the coherence geometry. AToM does not replace enactivism; it gives it the missing mathematical backbone.

3.4 Extended Cognition: Coupling, but Not Scaling

Extended cognition offers one of 4E’s most radical claims: cognitive processes can extend into tools, technologies, and social structures. Memory can be offloaded to notebooks, problem-solving can be distributed across teams, and identity can be scaffolded by digital artifacts and cultural systems. This extension thesis has been influential in cognitive science, distributed systems, and social epistemology.

But extended cognition lacks a formal account of how coupling scales from dyadic interactions to institutional and cultural systems. It tells us that cognition extends, but not how far, with what stability, or through what dynamics. It does not identify when extension preserves cognitive coherence and when it amplifies fragmentation, overload, or collective dysregulation.

AToM addresses this by introducing entrainment operators that characterize how coherence propagates across levels of organization. These operators formalize:

• cross-level synchronization (neural → interpersonal → institutional),
• mutual constraint enforcement,
• and coherence preservation across nested manifolds.

Through entrainment, AToM explains why some cognitive extensions become stable infrastructures (e.g., mathematical notation, scientific institutions) while others produce runaway feedback loops, polarization, or collapse. Extended cognition describes coupling; AToM explains its success or failure.

3.5 Why 4E Needs Coherence as Its Fifth Structure

Taken together, the limitations of the four Es point to the need for a structural completion. Embodiment identifies the locus of action, embedding maps the ecological surround, enaction characterizes the dynamics of interaction, and extension reveals the distributed architecture. But none identifies what makes these elements hang together.

AToM provides that missing structure. By treating coherence as the cross-scale invariant that governs stability, integration, failure, and regeneration, AToM transforms 4E cognition from a cluster of related insights into a unified, mathematically grounded framework. Without coherence, 4E explains where cognition happens; with coherence, it explains how cognition survives, how it collapses, and how it evolves across levels of complexity.

In this way, AToM does not replace the 4Es—it allows them to become, for the first time, a fully integrated science of mind.

4. Trauma as Dimensional Collapse

AToM reframes trauma not as a psychological category or a narrative wound but as a geometric deformation in the coherence structure of a cognitive–physiological–relational system. Traditional accounts describe trauma through symptoms—flashbacks, avoidance, hypervigilance, dissociation—but these emerge from deeper structural transformations. AToM identifies trauma as a collapse in the system’s capacity to maintain stable trajectories across its manifold of possible states. This collapse is detectable across neural connectivity, autonomic regulation, interpersonal functioning, and narrative organization. Instead of treating trauma as an event, AToM models it as a dimensional transition, marked by quantifiable changes in curvature, topology, and coupling.

Central to this reframing is the recognition that healthy cognition operates within a high-dimensional manifold: a space of flexible perceptual, emotional, relational, and behavioral possibilities. In health, this manifold supports multiple stable pathways for prediction, regulation, and sense-making. Trauma constricts this flexibility. The first transformation is a loss of dimensionality—the effective state-space narrows as the system abandons degrees of freedom that have become too costly or unpredictable to maintain. Behavioral repertoires shrink, emotional variability decreases, and perceptual filters harden. What appears clinically as rigidity, avoidance, or numbing corresponds mathematically to a collapse of the manifold onto a lower-dimensional subspace.

Dimensionality reduction has profound downstream effects. As the system’s degrees of freedom shrink, the remaining dimensions must carry a greater proportion of predictive, affective, and regulatory load. This heightened load produces curvature spikes, indicating regions of increased sensitivity to perturbation. In information geometry, curvature corresponds to how sharply the statistical manifold bends under small changes in input. High curvature means that slight deviations produce large variational gradients—in lived terms, small cues evoke disproportionately large autonomic and emotional reactions. Hypervigilance, startle responses, and affective volatility arise not from cognitive bias but from geometric amplification. The system has been reshaped so that modest inputs now sit in regions of steep curvature, making threat detection over-responsive and safety detection under-responsive.

In addition to curvature amplification, trauma induces topological bottlenecks. Topologically, healthy systems exhibit multiple pathways connecting cognitive, affective, and relational states. These pathways appear in persistent homology as stable cycles or high-dimensional features that remain robust across scale. Trauma restricts these pathways, creating narrow channels—or bottlenecks—between major subsystems. Information flow becomes constrained. The person can enter certain states easily (hyperarousal, shutdown, intrusive replay) but struggles to transition out of them. Rumination loops, repetitive relational ruptures, and intrusive memory attractors emerge from this topological narrowing. What clinical nosology calls “stuckness” or “reactivity” is, in AToM terms, a reduction in topological persistence and a constriction of viable manifold transitions.

When dimensional collapse, curvature spikes, and topological bottlenecks co-occur, the system enters a state of hysteresis, meaning that returning to prior levels of coherence requires far more energy than the initial collapse required. Trauma is path-dependent: once coherence geometry deforms, the system can remain trapped in this configuration even when external threat has ceased. This irreversibility is evident in neurobiological findings on PTSD, where hippocampal–prefrontal connectivity decreases, amygdala reactivity remains elevated, and autonomic flexibility diminishes. Hysteresis explains why trauma does not spontaneously resolve with time or in safe environments; the system cannot simply “pop back” to its prior configuration because the manifold itself has been reshaped.

This geometric framing clarifies why dissociation—often misunderstood as absence—is better conceptualized as a dynamic boundary condition. Faced with intolerable curvature spikes and bottlenecks, the system may partition its manifold into quasi-independent regions to prevent destabilizing cross-talk. Dissociation acts as a protective barrier that shields fragile subsystems from interacting in ways that would create catastrophic coherence loss. However, these boundaries come at a cost: they prevent integration, reduce dimensionality further, and make transitions across states more difficult. AToM thus treats dissociation as a functional but self-limiting form of boundary formation within a distorted coherence geometry.

This account also synthesizes findings from diverse scientific domains. In affective neuroscience, trauma correlates with disruptions in neural oscillatory coupling, particularly between prefrontal regulatory circuits and limbic threat systems. In autonomic physiology, trauma reduces vagal tone and impairs cardiorespiratory synchronization, reflecting a collapse in cross-system entrainment. In developmental psychology, trauma disrupts interpersonal attunement, creating relational patterns marked by either hyperactivation or withdrawal—signatures of curvature and dimensional instability. In narrative psychology, trauma fragments autobiographical coherence, producing oscillations between intrusive overintegration and narrative voids—linguistic manifestations of topological collapse.

AToM integrates these literatures by showing that they are not disparate symptoms but different measurement windows on the same underlying deformation. Whether viewed through neural connectivity, bodily regulation, narrative structure, or relational patterns, trauma manifests as coherence breakdown: a loss of integrative capacity that spans scales. Because coherence is multi-level, trauma must be multi-level as well. A purely psychological account cannot explain autonomic dysregulation; a purely physiological account cannot explain narrative fragmentation; a relational account cannot capture topological collapse. AToM’s coherence geometry unifies these dimensions by providing a common invariant that links them.

This structural perspective also clarifies why trauma can exhibit paradoxical dynamics. A system may simultaneously appear hyperactive and frozen, vigilant and numb, intrusive and dissociated. These are not contradictions but consequences of dimensional collapse paired with curvature amplification. Some regions of the manifold become hyper-responsive attractors, while others become inaccessible voids. The system oscillates between these extremes not because it is conflicted but because its geometry no longer supports smooth transitions across states. In this way, AToM reframes trauma not as disorder but as a reorganization—an adaptive but costly strategy for maintaining minimal coherence under overwhelming constraint.

Understanding trauma in this way has significant implications. First, it explains why trauma treatment cannot rely solely on cognitive reframing: cognition alone cannot repair geometric deformation. Second, it highlights why relational, somatic, and rhythmic interventions are effective: they re-expand dimensionality, reduce curvature, reopen topological pathways, and restore entrainment. Third, it clarifies why recovery is gradual and nonlinear: hysteresis and manifold reshaping require sustained scaffolding to reverse. Finally, it explains why neurodivergent individuals, particularly those with high coherence-sensing precision, may be more sensitive to trauma-induced curvature changes and experience more pronounced collapse.

In AToM, trauma is not a story about harm; it is a story about coherence. Trauma is the condition in which a system loses the geometric flexibility required to remain itself. It is the collapse of dimensional richness into brittle pathways, the amplification of local curvature into global instability, the constriction of topology into narrow attractors, and the onset of hysteresis that traps the system in a self-limiting configuration. By defining trauma as dimensional collapse, AToM provides a structural, measurable framework that explains its symptoms, predicts its dynamics, and guides its repair across neural, somatic, relational, and cultural scales.

Below is a ~1000-word, publication-ready expansion of Section 5, written in the mature AToM academic voice—no tables, no notes, no meta-commentary. Just clean theory.

5. Coherence as the Fifth Structural E

The four Es—embodied, embedded, enactive, and extended cognition—map the terrain of cognitive life but do not specify the structural invariant that organizes it. Each E identifies a necessary dimension of cognition, yet none identifies what keeps these dimensions integrated, stable, and mutually compatible under perturbation. A system may be embodied without being coherent; it may be embedded but dysregulated; it may enact meaning chaotically; it may extend into the world in ways that amplify rather than stabilize its functioning. What is missing is a unifying structural principle that determines when these processes converge to produce sense-making and when they diverge into dysfunction, fragmentation, or collapse.

AToM proposes coherence as that missing structural E—not a metaphorical coherence but a formally definable, cross-scale geometric construct. Coherence in AToM is the system’s capacity to maintain integrable trajectories across internal and external constraints, and it can be operationalized as a measurable tuple:

C = (curvature smoothness,dimensional stability,persistence lifespan, cross-frequency coupling ratio)

This tuple captures the four structural signatures present whenever cognitive systems function adaptively, maintain internal consistency, and propagate meaning across scales—from neural microdynamics to relational interactions to institutional processes. Coherence, in this sense, is not an emergent “feel” or narrative description but a measurable property of the system’s underlying manifold. It is the geometric invariant that links the four Es together into a unified theory of mind.

5.1 Curvature Smoothness: Stability Under Perturbation

Curvature smoothness measures how sharply the system’s manifold bends in response to small changes in input or internal state. In information geometry, curvature reflects sensitivity: high curvature means small deviations trigger large gradients, producing instability and volatility; low curvature indicates robustness and predictability. Healthy cognition maintains smooth curvature across perceptual, interoceptive, affective, and relational manifolds. This smoothness allows the system to anticipate change, modulate energy expenditure, and maintain predictive stability.

Curvature smoothness explains why coherent systems can tolerate ambiguity, integrate novelty, and remain flexible under stress. It clarifies why trauma generates hypersensitivity: curvature spikes amplify prediction error, making the world feel overwhelming and uncontrollable. It explains interpersonal attunement, where smooth relational curvature enables mutual regulation and shared sense-making. And it accounts for institutional stability, where curvature smoothness across procedural norms protects organizations from cascading failure.

In AToM, curvature smoothness is not a metaphor for calmness or clarity. It is the mathematical signature of predictive stability across scales.

5.2 Dimensional Stability: Maintaining Freedom to Move

Dimensional stability refers to the number of degrees of freedom a system maintains in its state-space. Cognitive health requires a manifold rich enough to support multiple pathways of action, interpretation, and regulation. Systems with high dimensionality can shift among modes flexibly; systems that lose dimensionality become rigid and fragile.

Dimensional stability explains why creativity, resilience, and emotional range depend on access to multiple cognitive and affective dimensions. It clarifies why rigid, avoidant, or dissociated states reflect collapsed dimensionality. It explains how attachment security emerges: not from perfect regulation but from the ability to maintain dimensional richness in the face of stress. It clarifies why cultural systems survive when they preserve institutional plurality and collapse when they shrink into narrow ideological attractors.

AToM treats dimensional stability as the structural backbone of adaptability. Without it, the embodied, embedded, enactive, and extended processes identified by 4E cognition cannot coordinate or support one another.

5.3 Persistence Lifespan: Topological Integrity Across Scales

Persistence lifespan refers to the duration and stability of topological features—clusters, cycles, connected components, loops—across multiple scales of analysis. Using persistent homology, one can measure whether meaningful structures remain intact as resolution, context, or load changes. Systems with long persistence lifespans maintain interpretive and behavioral stability; those with short lifespans exhibit fragmentation and incoherence.

Persistence lifespan explains why stable identity depends on the continuity of narrative, memory, and relational structures across time. It clarifies why interpersonal relationships require recurring patterns of attunement that persist across contexts. It explains why institutions require durable norms and feedback loops—topological features that withstand fluctuations in population, technology, and circumstance.

In neural dynamics, persistence lifespan corresponds to the stability of functional networks over time. In cultural systems, it corresponds to the durability of myths, rituals, and symbolic frameworks. In all cases, long persistence lifespans indicate coherent manifold structure; short ones indicate instability or collapse.

5.4 Cross-Frequency Coupling Ratio: Integration Across Timescales

Cross-frequency coupling ratio measures how well oscillatory processes at different tempos synchronize or constrain one another. Cognition is inherently multi-scale: neural gamma rhythms synchronize with theta cycles; autonomic oscillations coordinate with respiratory and cardiac rhythms; conversational pacing synchronizes with micro-gestural dynamics; institutional rhythms interact with cultural tempos. When cross-frequency coupling is balanced, systems integrate information across timescales; when it collapses, systems become either chaotic or inert.

Cross-frequency coupling explains why emotional regulation requires alignment among neural, autonomic, and behavioral tempos. It clarifies why teams and groups lose coherence when timing becomes misaligned. It explains why societies fracture when institutional tempos accelerate while cultural tempos lag, or when economic and technological cycles outpace governance structures.

In AToM, cross-frequency coupling ratio is the operational signature of entrainment: the ability of a system to align rhythms without collapsing variability. High coupling without rigidity yields coherence; high coupling with rigidity yields authoritarian synchrony; low coupling yields fragmentation.

5.5 Coherence as the Structural Completion of the 4E Framework

Each component of the coherence tuple corresponds to a domain that 4E cognition identifies but cannot formalize. Embodied cognition requires curvature smoothness to maintain interoceptive and sensorimotor stability. Embedded cognition requires dimensional stability to retain adaptive possibilities. Enactive cognition requires persistent topological structure to maintain sense-making trajectories. Extended cognition requires cross-frequency coupling to scale coherence across tools, institutions, and cultural systems.

Coherence is therefore not an optional addition to the 4E paradigm. It is the structural invariant that determines the success or failure of each E. Without coherence:

• embodiment becomes dysregulated,
• embedding becomes overwhelming or ineffective,
• enaction becomes chaotic or collapsed,
• and extension becomes destabilizing rather than supportive.

With coherence, the four Es become mutually constraining components of a unified cognitive architecture, capable of withstanding perturbation, integrating novelty, and scaling across layers of complexity.

5.6 AToM’s Definition of Cognition

By formalizing coherence as a measurable tuple, AToM defines cognition itself as:

the maintenance of stable, integrable trajectories across the system’s coherence geometry under changing constraints.

This definition transcends representational, computational, and interactionist theories while preserving their strengths. It allows cognition to be analyzed:

• in neural firing patterns,
• in bodily rhythms,
• in interpersonal dynamics,
• in institutional processes,
• and in cultural transformations— all through the same structural lens.

Coherence is thus the fifth structural E: the invariant that binds the embodied, embedded, enactive, and extended dimensions of cognition into a single, measurable, multi-scale system.

6. Information Geometry as the Structural Backbone of Coherence

Any claim that coherence is a structural invariant of cognition requires a mathematical foundation that is both rigorous and domain-appropriate. AToM does not borrow equations from physics or import metaphors from geometry; instead, it draws on a well-established lineage of information geometry, dynamical systems theory, and topological data analysis, each of which provides tools for describing the shape, curvature, dimensionality, and connectivity of complex systems. These tools have already been applied to neural activity, perceptual inference, physiological regulation, behavior, language, and organizational dynamics. AToM’s contribution is to integrate these mathematical resources into a single structural vocabulary capable of describing coherence across scales—from neural microdynamics to cultural systems.

Information geometry begins with the idea that systems navigating uncertainty can be represented as points on a statistical manifold whose geometry is defined by the relationships among probability distributions. Cognitive systems, understood through predictive processing and active inference traditions, continuously update internal generative models in response to sensory perturbations. These updates trace trajectories across an underlying manifold whose local and global geometry encodes the system’s constraints, expectations, and adaptive landscape. The Fisher Information Metric provides a way to measure curvature on this manifold, revealing how sharply the system’s internal model responds to small changes in input. High curvature corresponds to hypersensitivity, volatility, or instability; low curvature corresponds to robustness and coherence. This metric supplies the core of AToM’s notion of curvature smoothness.

Curvature alone, however, cannot capture the higher-order interactions that shape complex cognitive and relational dynamics. For that, information geometry employs the Amari–Chentsov tensor, a third-order geometric object that describes how probability distributions deform under transformation. This tensor allows AToM to articulate how coherence is preserved or eroded when systems integrate diverse signals—neural, autonomic, affective, linguistic, or social. When deformation is smooth and consistent across scales, coherence is preserved; when deformations produce sharp discontinuities, the manifold becomes unstable. The Amari–Chentsov framework therefore grounds AToM’s claim that coherence involves not just local curvature but the stability of transformation across nested manifolds.

Yet cognition is not merely statistical; it is also topological. Systems that maintain meaning across time must preserve the shape of their internal and relational dynamics. Topological data analysis (TDA) provides tools for detecting such structure. Persistent homology identifies stable features—clusters, cycles, bottlenecks—within high-dimensional data by tracking which features persist across multiple scales of observation. AToM leverages persistent homology to capture the persistence lifespan component of coherence: long-lived topological features correspond to stable patterns of neural connectivity, relational attunement, habitual actions, institutional routines, or cultural narratives.

TDA also enables AToM to diagnose failure modes. Topological bottlenecks reveal where information flow becomes constricted or where transitions between states become fragile. In trauma, persistent homology detects narrowing pathways that map onto hyperaroused or dissociative attractor states. In relationships, bottlenecks reveal ruptures in mutual regulation. In organizations, bottlenecks manifest as siloing, bureaucratic inertia, or breakdowns in communication. Persistent homology therefore supplies the topological backbone of AToM’s account of coherence collapse.

Dynamic coherence also requires the ability to propagate structure across time. For this, AToM employs the language of curvature flow, which models how manifolds evolve under constraints. Cognitive systems continuously adjust their internal geometry in response to new data, embodied changes, relational environments, and cultural scaffolds. Curvature flow formalizes how these changes occur: whether the manifold smooths toward stability or sharpens toward collapse. Curvature flow and information-geometric dynamics together provide a way to model the temporal evolution of coherence across the lifespan, across relationships, and across institutional or cultural contexts.

Another essential component of AToM’s mathematical architecture is cross-frequency coupling. Complex systems—from neural networks to interpersonal interactions to organizational communication—operate across multiple temporal scales. Coherence requires that these scales remain sufficiently synchronized to permit information flow without collapsing diversity. Phase-amplitude coupling, phase-phase coupling, and other cross-frequency measures quantify how rhythms at different tempos constrain or liberate one another. These metrics anchor AToM’s claim that cross-frequency coupling ratio is a structural component of coherence: systems with healthy coupling integrate information; systems with excessive coupling become rigid; systems with insufficient coupling become fragmented.

While AToM relies primarily on established mathematical tools, it proposes two new constructs to unify these measures into a single analytic framework. The first is the Coherence Operator, denoted \hat{C}, which integrates curvature, topological persistence, entropy gradients, and cross-frequency coupling into a composite functional. The purpose of \hat{C} is not to introduce new mathematics but to provide a unifying computational operator capable of mapping multi-modal data—neural signals, physiological measures, behavioral trajectories, linguistic patterns, and relational dynamics—into a coherence score or coherence field. Although formal development of the operator belongs to future technical work, its conceptual role is clear: \hat{C} acts as the transformation that reveals coherence as a measurable, domain-agnostic invariant.

The second construct is the Trauma Sensitivity Constant, denoted \tau. This scalar quantifies how rapidly curvature increases in response to perturbations; systems with high \tau experience trauma-like collapse when confronted with minimal destabilization. \tau therefore captures individual differences in susceptibility to coherence distortion—differences shaped by developmental history, neurodivergence, physiology, relational context, and environmental stress. Like \hat{C}, the constant is explicitly proposed rather than fully formalized. Its function is to render precise an idea long recognized in clinical and developmental research: some systems are geometrically more fragile than others, and trauma reflects an interaction between external perturbation and internal sensitivity.

Taken together, these mathematical tools give coherence the structural grounding required to serve as the fifth E. They provide a language capable of unifying 4E cognition with predictive processing, developmental psychology, trauma science, interpersonal neurobiology, organizational dynamics, and cultural theory. They make it possible to describe, compare, and intervene upon coherence at any level of human experience. Most importantly, they transform coherence from an intuitive or phenomenological notion into a computationally tractable, experimentally measurable variable.

In AToM, coherence is not an epiphenomenon of successful cognition; it is the geometry of cognition itself. Systems that maintain smooth curvature, stable dimensionality, persistent topological structure, and balanced cross-frequency coupling are capable of embodied, embedded, enactive, and extended functioning. Systems that lose coherence lose the capacity to coordinate these dimensions and drift into collapse, fragmentation, or rigidification. Information geometry, dynamical systems theory, and topological analysis together provide the scaffold for understanding why coherence is the structural invariant that the 4E paradigm requires—and how it can finally be made empirically measurable across neural, physiological, relational, and cultural scales.

7. Clarifying AToM’s Distinctiveness: Boundary Conditions and Anticipated Objections

Any framework that positions itself as a structural completion of 4E cognition must distinguish its contributions from nearby traditions while addressing the strongest conceptual objections. AToM does not propose an alternative to predictive processing, enactivism, interpersonal neurobiology, or cultural evolution; instead, it integrates insights from each into a single geometric vocabulary. This section delineates the conceptual boundaries of AToM, clarifies what it adds that existing theories do not, and directly confronts the three objections most likely to arise from expert reviewers.

7.1 AToM Is Not Predictive Processing Wearing New Clothes

Predictive processing (PP) and the free-energy principle (FEP) offer powerful accounts of how cognitive systems maintain local stability by reducing prediction error. AToM builds upon these insights, but its scope and formal commitments differ fundamentally from PP.

Most importantly, PP provides a micro-scale variational principle—minimization of error gradients within a generative model—but it does not specify the geometry of coherence across scales. PP lacks formal primitives for:

• dimensional collapse (PP does not define the manifold whose dimensionality changes),
• curvature spikes (Fisher curvature is not used to diagnose cognitive or affective instability),
• topological bottlenecks (no direct representation of persistent homology),
• hysteresis and irreversibility (PP lacks a mechanism for formally modeling path dependence),
• cross-scale entrainment (PP does not explain how micro-dynamics propagate through relational, institutional, or cultural systems).

PP models local dynamics of inference; it does not provide a structural theory of global coherence. AToM uses PP as the microscopic engine of coherence but extends it into a cross-scale framework using tools from information geometry and topology. In effect:

• PP ≈ local coherence maintenance
• AToM ≈ global coherence geometry

AToM therefore does not repackage PP; it completes it by adding the manifold on which prediction unfolds and the structural invariants that determine whether prediction remains stable across scales.

7.2 AToM Is Not Participatory Sense-Making in Mathematical Disguise

Enactivist theories of participatory sense-making (PSM) describe how meaning emerges in interaction. They emphasize coordination, breakdown, repair, and co-regulation within lived experience. AToM shares this focus on relational dynamics but diverges significantly in method.

PSM provides a phenomenological and interactional description of how sense-making becomes shared. It does not supply a formal invariant that distinguishes productive coordination from destabilizing forms of coupling. PSM lacks tools for measuring or predicting:

• when relational synchrony becomes over-coupling,
• when misattunement transitions into trauma,
• when dyads enter coherence regimes vs. dissociative regimes,
• how relational geometry maps onto neural or autonomic structure,
• how relational patterns scale into group, institutional, or cultural dynamics.

AToM addresses these omissions by grounding relational dynamics in a coherence manifold. On this manifold, sense-making trajectories can be evaluated by geometric criteria: curvature smoothness, dimensional stability, topological persistence, and coupling ratios. This allows AToM to formalize distinctions that PSM leaves intuitive, such as why some interactions lead to mutual enrichment while others lead to collapse or entrapment.

In short:

• PSM = description of relational dynamics
• AToM = geometry of relational stability and failure

AToM does not replace enactivism; it equips it with the invariants it requires to become empirically measurable and computationally tractable.

7.3 Coherence Is Not a Vague Concept; It Is a Measurable Tuple

A common objection is that “coherence” risks becoming an all-purpose descriptor. AToM avoids this problem by defining coherence as a sharply constrained tuple of structural invariants:

C = (\kappa,\ d,\ H_k,\ \rho)

where

\kappa = curvature smoothness,

d = dimensional stability,

H_k = persistent homology lifespan,

\rho = cross-frequency coupling ratio.

This tuple is not metaphorical. Each component can be measured in:

• neural data (functional connectivity, oscillatory coupling, attractor structure),
• physiological data (HRV, EDA, interoceptive volatility),
• behavioral data (movement smoothness, affective transitions),
• linguistic data (embedding curvature, semantic drift, narrative topologies),
• relational data (synchrony indices, rupture-repair cycles),
• institutional data (communication topology, incentive alignment),
• cultural data (narrative persistence, norm cycles, symbolic stability).

AToM’s coherence tuple ensures that coherence is as measurable and falsifiable as constructs like entropy, complexity, or free energy. This directly counters concerns about conceptual vagueness and supports empirical integration across disciplines.

7.4 Coherence Is Not the Same as Stability, Harmony, or Synchrony

Coherence, in AToM, is not synonymous with harmony, integration, or stability. Harmony may emerge from coherence, but coherence is a geometric property of a system’s manifold, not an aesthetic or experiential state. Stability may accompany coherence, but unstable systems can exhibit coherence during transition. Synchrony may appear coherent, but forced synchrony can collapse dimensionality and produce pathological rigidity.

Coherence has precise structural meaning:

• A system may be highly synchronized yet incoherent (e.g., authoritarian social control).
• A system may be richly diverse yet coherent (e.g., robust ecosystems or scientific communities).
• A system may be temporarily unstable yet coherent (e.g., phase transitions).
• A system may be predictable yet incoherent (e.g., rigid dissociative freeze states).

By distinguishing coherence from superficial uniformity, AToM avoids collapsing into functionalism or normalization.

7.5 Adjacent Theories and the Gaps They Leave Unfilled

AToM exists within a rich theoretical landscape. Several adjacent frameworks contribute essential insights, but none provides the cross-scale structure that coherence geometry requires.

• Gibsonian affordances reveal organism–environment coupling but lack topological criteria.
• Fristonian FEP explains local stability but not multi-level propagation or collapse modes.
• Coordination dynamics (Kelso) capture phase transitions but do not model multi-manifold coherence.
• Phenomenological integration (Gallagher, Thompson) illuminates lived stability but lacks measurable invariants.
• Interpersonal neurobiology (Siegel, Porges) identifies integration principles but not the geometry behind them.
• Distributed cognition (Hutchins) maps cognitive ecologies but not coherence constraints.
• Cultural evolution (Boyd, Henrich) tracks macro-dynamics without a coherence metric linking micro to macro.

AToM synthesizes these contributions by articulating the unifying geometric structure these theories implicitly rely on.

7.6 Why AToM Requires These Boundaries

AToM is a structural theory, not an interpretive or descriptive one. It introduces coherence as a measurable invariant and entrainment as the cross-scale dynamic that maintains it. Without clearly delineating the boundaries between AToM and its conceptual neighbors, the novelty of these contributions would be obscured, and the framework would risk being misunderstood as a reformulation of existing ideas rather than a structural unification.

By specifying these boundaries, AToM positions itself not simply as another entrant in the cognitive science landscape but as a next-generation integrative geometry—a framework capable of linking predictive processing, enactivism, relational dynamics, developmental trajectories, institutional organization, cultural evolution, and technological mediation within a single mathematically coherent paradigm.

8. Empirical Predictions and Falsifiability

A structural theory of cognition must generate not only conceptual clarity but also testable predictions. AToM’s central claim—that coherence is the cross-scale invariant underlying cognitive, relational, and cultural stability—yields a suite of empirically adjudicable hypotheses. These predictions span neural dynamics, physiological regulation, behavioral coordination, linguistic patterns, relational processes, institutional systems, and cultural evolution. Crucially, each prediction follows directly from the geometry of coherence and therefore offers a falsifiable implication of the theory. If the coherence tuple does not outperform competing constructs in these domains, the theory weakens; if it does, AToM gains empirical traction as a unifying framework.

AToM posits that coherence has four measurable components: curvature smoothness, dimensional stability, topological persistence, and cross-frequency coupling ratios. Each of these components should exhibit identifiable signatures that precede, accompany, or predict changes in cognitive functioning, emotional regulation, relational dynamics, organizational stability, or cultural patterns. This provides a rich empirical landscape, but several predictions stand out as particularly diagnostic.

8.1 Coherence Metrics Should Outperform Traditional Psychometric Measures

The first prediction is foundational: coherence metrics—derived from neural recordings, physiological signals, or linguistic embeddings—should correlate with behavioral and clinical outcomes more strongly than conventional psychometric instruments. Because coherence captures underlying manifold structure rather than retrospective self-report, it should provide earlier and more precise indicators of:

• affective instability,
• trauma reactivation,
• relational dysregulation,
• attentional fragmentation,
• executive overload,
• or depressive collapse.

For example, heart-rate variability (HRV), a proxy for autonomic dimensionality and cross-frequency coupling, should outperform anxiety questionnaires in predicting panic episodes or stress responses. Similarly, the curvature of linguistic embeddings—reflecting semantic and narrative coherence—should shift before conscious awareness of emotional or cognitive deterioration. If coherence metrics fail to outperform traditional measures, AToM would require revision; if they succeed, the theory gains empirical force.

8.2 Linguistic Curvature Should Shift Before Therapeutic Breakthroughs

AToM predicts that emotional or cognitive breakthroughs in therapy occur when the coherence manifold reorganizes—when curvature smooths, dimensionality expands, and topological bottlenecks reopen. These shifts should manifest linguistically before they manifest phenomenologically. Specifically:

• token–token transition entropy should decrease,
• embedding curvature should flatten,
• semantic drift should reduce,
• and narrative arcs should exhibit increased persistence and integrability.

These linguistic changes should precede self-reported insight, affective clarity, or behavioral change. If these predictions hold across therapeutic modalities—psychodynamic, CBT, somatic, EMDR—it would indicate that coherence geometry provides a trans-theoretical predictor of transformation. If linguistic curvature remains static despite subjective or behavioral change, AToM’s claim about multi-scale coherence propagation would be challenged.

8.3 Autistic–LLM Teams Should Outperform Neurotypical Raters at Coherence Detection

A central implication of AToM is that neurodivergent cognition—especially autistic cognition—constitutes a high-resolution coherence-sensing phenotype. Autistic individuals detect micro-level inconsistencies, pattern breaks, and structural instabilities that neurotypical smoothing obscures. When paired with large language models, which excel at detecting statistical irregularities in embedding space, these teams should outperform neurotypical raters in tasks requiring:

• logical consistency checking,
• narrative coherence evaluation,
• institutional process auditing,
• relational pattern detection,
• anomaly spotting in complex data.

The prediction is clear and falsifiable: autistic + LLM dyads should exhibit elevated detection accuracy relative to neurotypical controls. If they do not, AToM’s claim about precision coherence sensing would weaken; if they do, AToM would provide a new foundation for collaborative neurodiversity research.

8.4 Entrainment Breakdowns Should Precede Relational and Institutional Collapse

If coherence is the invariant that maintains stability across scales, then relational and organizational failures should be preceded by measurable entrainment breakdowns. In dyads, early signs should include:

• decreased autonomic synchrony,
• phase desynchronization in vocal or gestural rhythms,
• reduced repair frequency after micro-ruptures,
• increased variance in cross-frequency coupling.

In teams or organizations, early signs should include:

• divergence in communication rhythms,
• increased latency in response cycles,
• fragmentation of workflow patterns,
• inconsistent application of norms,
• and widening gaps between espoused values and enacted practices.

These entrainment failures should occur before visible conflict, burnout, turnover, or institutional dysfunction. If collapse occurs without preceding coherence degradation, AToM would require amendment; if early degradation consistently predicts failure, AToM becomes a powerful diagnostic framework for relational and organizational health.

8.5 Topological Bottlenecks Should Predict Trauma Relapse Risk

Trauma, in AToM, is characterized by manifold collapse, bottlenecking, and curvature amplification. If this is true, then topological signatures should precede trauma reactivation. Using persistent homology on physiological, linguistic, or behavioral data, one should detect:

• narrowing of state-space transitions,
• collapse of high-persistence cycles,
• re-emergence of rigid attractor basins,
• and loss of dimensionality in affective or interoceptive variability.

These signatures should predict relapse risk earlier than subjective distress or clinical interview. For example, a sudden decrease in the persistence of autonomic cycles should precede the onset of hyperarousal or dissociation. If persistent homology fails to predict relapse, AToM’s topological component would require revision; if it succeeds, the theory gains substantial empirical grounding.

8.6 Coherence Should Scale Across Modalities

Another key prediction is that coherence should align across multiple measurement channels. If coherence is truly cross-scale, then:

• neural coherence (oscillatory coupling)
• physiological coherence (HRV, EDA)
• behavioral coherence (rhythm, smoothness)
• linguistic coherence (embedding stability)
• relational coherence (synchrony, repair cycles)
• institutional coherence (communication topology)
• and cultural coherence (narrative persistence)

should correlate more strongly with one another than with any external variable.

If coherence is low-dimensional and cross-modal, AToM holds; if coherence fragments across measurement types, the theory must explain why.

8.7 AToM Should Predict the Success or Failure of Collective Systems

Finally, AToM predicts that large-scale cultural or institutional stability depends on maintaining coherence across nested manifolds. When technological, economic, relational, and cultural tempos diverge, coherence collapses. By quantifying coherence across these scales, AToM should be able to:

• forecast institutional paralysis,
• detect early signs of polarization,
• model collective sense-making breakdowns,
• and identify stabilizing interventions.

If coherence metrics fail to forecast large-scale instability, AToM’s cross-scale aspiration would be undermined; if they succeed, AToM becomes a general framework for understanding societal dynamics.

8.8 Summary: Coherence as a Falsifiable Construct

Each prediction above can be tested using existing methods: physiological sensors, linguistic embedding analysis, TDA pipelines, multimodal integrative models, behavioral synchrony tools, and organizational network metrics. This makes AToM a scientifically actionable theory, not a conceptual synthesis.

A single falsifying domain would weaken AToM; multiple confirmations would position it as the first coherence-centered, cross-scale science of meaning.

9. Information–Geometric Foundations of Coherence

A structural theory must be mathematically grounded without relying on borrowed metaphors or inappropriate extensions of physical laws. AToM meets this requirement by drawing from three mature mathematical traditions—information geometry, dynamical systems theory, and topological data analysis—while introducing only the minimal set of new constructs required to unify them. These tools provide the formal basis for coherence as the fifth structural E, allowing complex cognitive and social systems to be described in terms of curvature, dimensionality, topology, and coupling dynamics rather than folk-psychological intuition.

AToM’s mathematical commitments remain intentionally domain-appropriate: they do not claim that human systems follow physical laws, only that they exhibit patterns best described using geometric and topological language. The purpose is conceptual precision, not reductionism.

9.1 Statistical Manifolds and the Role of Curvature

Information geometry begins with the idea that cognitive systems can be represented as points on a statistical manifold, a space whose structure is determined by the relationships among probability distributions. Every generative model—whether neural, physiological, behavioral, relational, or institutional—defines such a manifold. The system’s internal state evolves as it updates these distributions in response to perturbations.

The Fisher Information Metric supplies a principled measure of curvature on this manifold. In plain text, we denote curvature as C_fisher. High values of C_fisher indicate that small changes in input produce disproportionately large gradients in the system’s prediction landscape, which corresponds to hypersensitivity, volatility, and instability. Low values of C_fisher reflect robustness and smooth predictive integration.

Curvature is therefore the mathematical foundation of AToM’s first coherence component, curvature smoothness. A system with smooth curvature can tolerate uncertainty; one with curvature spikes becomes reactive and fragile. This connects directly to AToM’s account of trauma, hypervigilance, interpersonal rupture, and institutional brittleness.

9.2 Higher-Order Structure: The Amari–Chentsov Tensor

Curvature alone cannot capture how a system transforms information across nested levels. For this, information geometry introduces the Amari–Chentsov Tensor, which we denote in plain text as T_amari. This is a higher-order tensor describing how probability distributions deform under changes in parameters.

The significance of T_amari for AToM is twofold:

1. It measures how internal models warp when integrating new signals, allowing dynamic deformation of coherence to be quantified.
2. It serves as the mathematical foundation for understanding how coherence propagates from local to global scales.

A system with stable T_amari maintains consistent transformations across perception, physiology, action, and social exchange. A system with unstable T_amari becomes inconsistent or fragmented. This tensor therefore grounds the second coherence component, dimensional stability, by identifying when transformations preserve or collapse degrees of freedom.

9.3 Dimensionality and Manifold Collapse

AToM identifies the number of effective degrees of freedom in a system’s state-space as a core feature of coherence. In plain text, we denote dimensionality as D_system. Healthy systems operate in high-dimensional manifolds, enabling flexibility, optionality, and resilience. Trauma, burnout, relational overwhelm, and institutional rigidity correspond to a reduction in D_system—the system collapses onto a narrower set of possible trajectories.

Dimensionality is measurable through:

• principal component structure of neural data,
• entropy of autonomic variability,
• linguistic embedding dimensionality,
• diversity of behavioral transitions,
• relational role flexibility,
• institutional response variety.

In each case, high D_system indicates coherence; low D_system signals collapse. This forms the second component of the coherence tuple.

9.4 Topological Persistence Across Scales

Topology provides tools for understanding not just local curvature but global structure. Using persistent homology, one can track how topological features—clusters, loops, bottlenecks, cavities—persist across changes in resolution or scale. AToM refers to this quantity as H_persistence.

Persistent features indicate stable patterns of coordination:

• stable neural assemblies,
• consistent autonomic cycles,
• recurring relational micro-rhythms,
• durable institutional procedures,
• resilient cultural narratives.

When H_persistence is high, the system retains its shape across perturbations. When H_persistence collapses, the structure becomes fragile or chaotic. This is the third coherence component, persistence lifespan.

Topological bottlenecks, denoted B_topo, correspond to narrowed pathways between states. These bottlenecks explain trauma flashpoints, relational traps, institutional gridlock, and cultural polarization.

9.5 Cross-Frequency Coupling and Multi-Temporal Integration

Human systems operate across multiple timescales simultaneously. Neural activity couples fast gamma rhythms to slower theta cycles; physiological systems couple respiratory cycles to cardiac rhythms; interpersonal interactions couple micro-gestural timing to conversational turns; institutions couple daily practices to annual cycles; cultures couple generational tempo to civilizational drift.

AToM captures this multi-temporal interaction through cross-frequency coupling ratios, denoted R_cfc. These ratios measure how rhythms at different frequencies constrain, echo, or stabilize one another.

Balanced R_cfc allows systems to integrate across scales without collapsing variability. Excessively high R_cfc leads to rigid synchrony; low R_cfc leads to fragmentation. This forms AToM’s fourth coherence component, cross-frequency coupling.

9.6 The Coherence Operator in Plain Text

To unify these measures, AToM introduces the Coherence Operator, written in plain text as C_OPERATOR.

C_OPERATOR takes as input:

• curvature (C_fisher),
• dimensionality (D_system),
• topological persistence (H_persistence),
• and cross-frequency coupling (R_cfc),

and produces a coherence field or score.

Formally, C_OPERATOR is not yet defined mathematically; it is a composite operator meant for future development. Its purpose is conceptual integration: to combine geometric, topological, and dynamical measures into a single variable that can be computed from multi-modal data. It is analogous not to a physical operator but to integrative constructs in computational neuroscience and machine learning, such as energy functions or latent embeddings.

9.7 Trauma Sensitivity Constant

AToM also proposes the Trauma Sensitivity Constant, written in plain text as TAU. TAU quantifies how rapidly curvature increases in response to perturbation. High TAU systems show trauma-like collapse under minimal stress; low TAU systems remain stable across broader ranges of perturbation.

TAU provides a mathematically grounded way to articulate long-standing clinical observations:

• early relational environments alter sensitivity to stress,
• neurodivergent systems may detect coherence fractures earlier,
• trauma histories modify curvature response landscapes,
• organizational cultures vary in fragility and resilience.

Like C_OPERATOR, TAU is proposed rather than fully defined. Its value lies in clarifying the concept of vulnerability in geometric terms.

9.8 Why These Mathematical Tools Are Necessary

The mathematical foundations of AToM accomplish three essential goals:

1. They ensure coherence is not metaphorical but measurable. Each component corresponds to an existing mathematical construct with empirical implementations.
2. They give 4E cognition a structural backbone. Embodiment, embedding, enaction, and extension require a manifold on which their interactions unfold.
3. They allow cognitive science to move beyond qualitative description. Curvature, dimensionality, persistence, and coupling ratios can be computed from neural, physiological, behavioral, linguistic, relational, institutional, and cultural data.

In sum, these foundations elevate coherence from an intuitive notion to a scientifically tractable invariant, capable of grounding a unified multi-scale theory of mind.

Below is a tight, disciplined, publication-ready Section 10 (~1000 words)—no looseness, no rhetorical drift, no repetition, only the clean, high-density AToM voice you established earlier.

10. Frontier Implications: Toward a Unified Science of Meaning

AToM’s central claim—that coherence is the cross-scale invariant organizing cognitive, physiological, relational, institutional, and cultural systems—opens a broad frontier of implications that extend far beyond theoretical integration. By providing a geometric vocabulary applicable from neurons to societies, AToM reframes long-standing scientific challenges as problems of coherence formation, maintenance, loss, and recovery. These implications are not speculative extensions; they follow directly from the structural commitments established in previous sections. They reveal how coherence geometry can guide empirical research, clinical intervention, organizational design, educational reform, cultural analysis, and the development of artificial intelligence. What unifies these frontiers is the recognition that systems at every scale must maintain stable trajectories through a manifold of constraints, and that coherence provides the only domain-general metric for assessing how well they do so.

In behavioral health, AToM suggests that diagnostic categories are coarse descriptions of underlying coherence geometries. Anxiety corresponds to curvature spikes; depression corresponds to amplitude collapse and loss of topological persistence; ADHD reflects unstable cross-frequency coupling; autism reflects high-resolution coherence sensing; and trauma reflects dimensional collapse and bottlenecking. This predicts that coherence metrics should outperform symptom checklists in forecasting clinical trajectories, relapse risk, treatment response, and rupture–repair cycles in therapy. It also suggests that interventions succeed not by modifying “content” but by reorganizing curvature, reopening dimensions, rebuilding persistence, and restoring multi-timescale entrainment. Somatic therapies, relational attunement protocols, breathwork, EMDR, and psychedelic-assisted treatments can be understood as coherence-restoration processes rather than domain-specific techniques. This shifts behavioral health from a symptom-based practice to a form of precision coherence engineering.

In education, AToM reframes learning not as the acquisition of information but as the progressive expansion of a learner’s coherence bandwidth. Learners differ not primarily in ability but in tempo, dimensionality, and entrainment thresholds. A learner with low autonomic flexibility or unstable coupling cannot integrate material delivered at an incompatible tempo; a learner with collapsed dimensionality cannot generalize across contexts; a learner with fragmented persistence cannot maintain conceptual structures long enough to build expertise. This implies that educational systems built on fixed pacing and standardized trajectories will misclassify children whose coherence geometries do not match institutional tempos. Adaptive instruction—which modulates pace, modality, and relational scaffolding to maintain coherence—should outperform traditional methods. This framework explains why certain environments unlock latent intelligence and others suppress it: the decisive factor is not innate capacity but alignment between manifold structure and instructional tempo.

In medicine and physiology, AToM interprets health as the maintenance of cross-system coherence. Circadian, ultradian, endocrine, autonomic, and interoceptive rhythms must remain entrained for homeostasis to persist. Chronic illness—whether metabolic, inflammatory, psychiatric, or cardiovascular—often reflects coherence breakdown across these oscillatory systems. HRV, respiratory sinus arrhythmia, interoceptive accuracy, and neural oscillatory coupling become coherence biomarkers rather than isolated physiological indicators. This suggests that interventions targeting rhythmic integration—sleep optimization, paced breathing, cold exposure, movement protocols, vagal stimulation, and stress patterning—may have broad-spectrum therapeutic effects because they restore multi-timescale coherence. The same logic applies to recovery from infection, long systemic stress, and chronic inflammatory states: resilience emerges when coherence across physiological manifolds is rebuilt.

In organizational and institutional dynamics, AToM offers a vocabulary for diagnosing alignment and dysfunction. Organizations function as multi-layer coherence machines: roles, norms, incentives, communication channels, and procedures must maintain curvature smoothness and stable topologies. When institutional rhythms diverge—e.g., when technological acceleration outpaces governance, when communication bottlenecks arise, or when incentive structures distort decision trajectories—coherence collapses. This yields polarization, inertia, contradictory policy, or cascading failure. AToM predicts that measuring coherence across institutional subsystems—via communication network topology, linguistic curvature, response-latency rhythms, and cross-departmental coupling—can reveal vulnerability long before failure becomes visible. Organizational reform, from this perspective, is not ideological but geometric: interventions succeed when they reopen dimensions, smooth curvature, increase persistence, and rebalance coupling across scales.

In cultural systems, AToM clarifies why shared myths, rituals, symbols, and narratives persist: they are coherence-preserving compression schemes. When cultural coherence collapses—due to rapid technological change, ecological disruption, demographic pressure, or informational overload—societies experience curvature spikes that manifest as anxiety, polarization, and epistemic fragmentation. Narrative convergence disappears; symbolic anchors lose persistence; topological bottlenecks form between subcultures. AToM predicts that coherent cultures exhibit long narrative persistence, stable ritual rhythms, and balanced cross-temporal coupling (between generational cycles, institutional tempos, and technological acceleration). Cultural renewal, therefore, requires restoring coherence across these temporal and symbolic layers—not simply promoting new slogans or policies.

In geopolitics, AToM interprets large-scale stability in terms of coherence across infrastructural, informational, economic, and cultural manifolds. Societal collapse occurs when cross-scale entrainment fails: when institutional tempos accelerate while cultural tempos slow; when climate shifts generate curvature spikes in resource distribution; when economic rhythms oscillate faster than regulatory frameworks; when communication technologies amplify misinformation in ways that bypass coherence-restoring mechanisms. AToM predicts that geopolitical fragility will be preceded by detectable coherence degradation in public discourse, institutional communication networks, demographic rhythms, and narrative cohesion. Conversely, states that maintain coherence across these layers—via slow variables such as norms, robust institutions, and symbolic stability—should exhibit resilience.

In artificial intelligence, AToM provides a framework for conceptualizing AI not as an emerging agent but as an entrainment prosthetic. Modern AI systems compress high-dimensional information into accessible patterns, smooth curvature in human sense-making, stabilize attention, detect inconsistency, and extend the coherence manifold of individuals and groups. AToM predicts that AI will be most beneficial when used to augment human coherence—through adaptive pacing, contextual modulation, narrative repair, and multi-modal integration—and most harmful when used to impose rigid synchrony or amplify high-frequency perturbations. Coherence geometry thus provides a rigorous basis for evaluating AI alignment: aligned systems are those that preserve and extend human coherence across scales rather than constrict or destabilize it.

Finally, AToM points toward a general science of meaning, wherein meaning is not a semantic property but the felt signature of coherence under constraint. Meaning emerges when systems maintain integrable trajectories across physiological, cognitive, relational, and cultural manifolds. Loss of meaning reflects coherence collapse. Restoration of meaning is coherence reconstruction. This reframing unifies phenomenology, neuroscience, developmental psychology, behavioral science, organizational theory, and cultural history within a single structural language. It suggests that disciplines long separated by methods and metaphors can converge on coherence geometry as the shared variable that renders human systems intelligible.

In sum, AToM’s frontier implications are not extensions into new domains but clarifications of how coherence geometry already structures them. The theory provides the conceptual and mathematical tools needed to build a unified science that spans mind, body, relationship, institution, and culture—a science oriented not around content, representation, or isolated processes, but around the dynamic maintenance and transformation of coherence across nested scales of complexity.

11. Conclusion: Coherence as the Structural Invariant of Mind

AToM advances a single unifying claim: cognition, at every scale, is organized by the geometry of coherence. Embodied processes, environmental affordances, enactive dynamics, and extended architectures provide the locations and modalities through which cognition unfolds, but none of these dimensions account for the structural conditions that determine when cognition holds together, why it collapses, or how it propagates across neural, physiological, relational, and institutional levels. Coherence fills this gap by providing a measurable, multi-scale invariant that describes how systems maintain integrable trajectories under constraint. In doing so, it transforms the 4E paradigm from a descriptive framework into a structurally complete theory capable of supporting empirical validation, mathematical modeling, and cross-domain generalization.

AToM’s account of coherence—as a tuple consisting of curvature smoothness, dimensional stability, persistence lifespan, and cross-frequency coupling ratio—clarifies that coherence is not a phenomenological impression or metaphor but a geometric property of the system’s underlying manifold. These components are directly measurable across neural activity (via functional connectivity and oscillatory coupling), physiological regulation (via HRV, EDA, interoceptive volatility), linguistic organization (via embedding curvature and narrative persistence), behavioral dynamics (via movement smoothness and transition entropy), relational patterns (via synchrony indices and rupture–repair structure), and institutional or cultural systems (via communication topology, symbolic persistence, and cross-temporal alignment). Because coherence is measurable across modalities, it allows cognition to be studied through a unified vocabulary that does not depend on domain-specific constructs or theoretical silos.

This structural grounding clarifies phenomena that have remained difficult to integrate. Trauma, for example, becomes legible as dimensional collapse, curvature amplification, and topological bottlenecking—the deformation of a manifold rather than a psychological category or narrative disruption. Neurodivergence appears not as disorder but as variation in coherence sensitivity, with some systems exhibiting higher resolution detection of pattern breaks and others exhibiting greater smoothing. Interpersonal dynamics can be evaluated by examining whether coupling patterns preserve or erode coherence. Institutional resilience becomes a question of cross-scale entrainment, where mismatch among tempos or loss of persistence across processes predicts collapse. Cultural stability becomes a question of symbolic, ritual, and narrative persistence across generational and civilizational manifolds.

AToM also clarifies how meaning emerges. Meaning is not located in symbols, internal representations, or subjective impressions; it is the felt signature of coherence. When curvature smooths, dimensions reopen, topological structures persist, and rhythms align across scales, the system experiences orientation, intelligibility, and viability. When coherence collapses—through overload, contradiction, misalignment, or trauma—the system experiences confusion, fragmentation, or loss of meaning. Meaning is therefore not an additional property layered onto cognition but the experiential correlate of coherence maintenance across internal and external constraints. This reframing dissolves longstanding divides between phenomenology and computation, subjective experience and objective structure, narrative coherence and physiological regulation.

Because coherence is structural, AToM connects cognitive science to adjacent disciplines without requiring reduction. Predictive processing describes local stability mechanisms but not global coherence geometry; AToM supplies the manifold on which prediction becomes stable or unstable. Enactivism describes interactional and relational dynamics but lacks formal invariants; AToM provides the geometric criteria that distinguish productive coupling from collapse. Interpersonal neurobiology articulates the importance of integration across subsystems but lacks the topological and information-geometric vocabulary needed to formalize integration. Cultural evolution describes macro-scale patterns without modeling coherence propagation across levels. AToM integrates insights from each of these traditions by offering a domain-general structural scaffold.

AToM’s mathematical foundations—drawn from information geometry, dynamical systems theory, and topological data analysis—ensure that coherence can be formalized without importing inappropriate analogies from physics. The Fisher Information Metric, the Amari–Chentsov tensor, persistent homology, and cross-frequency coupling measures provide the minimal set of established constructs required to describe coherence. AToM’s two proposed additions—the Coherence Operator (C_OPERATOR) and the Trauma Sensitivity Constant (TAU)—serve as conceptual integrators rather than speculative equations, offering future directions for formal development without compromising the theory’s current empirical grounding.

This structural clarity yields falsifiable predictions across domains. Coherence metrics should outperform psychometric assessments; linguistic curvature should shift before subjective insight; autistic–LLM teams should detect coherence fractures more reliably than neurotypical raters; entrainment breakdowns should precede relational and institutional collapse; and topological bottlenecks should predict trauma relapse risk. These predictions provide routes for empirical adjudication and ensure that AToM remains accountable to data rather than philosophical elegance alone.

By offering a multi-scale, measurable, mathematically grounded account of coherence, AToM moves cognitive science toward a unified science of meaning. Meaning is coherence under constraint; coherence is the organization of trajectories across nested manifolds; entrainment is the dynamic class that maintains coherence across timescales and subsystems. This framework allows disciplines traditionally separated by method, vocabulary, or epistemology to converge on a shared structural variable. Neuroscience, psychotherapy, education, organizational theory, anthropology, political science, and artificial intelligence can all be linked through coherence geometry without collapsing their unique phenomena into a single level of description.

In its final form, AToM is not a new paradigm that replaces existing ones, but a structural completion of those already in place. It clarifies the underlying geometry that predictive processing minimizes, that enaction traverses, that embodiment grounds, that interpersonal attunement stabilizes, that institutions scaffold, and that cultures preserve. By identifying coherence as the fifth structural E, AToM provides a mathematically principled, empirically testable, and conceptually parsimonious model of how meaning is generated, maintained, lost, and restored across the full architecture of human life.

Meaning is not mysterious.

It is what coherence feels like from the inside.