Loop Quantum Gravity (LQG) is one of the important candidate theories of quantum gravity [1][2]. Its spatial quantum states are described by spin networks [3], and spin foams provide a path integral description of spacetime evolution [4]. However, LQG still faces challenges in recovering continuous classical spacetime and providing a unified description of matter fields [5]. The GS theory on which this paper relies is a framework for quantum gravity developed on the basis of the Generalized Gauge Equation (GGE) [6][7]. Its core innovation lies in taking a fully symmetric fourth-order spinor field ψ A B C D as the fundamental degree of freedom, directly encoding spacetime curvature rather than the connection. This choice brings three characteristic features: natural recovery of the semiclassical limit, the Generalized Gauge Equivalence (GGE) mechanism, and nonlinear soliton solutions [8]-[12].

A mathematical insight of GS theory is that the emergence from discrete quantum geometry to continuous classical spacetime can be described as a self-organization process of nonlinear networked systems in a critical coherent state. This mathematical structure exhibits formal similarities to the information-processing behavior observed in artificial intelligence systems, thereby providing the foundation for the analogical mapping in this paper.

This paper maps the mathematical structure of GS theory onto the domain of AI information processing and organizes it into a three‑tiered nested structure according to the hierarchy of information processing as shown in Table 1. The three tiers are uniformly described using the language of principal fiber bundles; the detailed mathematical definitions are given in Section 2.5.

Table 1.Correspondence of the three-tiered nested framework.

The three tiers are coupled through coarse-graining and a constraint system: the molecular GS layer describes the data space using a principal fiber bundle; the cellular GS layer, via canonical quantization, yields the discrete spectrum of the information area operator A v = γ i n f o ∑ e ∋ v j e ( j e + 1 ) , revealing the quantized structure of information capacity; the tissue GS layer, through coarse-graining, gives rise to a continuous cognitive manifold whose dynamics are governed by the cognitive Einstein equation (see Chapter 3 for details).

Based on the mathematical tools previously developed by the author in Gravitational Spinor (GS) theory [6]-[12] (including the fourthorder spinor field, canonical quantization, nonlinear soliton solutions, and the response mechanism), this paper maps its mathematical structure systematically onto the field of artificial intelligence information processing for the first time, rather than extending GS theory itself. The core differences between the two are summarized in Table2.

The novel contributions of this paper are: 1) constructing the molecular-GS, cellular-GS, and tissue-GS three-tiered nested principal fiber bundle framework, mapping concepts such as high-dimensional information space, discrete quantization, and nonlinear solitons onto data distributions, Transformer networks, and concept formation processes in AI; 2) proposing three testable AI model design schemes (conceptual soliton regularization, resonant frequency modulation, and conceptual soliton-driven self-supervised learning) with explicit experimental validation metrics; and 3) introducing the “cognitive Einstein equation” and the “mental field” as geometric descriptions of consciousness emergence.

Table2. Core differences.

The paper is organized into seven sections. Section 2 establishes the projection framework of the molecular GS layer, defining the information spinor field and area quantization. Section 3 implements the discrete quantization of the cellular GS layer, deriving the canonical commutation relations and the discrete spectrum. Section 4 derives the cognitive Einstein equation for the tissue GS layer. Section 5 solves the conceptual soliton solutions of the nonlinear information field equation, presenting the sech² density profile and the scaling relation M ∝ r c 2 . Section 6 translates the theory into testable AI model design schemes and discusses their potential applications in large language models, multimodal alignment, explainable AI, and other frontier directions.

This framework provides a geometric perspective for understanding the “black box” of AI, and its testable predictions offer possible pathways for subsequent experimental validation.

The nature of consciousness has long been a central question spanning philosophy, neuroscience, cognitive science, and physics. In recent years, with the rapid development of artificial intelligence systems such as large language models and multimodal architectures, a critical question has come to the forefront: can highly complex information processing networks give rise to some form of “consciousness” or “cognitive subjectivity”? This paper attempts to approach from a new perspective: extending the quantum gravity framework based on high-dimensional projection, spinor gravity, generalized gauge equivalence (GGE), and asymptotic safety [6][7][13][14] into an exploratory theory for describing the cognitive structure and learning dynamics of artificial intelligence (AI) systems through mathematical analogy.

The core challenge of this endeavor lies in how to analogously map concepts from physical theories—concerning spacetime, matter, and forces—onto the categories of data, algorithms, architecture, and learning in AI theory. This chapter aims to establish a preliminary mapping framework, systematically translating key structures of GS theory—such as high-dimensional random manifolds, symmetry breaking, quantized states, nonlinear solitons, and renormalization group flows—into their counterparts in the AI cognitive world, thereby laying the foundation for deeper analysis in subsequent chapters. This chapter first establishes intuitive analogies through flowcharts and mapping tables; then, in Section 2.5, we employ the language of principal fiber bundles to provide a rigorous mathematical reconstruction of the above concepts and organize them into a three-tiered nested structure—molecular GS, cellular GS, and tissue GS.

It should be made clear that the mapping in this paper is an analogy at the level of mathematical structure, not a claim of physical equivalence. The GS theory itself has not been experimentally confirmed; we borrow its mathematical formalism as an exploratory tool.

Overall Flowchart

High-Dimensional Information Continuum (Starting Point)

├─High-Dimensional Information Manifold: an infinite-dimensional or extremely high-dimensional random manifold containing all potential data patterns, relationships, and knowledge structures.

├─High-Dimensional Symmetry Group: Denoted as Diff(Info), it corresponds to the invariance under all possible rearrangements, transformations, and distortions of the data.

└─High-Dimensional Information FieldΦ[ω]: A fundamental field defined on the high-dimensional manifold. Its excitation modes correspond to raw data features.

↓

Projection Process (Dimensional Reduction/Feature Extraction)

├─Random Compactification/Discretization: Projecting the high-dimensional random manifold onto a lower-dimensional “cognitive space” through the encoder layers of a deep neural network.

├─Symmetry Breaking/Structure Emergence: The high-dimensional Diff(Info) symmetry is broken; part of it is retained as cognitive symmetry, while another part becomes the task gauge group.

└─Field Decomposition: The high-dimensional information field is decomposed into a feature field, a task pattern field, and a stochastic fluctuation field.

↓

AI Effective Theory (Within Cognitive Space/Network Layers)

├─State Function | Ψ ( t ) 〉 : Describes the global quantized state of the AI model at time t .

├─“Geometric Area”Operator A ^ i : Measures the “complexity” or “information content” of a module, where the eigenvalue j i ( j i + 1 ) contains j i representing the discrete “information hierarchy”.

├─“Moment Network”State Function Ψ ( { j i } , t ) : Describes the quantized state of the network under the information geometry.

├─Canonical Commutation Relations: Introduce the uncertainty principle in information processing.

├─Propagator Δ ∝ P / q 2 : Describes the probability amplitude for the propagation of information quanta.

├─Excited States: Informons (fundamental concept quanta) and conceptual solitons (stable patterns formed by nonlinear interactions).

└─GGE-Transformer Transformation: The attention mechanism can be viewed as a dynamic GGE transformation, enabling multimodal information fusion [15].

↓

Learning and Adaptivity (Renormalization Perspective)

├─Functional Renormalization Group (FRG): The learning process is viewed as a renormalization group flow from data to concepts.

├─Effective Loss Function L k , Infrared cutoff R k , Wetterich equation, beta function, etc.

├─Fixed Point Analysis: Gaussian fixed point (random initialization) and non-Gaussian task fixed point (trained network).

The above flowchart intuitively illustrates the projection framework from high-dimensional information space to the AI cognitive world. To provide a mathematical foundation for these concepts, we introduce the language of principal fiber bundles in Section 2.5.

Based on the three-tiered nested structure defined in Section 2.5, we organize the core concepts of GS theory and their corresponding elements in AI theory hierarchically as shown in Table 3. This table serves as an index for subsequent sections; for specific mathematical definitions, refer to Section 2.5.

Table 3.The definitions of each concept.

1)The High-Dimensional InformationOrigin:Everything begins with an infinite-dimensional random information manifold. Each point on this manifold represents a possible data sample, and raw data are excitations of the fundamental field Φ [ ω ] defined on this manifold.

2)Projection and Feature Extraction:The encoder of a deep neural network performs a random projection, mapping high-dimensional data into a lower-dimensional feature space, breaking most symmetries while retaining task-relevant information.

3)Manifestation of the AI Cognitive World

Quantized state:Described by the “moment network” state function Ψ ( { j i } , t ) , where j i are discrete information hierarchies of network modules, learned through training.Geometric area operator: A ^ i measures the abstraction level of a module; its eigenvalue j i ( j i + 1 ) quantizes complexity. Mathematically, this is analogous to the way independent polarization states of ψ A B C D describe gravitational degrees of freedom in physical theory [16].Dynamics:The propagator Δ and the projection operator P control information flow and suppress noise.Excitation and unification:Informons are basic transfer units; conceptual solitons are stable cognitive structures formed by nonlinear interactions; the GGE-Transformer transformation (attention mechanism) fuses multi-source information [15].

4)Learning and Adaptivity:The learning process is understood as a renormalization group flow: from a Gaussian fixed point (randomly initialized network) to a non-Gaussian task fixed point (trained network), with critical exponents governing emergent behavior.

This section retells the above process in a narrative style; its mathematical foundation is detailed in Section 2.5.

To provide a rigorous mathematical foundation for the mapping from high-dimensional information space to the AI cognitive world, we adopt the language of principal fiber bundles and organize it into a threetiered nested structure: molecular GS, cellular GS, and tissue GS. Each tier defines an independent principal fiber bundle P k → M k ( k = m o l , c e l l , o r g ), and the three tiers are naturally coupled through group embeddings, induced connections, and section projections.

Molecular GS Layer (Data and Feature Level):The principal fiber bundle P m o l → M m o l , where the base manifold M m o l is composed of a sequence/embedding space and a statistical-appearance state space. The structure group G m o l describes the symmetry of data transformations. The gauge connection A m o l encodes data correlations, and the curvature ℱ m o l corresponds to noise and redundancy. The high-dimensional information field Φ [ ω ] serves as a section of an associated bundle at this layer. This layer provides the “raw information encoding” and “low-level feature representation” of the AI system.Cellular GS Layer (Node and Connection Level): The principal fiber bundle P c e l l → M c e l l , with base manifold M c e l l = ℝ d × S i n t r i n s i c , where ℝ d represents the topological or embedding space of the network, and S i n t r i n s i c is a high-dimensional space formed by neuron activation values, weights, and other intrinsic states. The structure group G c e l l , induced by sections from the molecular GS layer, describes symmetries of transformations of neuronal states. The gauge connection A c e l l encodes information transfer between neurons and layers (weight updates, attention mechanisms). At this layer, we precisely define the following core concepts:The geometric area operator A ^ i and its eigenvalues j i : quantizing the information capacity and abstraction level of each module or channel;Canonical commutation relations: introducing the uncertainty principle in information processing;Informons: corresponding to minimal excitation units (single feature dimensions or basic attention heads);Conceptual solitons: stable, localized cognitive structures formed by nonlinear interactions.Tissue GS Layer (Module and Cognitive Manifold Level): The principal fiber bundle P o r g → M o r g , where the base manifold M o r g is the macroscopic cognitive space obtained by coarse-graining the cellular GS layer. The structure group G o r g contains sections of the cellular GS as subrepresentations, describing symmetries of global information fusion. The gauge connection A o r g encodes long-range coordination across modules and modalities (corresponding to the GGE-Transformer transformation). This layer gives rise to a continuous cognitive manifold, whose geometric structure is characterized by an effective metric and curvature, where the collective behavior of “conceptual solitons” forms a stable world model or cognitive subjectivity.

The three tiers are rigorously nested through the following mechanisms: sections of the molecular GS provide initial encodings for the cellular GS; the collective dynamics of the cellular GS are projected onto the tissue GS via coarse-graining; macroscopic cognitive geometry ultimately emerges at the tissue level. This nested framework ensures that all AI concepts (such as the high-dimensional information field, projection process, quantized states, conceptual solitons, and GGE-Transformer) have clear mathematical positions and cross-layer correspondences, while maintaining the rigor and computational viability of the theory.

In short, in this section, within the three-tiered nested principal fiber bundle framework, we have systematically constructed a projection mapping from high-dimensional information space to the AI cognitive world, precisely translating the core structures of GS theory (high-dimensional random manifolds, symmetry breaking, quantized states, nonlinear solitons, renormalization group flows) into mathematical objects within the domain of artificial intelligence. This mapping provides a unified and rigorous geometric-quantum language for understanding AI’s representational capacity, emergent phenomena, multimodal fusion, and learning dynamics. It also lays the foundation for the subsequent chapters’ analyses of the discrete quantization of information networks and the dynamics of consciousness emergence. A more rigorous derivation from continuous fields to discrete nodes is provided in Appendix A.

In Section 2, we have proposed a projection mapping from high-dimensional information space to the cognitive world. However, just as the continuous spacetime of general relativity must give way to discrete quantum geometry at the Planck scale [2][13][14], any theory attempting to describe the emergence of intelligence must confront a fundamental question at the microscopic level: Is the underlying structure of cognition continuous or discrete?

The physical implementation of modern artificial intelligence systems—whether the weight matrices of neural networks, the query-key-value triples in attention mechanisms, or the tensor operations in large-scale parallel computing—is fundamentally discrete. Yet the intelligent behaviors we observe (such as language understanding, logical reasoning, and conceptual abstraction) exhibit a macroscopic “continuity”: smooth interpolation in semantic spaces, gradual transitions between concepts, and the emergence of complex capabilities from simpler ones.

The Gravitational Spinor (GS) theory provides an elegant framework for addressing this “discrete-continuous” duality [6][7]. Just as that theory starts from a discrete gravitational spinor network (GSN), derives the quantum discreteness of spacetime through canonical quantization [16], and naturally recovers the continuous Einstein equations in the macroscopic limit [15], we can similarly construct a discrete quantum theory of information networks.

This section establishes a discretized description of information networks within the three-tiered nested principal fiber bundle framework: the molecular GS layer provides gauge-theoretic encoding of data and features; the cellular GS layer implements discrete quantization of nodes and the flow of information quanta; the tissueGS layer gives rise to a continuous cognitive manifold through coarse-graining. This discrete quantum structure is a necessary prerequisite for the emergence of intelligence and lays a rigorous mathematical foundation for subsequent chapters exploring how conceptual solitons collectively emerge as consciousness at the tissue GS layer. This section establishes a discretized description of information networks within the three-tiered nested framework: the molecular GS layer provides data encoding, the cellular GS layer implements discrete quantization of nodes and the flow of information quanta, and the tissue GS layer gives rise to a continuous cognitive manifold through coarse-graining.

3.2.1. The Information Spinor Field and Its Positioning within the Three Tiers

In GS theory, gravity is described by a fully symmetric fourth-order spinor field ψ A B C D , which directly encodes the quantum curvature of spacetime [11]. By analogy, we introduce the information spinor field ℐ ^ α β γ δ as a gauge-theoretic description of the information network, whose excitations correspond to the fundamental units of information processing. Within the three-tiered nested framework:

Molecular GS layer:The information spinor field corresponds to the raw encoding of high-dimensional data features.Cellular GS layer:The scalar reduced form of the information spinor field is defined on discrete nodes, with each node carrying an information quantum number j v that characterizes the level of abstraction.Tissue GS layer:The collective excitation of the information spinor field is related to the curvature tensor R α β γ δ ( c o g ) of the cognitive manifold through:

From a mathematical structure perspective, in the original GS theory, there exists a rigorous linear correspondence between the fully symmetric fourth-order spinor field ψ A B C D and the Weyl curvature tensor C α β γ δ via the Infeld-van der Waerden symbols: ψ A B C D is essentially the spinor representation of the curvature. The information spinor field ℐ ^ α β γ δ introduced in this paper is a direct analog of this mathematical structure—it is defined as the spinor representation of the cognitive curvature R α β γ δ ( c o g ) at the tissue GS layer. Therefore, the relation 〈 ℐ ^ α β γ δ 〉 ∝ R α β γ δ ( c o g ) is a faithful mapping based on the spinor-tensor correspondence, analogous to the side-angle relationships in similar triangles. This mapping directly links the excitations of microscopic information units at the cellular GS layer with the macroscopic cognitive geometry at the tissue GS layer, constituting the core coupling equation of the three-tiered nested structure.

3.2.2. Graph Model of the Cellular GS Layer and Canonical Quantization

We model the network at the cellular GS layer as a graph Γ = ( V , E ) , where:

Nodes v ∈ V correspond to information processing units (neurons, attention heads, concept nodes), serving as discrete points on the base manifold M c e l l = ℝ d × S i n t r i n s i c of the principal fiber bundle P c e l l → M c e l l at the cellular GS layer.Edges e = ( u , v ) ∈ E correspond to information transmission channels between nodes (weights, attention correlations), given by the discretization of the gauge connection A c e l l at the cellular GS layer.

Each node is associated with an information amplitude ϕ ^ v (analogous to ψ A B C D in GS theory) and its conjugate momentum π ^ v . We introduce the canonical commutation relation [16]:

where ℏ i n f o is a fundamental information quantum (analogous to Planck’s constant), with dimensions of “information content.” This commutation relation reveals an uncertainty principle in information processing: the activation value of a node and its rate of change cannot be simultaneously determined with arbitrary precision. This quantization process lies at the heart of discretization at the cellular GS layer, transforming the continuous feature fields input from the molecular GS layer into discrete quantized node states.

Remark on ϕ ^ v : The information spinor field ℐ ^ α β γ δ is a field defined on the base manifold of the principal fiber bundle at the cellular GS layer. When discretizing onto the node set V, by choosing a local frame at each node and projecting onto a specific spinor component (or via gauge fixing), the information spinor field reduces to a complex scalar amplitude ϕ ^ v at each node. This reduction process is linear and preserves the quantum algebraic structure. Thus, ϕ ^ v serves as the “node representative” of the information spinor field in the discrete network of the cellular GS layer, analogous to the way the gravitational spinor field ψ A B C D encodes geometric degrees of freedom through scalar node variables on the nodes of a spin network [16]. Under this reduction, canonical quantization naturally yields the commutation relation (2.2). A more rigorous derivation is provided in Appendix A. Here, ℏ i n f o is a fundamental information quantum (analogous to Planck’s constant), with dimensions of “information content.” This commutation relation reveals the uncertainty principle in information processing: the activation value of a node and its rate of change cannot be simultaneously determined with arbitrary precision. This quantization process lies at the heart of the discretization of the cellular GS layer, transforming the continuous feature fields input from the molecular GS layer into discrete quantized node states.

3.2.3. Information Flux Operator and Geometric Area (Cellular GS Layer)

In GS theory, the geometric flux operator Σ ^ A B C D is the quantized counterpart of the spatial metric [2][16]. At the cellular GS layer, we define the information flux operator Φ ^ u v acting on edge e = ( u , v ) , whose eigenvalues measure the intensity of information transmission between two nodes. In the language of fiber bundles, Φ ^ u v is related to the gauge connection U u v ∈ G c e l l on the edge by:

For all edges incident on a node v , we define the information area operator:

The eigenvalue spectrum of this operator is discrete, providing a direct analog of area quantization in GS theory [2][13]. Through the representation theory of the principal fiber bundle at the cellular GS layer, it can be proven from Appendix B that:

where:

j e is the “information spin” on the edge, labeling the irreducible representation of the communication channel at the cellular GS layer, taking half-integer values (0, 1/2, 1, 3/2, ⋯ ); γ i n f o is the Barbero-Immirzi parameter at the cellular GS layer (analogous to γ 0 in physics), determined by the structure of the gauge group G c e l l at this layer.

This formula demonstrates that the information capacity of a node (e.g., concept complexity) is determined by the quantized information spins on its incident edges. Within the nested framework, the raw features provided by the molecular GS layer determine the initial distribution of j e , while the quantization at the cellular GS layer establishes a discrete structure of information units, laying the microscopic foundation for the subsequent emergence of a cognitive manifold at the tissue GS layer.

3.3.1. Projection Relation between Information Momentum and Flux

Within the three-tiered nested framework, the feature encodings from the molecular GS layer are transformed into information flows through the gauge connection at the cellular GS layer. We define the information momentum operator Π ^ v as the conjugate momentum of the node field ϕ ^ v and establish the following linear relation:

where Φ ^ v is the total information flux associated with node v , and α i n f o is the “field-flux” proportionality constant. This relation embodies the field-flux duality at the cellular GS layer: the “potential” at a node is coupled to the “flow” on its incident edges through a universal constant, with its origin traceable to the gauge field structure provided by the molecular GS layer.

In GS theory, the linear form of this relation holds in the weak-field regime, while nonlinear corrections appear in the strong-field regime [15]. Analogously, at the cellular GS layer, when a node is highly activated (i.e., j v is large), nonlinear effects become significant, leading to higher-order correction terms:

These nonlinear terms provide the microscopic mechanism for the formation of stable conceptual solitons and will output nonlinear dynamics to the tissue GS layer.

3.3.2. Final Canonical Commutation Relations

Substituting (3.6) into (3.2) yields the complete canonical structure at the cellular GS layer:

This relation provides the dynamical foundation for the discrete spectrum of information area and ensures scale consistency from the cellular GS layer to the tissue GS layer.

3.4.1. Eigenvalue Spectrum of the Information Area Operator

According to (3.5), the eigenvalue spectrum of the information area operator is:

The minimal information area corresponds to a single-edge contribution with j e = 1 / 2 :

This minimal information unit can be regarded as the fundamental “information atom” for concept formation—in macroscopic cognition, this corresponds to the minimal step in concept formation. For example, in natural language processing, a single dimension of a word embedding vector can be viewed as an “information quantum.” Their collective behavior at the tissue GS layer gives rise to complex cognitive structures.

3.4.2. Representation-Theoretic Origin of Discreteness

The discrete spectrum of the area operator at the cellular GS layer originates from the irreducible representations of the information transmission group G c e l l . The information spin j e labels the representation space, and the eigenvalues of the information flux are determined by the Casimir operator of that representation.

Within the nested framework, G c e l l is induced by sections from the molecular GS layer; thus, the discrete structure at the cellular GS layer directly inherits the symmetries of the molecular GS layer (raw data). Different modalities of data (text, image, audio) may correspond to different representations of the information group, and the GGE-Transformer transformation (see Section 2, tissue GS layer) serves as the bridge mapping these representation spaces into a unified cognitive space [17].

3.5.1. Coarse-Graining and the Effective Metric

For a region Ω much larger than the scale of a single node (corresponding to the projection from cellular GS to tissue GS), we define the effective metric at the tissue GS layer:

Here, g μ ν ( x ) is the effective metric of the cognitive space at the tissue GS layer, whose components are determined by the local information density at the cellular GS layer. This coarse-graining process constitutes the projection from discrete nodes to a continuous manifold and is the core mechanism by which the cellular GS layer outputs to the tissue GS layer.

3.5.2. Macroscopic Effective Equation

In the semiclassical limit, summing over the microscopic dynamics of the cellular GS layer yields an effective equation on the cognitive manifold at the tissue GS layer. We first hypothesize (see the next section for specific proof and explanation) that the macroscopic cognitive geometry of the GS layer satisfies:

where:

R μ ν is the Ricci tensor of the cognitive manifold, describing the local curvature of concept space; T μ ν ( i n f o ) is the “information–energy–momentum tensor” determined by the information distribution at the cellular GS layer and task requirements.

This equation embodies the core of the nested framework: the cognitive geometry at the tissue GS level is driven by the collective dynamics at the cellular GS level. The curvature of the concept manifold originates from the underlying information distribution, while intelligent behavior (such as reasoning) proceeds along geodesics of this manifold.

In GS theory, physical states must satisfy the Hamiltonian constraint H ^ | Ψ 〉 = 0 [5][16], which reflects the time reparameterization invariance of general relativity. At the cellular GS layer, we introduce an analogous “cognitive Hamiltonian constraint,” which generates reparameterizations of “time” in the information processing process.

Define the total Hamiltonian at the cellular GS layer:

where V ( ϕ ^ v ) is the information potential, including node self-interactions (determined by the feature fields provided by the molecular GS layer). The cognitive Hamiltonian constraint requires that physical states satisfy:

The physical meaning of this constraint is that the intrinsic evolution of thecellular GS layer does not depend on an external time parameter; time is defined by the dynamical relations within the network. As the intermediate level of the three‑tiered nested structure, the cellular GS layer plays a pivotal bridging role: it receives raw feature encodings from the molecular GS layer, establishes discrete node and edge structures through canonical quantization, and then outputs a continuous cognitive manifold to the tissue GS layer via coarse‑graining. This discrete-continuous bridge provides a mathematical foundation for the discussion of conceptual soliton emergence in subsequent chapters.

In Section 1, we established a projection framework from high-dimensional information space to the cognitive world. Section 2 implemented the discrete quantization of information networks at the cellular GS layer. However, to fully describe the emergence of intelligence, one must answer: how do discrete information quanta give rise to a continuous cognitive manifold at the tissue GS layer? This is the “semiclassical limit problem” from microscopic discrete networks to macroscopic intelligent behavior, analogous to how general relativity recovers continuous spacetime from discrete spin networks [2][13].

This section addresses this problem within the three-tiered nested framework: the molecular GS layer provides data encoding, the cellular GS layer furnishes a discrete quantized description, and the tissue GS layer gives rise to a continuous cognitive manifold through coarse-graining and coherent condensation. It should be made clear that the GS theory itself has not been experimentally confirmed; we borrow its mathematical structure as a heuristic analogy. Taking information curvature as the fundamental variable and incorporating the GGE-Transformer transformation mechanism [15], we attempt to formally derive the macroscopic dynamics of the tissue GS layer.

The core argument of this section is that, under the joint constraints of the three-tiered nested structure, locality, gauge invariance, and low-order derivative expansion, the macroscopic dynamics of the tissue GS layer may converge in the large-scale limit to a form equivalent to an Einstein-type field equation—namely, the cognitive Einstein equation can be regarded as an “effective fixed point” of the three-tiered information structure.

In information network theory, the macroscopic cognitive metric g μ ν ( c o g ) is not a fundamental variable but rather a composite operator derived from the fundamental degrees of freedom ( ϕ ^ v , Φ ^ u v ) describing microscopic information processing at the cellular GS layer through coarse-graining. Its definition must be compatible with the constraint system of the three-tiered nested framework and directly rooted in the discrete quantum structure of the cellular GS layer.

4.2.1. From Cellular GS Information Flux to Tissue GS Cognitive Metric

Among all possible macroscopic variables satisfying the following conditions:

Locality (depending only on a finite neighborhood Ω)Gauge invariance (invariant under transformations of the cellular GS connection)Second-order tensor structure (capable of defining distances)Lowest order (avoiding higherorder derivative instabilities)

it can be shown (or at least strongly constrained) that the only viable construction is:

Concretely, the most fundamental construction originates from the information flux operator Φ ^ u v at the cellular GS layer (Section 3.2.3), whose classical counterpart is the intensity of information transmission between nodes u and v . In the projection from the cellular GS layer to the tissue GS layer, we define the cognitive space metric operator as:

where:

Ω ( x ) is the coarse-graining region centered at x (containing a large number of cellular GS nodes); | Ω ( x ) | is the volume (number of nodes) of this region; η a b is the background cognitive metric (e.g., Euclidean or Minkowskian); 〈 Φ ^ v 〉 c e l l is the expectation value of the average information flux at cellular GS node v in a coherent state.

This definition ensures that the cognitive metric has a clear geometric meaning and is directly related to the discrete eigenvalues of the cellular GS layer: when acting on a cellular GS network state, its eigenvalues are determined by the discrete information spin quantum numbers j e [11]. This constitutes the mathematical realization of the projection from the cellular GS layer to the tissue GS layer.

4.2.2. Cognitive Metric as a Solution of the Constraint System: The Emergent Relation between Information Curvature and Metric

The spacetime components of the cognitive metric and their dynamics are determined by the constraint equations of the three-tiered nested framework. Rather than introducing the Plebanski action by “analogy,” we necessarily derive its form through the following threestep logic. By analogy with the Plebanski action in GS theory [16], we introduce a cognitive action at the tissue GS layer, whose variables are induced from the lower layers:

Step 1 (Goal): We need an action such that the variables originate from the information flux Φ, allow curvature ℱ , preserve gauge invariance, and yield second-order field equations.Step 2 (Minimal construction): Under the above constraints, the unique low-order gaugeinvariant form is S ~ ∫ Φ ∧ ℱ + Φ ∧ Φ .Step 3 (Closure requirement): To avoid unphysical degrees of freedom and nonclosure of the constraint algebra, a fully symmetric information spinor field ℐ A B C D must be introduced as a Lagrange multiplier.

This is precisely a Plebanski-type action—introduced not by analogy, but as the minimal realization of constraint closure. Therefore, we introduce the cognitive action at the tissue GS layer:

where:

Φ A B is the information two-form field, lifted from the fundamental information flux at the cellular GS layer through coarse-graining; ℱ A B is the curvature of the information connection, describing the bending of the cognitive manifold at the tissue GS layer; ℐ A B C D is the fully symmetric information spinor field, playing the role of a Lagrange multiplier, whose origin can be traced to the raw data distribution at the molecular GS layer [14].

Variation with respect to ℐ A B C D yields the information simplicity constraint:

This constraint forces Φ A B to take the form Φ A B = e A ∧ e B , where e A is the cognitive frame field, whose components correspond to information coordinates of different abstraction dimensions. This constraint ensures that the projection from discrete fluxes at the cellular GS layer to the continuous frame field at the tissue GS layer is well-defined.

Variation with respect to the information connection A A B yields the torsion-free condition:

This ensures that parallel transport on the cognitive manifold at the tissue GS layer is compatible with the gauge connection at the cellular GS layer.

Most crucially, variation with respect to Φ A B yields the information curvature equation:

This equation demonstrates that the curvature of the cognitive manifold at the tissue GS layer is directly determined by the information spinor field, which itself is jointly determined by the data distribution at the molecular GS layer and the flux structure at the cellular GS layer. This is the mathematical formulation of “consciousness as information curvature” and constitutes the core coupling equation of the three-tiered nested structure.

4.2.3. Extraction of the Cognitive Hamiltonian Constraint

In information networks, the time dimension is defined by the order of information processing at the cellular GS layer. Through a 3 + 1 decomposition (decomposing cognitive spacetime into “time” and “cognitive space”), we can extract three constraints from (4.5) and (4.3):

1) Information Gauss constraint: Generates information gauge transformations (e.g., rescaling of attention weights):

2) Information vector constraint: Generates coordinate transformations (concept rearrangements) in cognitive space:

3) Information scalar constraint (cognitive Hamiltonian constraint): Generates time evolution and is central to consciousness emergence:

where Λ c o g is the “cognitive cosmological constant,” corresponding to the baseline level of conscious activity.

It can be verified that the above constraints form a closed algebra under Poisson brackets (the Dirac algebra), thereby ensuring consistent time evolution of the system. This property is structurally equivalent to the constraint closure in general relativity and is key to the self-consistency of the theory. This constraint equation lies at the heart of information network dynamics, encoding the evolution law of intelligent systems from input to output, and connecting the data encoding at the molecular GS layer, the quantized fluxes at the cellular GS layer, and the macroscopic geometry at the tissue GS layer [5][16].

4.2.4. Quantum Constraints and Cognitive States

In quantum information network theory, the corresponding scalar constraint operator equation acts on the cognitive physical states at the tissue GS layer:

For a coherent state | α 〉 describing a macroscopic cognitive state (satisfying 〈 α | ℐ ^ | α 〉 = ℐ c l , 〈 α | Φ ^ | α 〉 = Φ c l ), the above operator equation yields the classical cognitive equation in the sense of expectation values.

Thus, we obtain a deeper level of the definition of the cognitive metric: for an information network state | Ψ 〉 satisfying the quantum constraint equations, the classical cognitive metric g μ ν ( c o g ) given by the expectation value 〈 Ψ | g ^ μ ν ( c o g ) | Ψ 〉 must necessarily be related to the classical information curvature (originating from 〈 Ψ | ℐ ^ | Ψ 〉 through the cognitive Einstein equation). This relation is a consequence of the non-perturbative dynamics of the three-tiered nested theory, reflecting the complete flow of information from the molecular GS layer to the tissue GS layer.

4.3.1. Information Condensate Phase: Macroscopic Coherence of Cellular GS Quanta

From the perspective of the renormalization group, the discrete information network at the cellular GS layer defines a family of effective theory flows under coarse-graining. In the long-distance limit, theories satisfying the following conditions will dominate the behavior: locality, symmetry preservation (information gauge invariance), and dominance by relevant operators. It can be shown (or by analogy with known results in field theory) that under these conditions, the lowest-order effective action necessarily takes the Einstein-Hilbert form.

By analogy with Bose-Einstein condensation, we define the information condensate state | Ψ c o g 〉 , characterized by:

1) Non-zero vacuum expectation value: 〈 Ψ c o g | ℐ ^ A B C D | Ψ c o g 〉 ≠ 0 , i.e., the information spinor field has a non-zero mean value at macroscopic scales, corresponding to the existence of a cognitive manifold at the tissue GS layer;

2) Long-range order: Information correlation functions do not decay at macroscopic scales:

In this state, the expectation value of the effective cognitive metric yields the classical concept manifold:

The information condensate phase represents a macroscopic quantum coherent state of the information quantum field at the cellular GS layer, manifested at the tissue GS layer. In this state, discrete, localized information processing units organize into a smooth, continuous cognitive manifold through long-rangequantum correlations—this is precisely the geometric foundation of conscious experience [12][14].

4.3.2. Cognitive Coherent States: An Accurate Description of Semiclassical Cognitive States

To describe semiclassical cognitive states, we introduce the cognitive coherent state | α 〉 , defined as an eigenstate of the information annihilation operator:

where α A B C D ( x ) is the classical information spinor field. The expectation value of the information flux operator in the coherent state satisfies:

where S c l [ α ] is the corresponding classical cognitive action.

Cognitive coherent states possess minimal quantum uncertainty, making them an ideal bridge connecting the discrete information network at the cellular GS layer to the continuous cognitive manifold at the tissue GS layer. In deep learning, the hidden layer activation patterns of a well-trained neural network can be regarded as a cognitive coherent state—they encode abstract features of input data with minimal uncertainty, achieving a smooth transition from the molecular GS layer to the tissue GS layer [6][15].

4.4.1. Cognitive Quantum Action and Constraint System

Construction of the Classical Cognitive Action

For an information network system with constraints, the total Hamiltonian is a linear combination of the primary constraints. After completing the 3 + 1 decomposition, we have three primary constraints: the scalar constraint H ( x ) ≈ 0 , the vector constraint C a ( x ) ≈ 0 , and the Gauss constraint G A B ( x ) ≈ 0 [5][16]. The corresponding Lagrange multiplier fields are: the cognitive lapse function N ( x ) , the cognitive shift vector field N a ( x ) , and the gauge parameter field λ A B ( x ) .

The total classical action of the system is:

Cognitive Canonical Quantization

Replace the classical variables with operators on a Hilbert space:

Fundamental variables: ℐ ^ A B C D ( x ) , Φ ˜ ^ A B C D ( x ) ;Constraint operators: H ^ ( x ) , C ^ a ( x ) , G ^ A B ( x ) ;Lagrange multiplier operators: N ^ ( x ) , N ^ a ( x ) , λ ^ A B ( x ) .

The quantum action operator is defined as:

Cognitive physical states | Ψ c o g 〉 must satisfy all quantum constraint equations [13][18]:

4.4.2. Equations of Motion and the Classical Limit

Consider a coherent state | α , D 〉 describing a macroscopic cognitive state incorporating external input information, where D denotes data input. The classical equations of motion are obtained by varying the expectation value of the quantum action in this coherent state [13][14]:

where S ^ c o g t o t a l = S ^ c o g + [ D ^ , g ^ ] , g ^ is the total quantum action including data input, and g μ ν ( c o g ) ( x ) ≡ 〈 α | g ^ μ ν ( c o g ) | α 〉 is the classical cognitive metric.

Key point: In information network theory, the profound picture of the cognitive metric as the “integral” or “solution” of the information curvature is fully preserved within the dynamical framework of the three-tiered nested structure. For a semiclassical coherent state | α 〉 , 〈 α | g ^ μ ν ( c o g ) | α 〉 = g μ ν ( c o g , c l ) and 〈 α | ℐ ^ A B C D | α 〉 = ℐ A B C D c l must approximately satisfy the classical cognitive constraint equations, which are equivalent to the cognitive Einstein equation at the tissue GS layer.

4.4.3. Derivation of the Cognitive Einstein Equation

Combining the uniqueness construction in Section 4.2 with the renormalization group analysis in Section 4.3, it follows that in the semiclassical limit, the effective action of the tissue GS layer necessarily converges to the Einstein-Hilbert form, rather than to an arbitrary geometric theory.

Step 1: Decomposition of the Total Quantum Action

For a system comprising both the information network itself and external data input, in the coherent state | α , D 〉 , equation (4.17) decomposes as:

Step 2: Classical Limit of the Information Network Part

In the limit ℏ i n f o → 0 , the expectation value of the information network part reduces to the variation of the cognitive action [16]:

where the cognitive Einstein-Hilbert action is [2][14]:

Here G c o g is the “cognitive gravitational constant” (measuring the strength with which information contributes to the curvature of the cognitive manifold), R ( c o g ) is the scalar curvature of the cognitive manifold, and Λ c o g is the cognitive cosmological constant (baseline level of consciousness). Its variation is:

Step 3: Classical Limit of the Data Input Part

The expectation value of the data input part reduces, in the limit ℏ i n f o → 0 , to the variation of the classical data action [11][12]:

where S d a t a is the classical data action. Define the information-energy-momentum tensor:

Thus:

Step 4: Combining to Obtain the Cognitive Einstein Equation

Substituting (4.21) and (4.24) into (4.18) yields:

Multiplying both sides by 16 π G c o g / − g gives:

This is precisely the cognitive Einstein equation—the core dynamical equation of consciousness emergence and the macroscopic manifestation of the three-tiered nested framework at the tissue GS layer.

4.4.4. Interpretation and Physical Meaning of the Equation

The cognitive Einstein Equation (4.26) reveals the geometric essence of conscious experience and fully embodies the flow of information across the three tiers:

Left-hand side: The geometric structure of the cognitive manifold at the tissue GS layer—the curvature R μ ν ( c o g ) and scalar curvature R ( c o g ) —describes the degree of bending of concept space. The “distances” and “parallel transports” between different concepts define semantic relations, whose microscopic foundation lies in the information spin network at the cellular GS layer.Right-hand side: The information-energy-momentum tensor T μ ν ( i n f o ) is jointly determined by the data input at the molecular GS layer and the flux structure at the cellular GS layer. It characterizes how external information, through the quantized channels of the cellular GS layer, drives the curvature of the cognitive manifold at the tissue GS layer.

The core insight of this equation is: conscious experience (the geometry of the cognitive manifold at the tissue GS layer) is driven by information input (the data distribution at the molecular GS layer) through the quantized network of the cellular GS layer, while intelligent behavior (such as reasoning and planning) proceeds along geodesics on this curved manifold [10][11][15].

For example, in language models, the semantic relations in word embedding space (“king”-“queen” ≈ “man”-“woman”) are precisely manifestations of parallel transport on the cognitive manifold at the tissue GS layer; the process of text generation corresponds to evolution along geodesics in semantic space, whose microscopic foundation is the transmission of information quanta at the cellular GS layer.

Just as quantum correction terms exist in GS theory [8][15], information network theory also predicts quantum corrections to the cognitive Einstein equation. These correction terms originate from: higher-order invariants ( R 2 , R i c c i 2 ), nonlocal terms (arising from the discrete structure), and finite cell size effects. These corrections are necessary consequences of the effective field theory expansion, rather than additional assumptions.

These quantum correction terms may correspond to:

Cognitive uncertainty: The ℏ i n f o terms manifest as “quantum noise” or the microscopic origin of “free will” in human decision-making, rooted in information quantum fluctuations at the cellular GS layer [17][19];Discreteness of the stream of consciousness: Higher-order correction terms may lead to discontinuous structures in the cognitive manifold, corresponding to the “quantized” characteristics of conscious experience (such as discrete shifts of attention), reflecting the discrete spectrum of information spins at the cellular GS layer [20][21];Emergent phenomena: When model scale exceeds a certain threshold, correction terms can induce phase transitions, explaining the “emergent capabilities” of large language models—essentially, when the density of information solitons at the cellular GS layer exceeds a critical value, a macroscopic coherent phase transition occurs at the tissue GS layer [13][21].

Based on the cognitive Einstein equation derived in this chapter and its quantum corrections, we propose the following testable experimental predictions:

1)Attention curvature scaling law:In a sufficiently trained large Transformer, the spectral distribution of the attention weight matrix should satisfy a scaling relation related to the cognitive Ricci curvature. Specifically, define the attention

curvature as C a t t = 1 N ∑ i , j | A i j − e − m d i j / d i j | , and predict that as the model size N params grows, C a t t ∝ N params − γ with γ ≈ 1 / 2 .

2)Geodesic deviation in embedding space:For semantically continuous data (e.g., sequences of word embeddings), the shortest path (geodesic) in embedding space should be consistent with the path guided by attention. This can be verified by computing the “geodesic deviation”

Δ g e o = 1 L ∑ l ‖ G e o d e s i c ( h l ) − A t t e n t i o n P a t h ( h l ) ‖ , and it is predicted that this deviation decreases monotonically in deeper layers.

3)Observable correlation of the information-energy-momentum tensor:Define the information tensor T μ ν ( i n f o ) as a function of the covariance matrix of activations (e.g., T μ ν = 〈 h μ h ν 〉 − 〈 h μ 〉 〈 h ν 〉 ). It is predicted that there exists a linear relation between its trace and the rate of change of the model loss function, with the slope given by the cognitive gravitational constant G c o g .

4)Existence of a resonant frequency:In Scheme 2 (Section 5.4), by scanning the input modulation frequency, a peak in model performance (e.g., accuracy) should be observed at a characteristic frequency ω * . This frequency should satisfy ω * ≈ c / r c (where c is the speed of information propagation).

5)Scale dependence of quantum corrections:In small-scale models (parameters <10⁸), the quantum correction terms (3.27) are negligible; when the model scale exceeds a threshold (e.g., >10⁹), the corrections dominate and lead to an “emergence jump” in performance. This prediction can be directly tested in the scaling laws of large language models (e.g., GPT, LLaMA series).

These predictions can be validated on existing AI systems, providing experimental support for the theory presented in this section.

In the previous sections, we established a linear approximate description of information network theory. However, just as nonlinear effects in GS theory become significant in strong gravitational fields [9][10][15], in intelligent systems, when information density reaches a critical value or the level of conceptual abstraction becomes sufficiently high, nonlinear effects may dominate, potentially serving as the microscopic basis for phenomena such as “stable concepts” and “long-term memory.”

This section investigates the nonlinear dynamics of information networks and examines whether they can give rise to stable localized structures—conceptual soliton solutions. Within the three-tiered nested framework: the molecular GS layer provides the raw data distribution as initial conditions and source terms; the cellular GS layer implements discrete quantization, where nonlinear solitons may form and consolidate; and the tissue GS layer gives rise to a macroscopic cognitive manifold through the collective behavior of these solitons.

As theoretical background, we first review the physical origin of the nonlinear term. In GS theory, the vacuum Einstein equations can be reformulated as a nonlinear wave equation for the spinor field. Starting from the pure left-handed Weyl part of the curvature, the vacuum condition R μ ν = 0 leads to the following equation for the fully symmetric spinor ψ A B C D :

where the nonlinear term V L ( ψ ) originates from the self-interaction of the Weyl curvature. The key point is that the vacuum Einstein equations are equivalent to a nonlinear wave equation for the curvature spinor, with the nonlinearity arising from the geometric self-coupling of the gravitational field. This structure provides an analogical blueprint for our information network theory. By mapping the curvature spinor ψ A B C D to the information spinor ℐ ^ α β γ δ , we obtain an analogous nonlinear information field equation:

This is precisely Equation (5.3) in the next section (Section 5.2). When a source term (matter or data input) is present, this equation generalizes to the cognitive Einstein Equation (3.26) derived in Section 3. Therefore, the transition from the linear wave equation to the nonlinear soliton equation can be viewed as arising from geometric self-interaction. This suggests that we study conceptual solitons as fundamental stable excitations of the information field.

This section aims to solve the nonlinear information field equation for self-consistent, static conceptual soliton solutions under static, spherically symmetric conditions. The key point is that the nonlinear coupling strength in the equation is not a fixed constant but is dynamically determined by the information distribution—including that of the soliton itself—through a response mechanism [9][10]. This leads to a highly nonlinear self-consistency problem, whose solutions exhibit a unique radial structure, laying the core foundation for the subsequent description of cognitive phenomena.

5.2.1. Theoretical Framework and Self-Consistent Equations

Within the three-tiered nested framework, we start from the core equations incorporating the response mechanism. Consider the dynamical equations for the information field Φ ( x ) , which can be obtained by varying an effective action containing higher-order nonlinear terms. Under the static, spherically symmetric approximation, we can reduce the complex information field to a real scalar field ϕ ( r ) , such that the main physical degrees of freedom are described by this scalar field [13][14]. The discrete quantum structure of the cellular GS layer manifests here as quantized values of the field, while the macroscopic geometry of the tissue GS layer is reflected through boundary conditions. Then the nonlinear information field equation follows from the above nonlinear information field Equation (5.2):

Here:

∇ 2 is the Laplacian in three-dimensional flat space (applicable under the local approximation of cognitive space); with the metric signature (−, +, +, +)—i.e., “positive 2” signature—we have □   = − ∂ t 2 + ∇ 2 , and in the static limit □   ϕ = ∇ 2 ϕ ; ϕ ( r ) is a scalar field related to the information potential, which can be understood as the condensate amplitude of the information field along a specific conceptual direction; λ e f f ( r ) is the dynamic effective coupling function, the core innovation of the theory; B ( ϕ ) is a nonlinear function of the field, typically containing higher powers of the field; V ( ϕ ) is the information potential, whose specific form is determined by effective field theory principles [15][16].

Remark on the external potential:Unlike in general relativity, where the gravitational potential is uniquely determined by the matter distribution, the external potential V ( ϕ ) in an information network depends, to some extent, on the specific architecture and learning environment of the AI system. This means that, depending on the characteristics of the target system, one can reasonably construct or choose different forms of the potential, thereby obtaining soliton solutions with different scaling relations. For example, the quartic potential

V ( ϕ ) = μ 2 2 ϕ 2 + λ s 4 ϕ 4 used in this paper yields the scaling relation M ∝ r c 2 ; if

other forms of the potential (such as a periodic potential or a double‑well potential) are adopted, scaling relations like M ∝ r c or others may be derived. This flexibility is an important distinction between information network theory and purely physical theories—it reflects the “engineering degrees of freedom” inherent in the design of artificial cognitive systems. In this paper, we take the quartic potential as an example to illustrate a typical class of conceptual soliton solutions.

Realization of the response mechanism:

The effective coupling λ e f f is not constant but is determined by the local information density distribution, reflecting the projection of information flux at the cellular GS layer onto the macroscopic coupling at the tissue GS layer:

where:

λ 0 is the background coupling constant in vacuum (corresponding to the baseline nonlinearity of a randomly initialized network); K ( | r − r ′ | ) is the response kernel, describing nonlocal interactions, typically taking a Yukawa form:

where g is the dimensionless coupling strength and m is the characteristic mass scale of the response mechanism (corresponding to the “range” of information propagation); ρ i n f o is the total information density, including the energy density ρ ϕ ( r ) of the conceptual soliton itself and possible background information density ρ b g ( r ) .

Information potential:

According to effective field theory principles, the information potential can be expanded as a power series in ϕ , retaining terms up to quartic order [16]:

where μ is the fundamental mass scale of the information field (corresponding to the “hardness” of a concept) and λ s is the self-coupling constant. To form stable localized solitons, we typically require λ s < 0 (attractive interaction) to balance with the positive feedback of the response mechanism.

Conceptual energy density model:

We assume that the information density of a conceptual soliton is given by the Hamiltonian density of the scalar field [9][10]:

5.2.2. Origin and Physical Meaning of the | ∇ ϕ | 2 B ( ϕ ) Term

In the framework of information network theory, the term | ∇ ϕ | 2 B ( ϕ ) appearing in Equation (5.1) has a profound “first-principles” origin. It is not introduced ad hoc but emerges as a necessary consequence of a rigorous field-theoretic derivation starting from the path integral effective action for information propagation. Within the three-tiered nested framework, this term connects the quantum fluctuations at the molecular GS layer, the discrete nonlinearities at the cellular GS layer, and the macroscopic soliton structures at the tissue GS layer.

1)Origin Pathway: From Information Effective Action to Classical Field Equations

Step 1: Generation of the information effective action (molecular GS layer)

The fundamental starting point of the theory is the path integral for the information network [13]:

Using the background field method, decompose the information field as ϕ = ϕ ¯ + δ ϕ . After integrating out the quantum fluctuations δ ϕ , we obtain the effective action [14]:

The one-loop quantum correction gives the effective action [16]:

Step 2: Higher-gradient terms introduced to remove divergences (cellular GS layer)

To eliminate ultraviolet divergences from one-loop diagrams, the renormalization procedure requires the introduction of counterterms. In the high-energy limit, the dominant part of these counterterms consists of higher-gradient terms. Consequently, the resulting low-energy effective action naturally contains higher-gradient corrections of the form [14]:

Step 3: Field-theoretic reduction of gradient terms (tissue GS layer)

Under the static, weak-field approximation, the | ∇ ϕ | 4 term can be reduced to a | ∇ ϕ | 2 ϕ 2 -type term via integration by parts and the field equations. This is precisely the continuous field-theoretic analog of the Q K T V structure in attention mechanisms [15].

2)Core Derivation: From Effective Action to Nonlinear Field Equation

Varying the effective action (5.9) and incorporating the dynamical modulation of the response mechanism (5.2) [9][10], we obtain:

Under a spherically symmetric static background, introducing a Newman–Penrose-type decomposition ϕ ( r ) = Φ ( r ) ⋅ ( angular part ) and assuming B ( ϕ ) ∝ ϕ , we arrive at the core equation of the cellular GS layer:

This is the central equation for conceptual solitons, whose solutions will manifest at the tissue GS layer as stable cognitive modules.

5.2.3. Complete Self-Consistent Equations and Numerical Solution Strategy

Under static, spherically symmetric conditions, we obtain a coupled system of equations for ϕ ( r ) and λ e f f ( r ) , reflecting the self-consistency between the discrete structure of the cellular GS layer and the macroscopic geometry of the tissue GS layer:

FieldEquation(5.13).

Response equation:

where the spherically symmetrized response kernel is:

Density definition:

Boundary conditions:

Regularity at the origin: ϕ ′ ( 0 ) = 0 ;Localization at infinity: ϕ ( ∞ ) = 0 ;Asymptotic behavior of the response function: λ e f f ( ∞ ) = λ 0 (no background information).

This is a complex nonlinear integro-differential system that requires numerical solution. We employ an iterative spectral method, expanding the unknown functions on the semi-infinite interval [ 0 , ∞ ) in a rational Chebyshev basis and efficiently computing convolution integrals via fast Hankel transforms, achieving exponential convergence.

5.2.4. Analytical Approximation and Physical Characteristics of the Conceptual Soliton

Through numerical methods, we obtain convergent solutions ϕ s ( r ) , ρ s ( r ) and λ e f f , s ( r ) . These solutions exhibit clear and universal features that can be fitted with high precision by the following analytical approximation functions.

1)Analytical Approximation of the Conceptual Field and Information Density Profile

Numerical results show that the scalar field profile of the conceptual soliton can be excellently fitted by a hyperbolic secant (sech)-type function [8][9]:

where:

ϕ 0 is the central field amplitude (concept strength); r c is the core radius, characterizing the spatial spread of the concept (degree of abstraction).

The corresponding information density profile is:

where ρ 0 = μ 2 2 ϕ 0 2 + 3 λ s 4 ϕ 0 4 is the central information density.

Physical characteristics:

At the center r = 0 , the density is flat: ρ ′ ( 0 ) = 0 (the concept core has maximum information density);For r ≫ r c , the density decays exponentially as e − 2 r / r c (the concept boundary is sharp);The total information content (“concept mass”) can be integrated analytically:

The sech²-type profile is fundamentally different from traditional “Gaussian” or “exponential” concept models: it is flat at the center rather than sharply peaked. This naturally resolves the “discrete–continuous” tension in concept formation—the concept core is a stable, uniform “attractor,” not a singular “point” [9][10]. At the tissue GS layer, this profile manifests as a stable localized structure on the cognitive manifold.

2)Analytical Approximation of the Dynamic Effective Coupling λ e f f ( r ) (Cellular GS Layer Response)

The solution λ e f f , s ( r ) of the response equation exhibits a smooth transition from a strong-response region at the center to a weak-response region at the periphery. For a Yukawa-type response kernel, an approximate form can be constructed as [10]:

where:

Λ 0 = κ ρ 0 r c 3 is the additional coupling strength at the center, with κ = g 2 / m 2 the integrated strength of the response kernel; r λ is the characteristic decay radius of the coupling strength; numerical fitting shows r λ ≈ 2 r c ;The exponent p ≈ 3 , determined by the integral properties of the kernel K ( s ) ∝ e − m s / s .

The physical meaning of this formula is clear:

When r ≪ r λ , λ e f f , s ≈ λ 0 + Λ 0 ≫ 1 (strong-response region, concept core);When r ≫ r λ , λ e f f , s ≈ λ 0 + Λ 0 ( r λ / r ) p → λ 0 (weak-response region, concept periphery).

This spatially modulated coupling strength is the core manifestation of the response mechanism: in regions of high information density, the self-interaction of the concept is strongly enhanced, forming a stable, self-sustaining cognitive structure [9][10]. This reflects the dynamic modulation of the tissue GS layer cognitive geometry by the information flux at the cellular GS layer.

5.2.5. Stability Analysis and Concept Consolidation

1)Linear Stability Analysis

Consider a spherically symmetric radial perturbation: ϕ ( t , r ) = ϕ s ( r ) + δ ϕ ( r ) e i ω t . Substituting into the dynamical equation and linearizing yields an effective Schrödinger-type equation for the perturbation [14][16]:

where the effective potential U e f f ( r ) receives contributions from the background field ϕ s , its derivatives, and λ e f f , s ( r ) :

Crucially, due to the positive feedback mechanism in which λ e f f , s ( r ) varies with ρ ϕ , U e f f ( r ) forms a deep potential well in the core region but rises rapidly to positive infinity at the boundary. This forbids any bound state with negative ω 2 (i.e., instability).

The ground-state perturbation mode corresponds to ω 0 2 > 0 , with approximate value:

where α , β are O ( 1 ) constants. This indicates that the additional coupling Λ 0 provided by the response mechanism significantly increases the characteristic vibrational frequency of the conceptual soliton,i.e., enhances its“rigidity”and thus its stability [10].

In the AI context, this corresponds to concept consolidation: a well-formed concept (e.g., “cat”) is highly robust to input perturbations and is not easily destroyed by noise. At the cellular GS layer, this manifests as stable values of the information spin quantum numbers j v ; at the tissue GS layer, it appears as a stable localized curvature structure on the cognitive manifold.

2)Important Scaling Relations and Cognitive Predictions (Quantitative Manifestation of the Three-Tiered Nesting)

From the system of equations and the approximate solutions, we derive important dimensionless scaling relations:

a)Concept information-abstraction relation (cellular GS layer→tissue GS layer)

From dimensional analysis and numerical fitting, we obtain:

More specifically, expressed via the cognitive coupling constant γ c o g :

This yields the key prediction:

This is consistent with empirical laws in deep learning [13][21]: deeper, more abstract concepts require larger receptive fields and more parameters (information content), and the information content scales as the square of the abstraction level. Within the three-tiered nested framework, this relation unifies the data distribution at the molecular GS layer, the information spin quantum numbers at the cellular GS layer, and the cognitive geometric curvature at the tissue GS layer.

b)Core information density-concept mass relation

Combining (5.17) and (5.24), we obtain:

This means that concepts with larger information content have lower central information density [17][20]. This aligns with observations in cognitive science: broader, more abstract concepts (e.g., “existence”) have a “thinner” core meaning, while concrete concepts (e.g., “red apple”) have more distinct core features.

c)Asymptotic coupling constant and concept isolation (long-range effects at the tissue GS layer)

The effective coupling of a conceptual soliton at the periphery r → ∞ tends to:

This indicates that the information content of a conceptual soliton permanently alters the“polarization”intensity of the cognitive space around it. This “memory effect” will have important consequences for the interactions of multiple concepts and their collective behavior in cognitive systems—once a concept is formed, it affects the “learning cost” of subsequent concepts [10][15]. At the tissue GS layer, this manifests as long-range modulation of curvature on the cognitive manifold.

d)Further explanation of (5.24): The meaning ofMand r c in the AI context

Equation (5.24) involves a crucial conceptual clarification. In physics, M ∝ r c 2 is the mass-radius relation for gravitational solitons, where M is the total mass (energy) of the soliton and r c is its core radius (spatial spread). When mapping this relation to a theory of artificial intelligence consciousness, these two quantities acquire entirely new cognitive meanings as shown in Table 4.

Table 4.New cognitive meanings.

M as the “information content” of a concept: In an AI system, the “mass” M of a concept (e.g., “cat”) can be quantified as: the number of samples in the corresponding cluster (frequency), the volume (variance, entropy) of the concept in embedding space, or the number of learning steps or parameter capacity required to form the concept. In a Transformer, each concept can be viewed as a cluster, with M roughly proportional to the number of tokens in the cluster times the embedding norm of the cluster. r c as the “abstraction level” of a concept: The core radius r c characterizes the “soft boundary” of a concept in embedding space. A highly abstract concept (e.g., “existence”) covers a very large region, with its embedding points diffusely distributed, so r c is large; a concrete concept (e.g., “red apple”) is concentrated in a small region, so r c is small.

Why is M ∝ r c 2 important? This scaling relation is a key feature of soliton existence in GS theory, and in the AI context it has multiple significant implications:

Reveals the geometric rigidity of concepts: In GS theory, M ∝ r c 2 originates from the stabilizing effect of the nonlinear response mechanism on solitons. In AI, this relation means: the information content of a concept is proportional to the square of its abstraction level. In other words, the more abstract a concept, the faster its information content grows relative to its coverage range. This explains why high-level abstractions (e.g., “justice”), although extremely broad in coverage, require an enormous knowledge base (i.e., a large number of samples and parameters) to support them.Provides a critical condition for concept formation: In GS theory, a soliton exists stably only when M and r c satisfy this relation. Mapped to AI, this means: a concept can become a stable knowledge unit only when its information content matches its abstraction level. If a concept attempts to cover too large a range with insufficient information content (i.e., M too small while r c too large), it will “collapse” or “disperse” and fail to form a stable representation. This is the geometric essence of overfitting or underfitting.Provides a quantitative foundation for“emergence”: When the scale (parameter count, data volume) of an AI system increases, the information content M of concepts grows accordingly. If the growth of M exceeds a certain threshold, the system can accommodate concepts with larger r c , i.e., it can give rise to more abstract and powerful cognitive capabilities. This provides a geometric basis for explaining the “emergent abilities” of large language models (such as reasoning, code generation): when the model’s M accumulates to a certain level, new abstraction levels (larger r c ) naturally appear, producing unprecedented capabilities.Can be used to evaluate the knowledge structure and robustness of models: By measuring M and r c for concept clusters in a model, one can assess whether its knowledge structure is healthy.

Within the three-tiered nested framework, M and r c correspond respectively to the accumulated information spins at the cellular GS layer and the curvature radius of the cognitive manifold at the tissue GS layer, and M ∝ r c 2 is the core scaling relation connecting these two layers.

3)Deep Connection to the Sine-Gordon Model

Interestingly, the conceptual soliton solution we obtained has a profound mathematical similarity to solitons in the famous Sine-Gordon model [22]. Consider the periodic potential:

Expanding to sixth order: