Prior to constructing the non-perturbative path integral, we establish the foundational premises of Gravitational Spinor (GS) theory that motivate a background-independent quantization and naturally lead to the spin foam formalism.

2.1.1. Classical Foundation and Variables

The fundamental classical variable is a completely symmetric 4-index spinor field, ψ A B C D ( x ) , posited to encode the conformal (Weyl) curvature of spacetime [10]. In the Hamiltonian formulation, its canonically conjugate momentum is Π A B C D . A key result, derived from the symplectic structure of the first-order GS action, is that this momentum is linearly proportional to a geometric density—the spinorial area flux density Σ A B C D :

Here, α is a constant involving Newton’s constant G and the Barbero-Immirzi parameter γ , and Σ A B C D is constructed from the spatial triad E i a :

where E i a is the spatial triad (3-dimensional frame field) on the spatial hypersurface. This object is the spatial restriction of the full spacetime tetrad E A A ' introduced later in the GGE constraint (Sec. 2.2.2), projected to the time-gauge-fixed slice. The relation (2.1.1) identifies the dynamical momentum with a geometric quantity, analogous to the variable transformation in Ashtekar’s formulation. Although this proportional relation Π ∝ Σ can be derived most straightforwardly in a linearized approximation, our detailed analysis confirms it remains valid with extremely high accuracy within the full nonlinear framework of General Relativity. Any potential nonlinear corrections are of the order O ( l P 2 / R 2 ) , where l P is the Planck length and R is the characteristic curvature radius. In all astrophysical and cosmological regimes (e.g., the solar system, neutron stars, cosmological scales), this correction is phenomenologically negligible (typically 10⁻⁸⁰ to 10⁻¹²²). The essence of the resulting discrete quantum geometry is preserved non-perturbatively, as evidenced by the existence of gravitational soliton solutions.

2.1.2. Canonical Quantization and Discrete Geometry

The standard procedure of canonical quantization is applied to the conjugate pair ( ψ , Π ) . Promoting them to operators and imposing the commutation relation [ ψ ^ , Π ^ ] = i ℏ δ leads, via (2.1.1), to the fundamental quantum commutator between the spinor field and the geometric flux operator:

The physical interpretation of Σ ^ as a geometric flux density operator is now central. This commutation relation defines the kinematical algebra of observables. A standard representation of this algebra is obtained by considering wavefunctionals Ψ [ A ] of an s l ( 2 , ℂ ) -valued connection A . In this connection representation, the Hilbert space H k i n is defined as the space of square-integrable functionals on the space of (generalized) connections, equipped with the inner product

where D [ A ] is a diffeomorphism-invariant (projective) measure [16][22]. This inner product is manifestly positive-definite, establishing H k i n as a genuine Hilbert space.

The representation theory of this algebra dictates that the spectrum of associated geometric observables (e.g., area) is discrete [16]. The eigenstates form an orthonormal basis of the kinematical Hilbert space H k i n and are realized as Gravitational Spinor Networks (GSNs)—abstract graphs Γ whose edges e are labeled by half-integers j e (spins) and whose nodes v are labeled by intertwiners i v . These quantum numbers arise as eigenvalues of geometric operators (e.g., area, volume) constructed from Σ ^ . Crucially, these GSN basis states are orthonormal:

This orthonormality follows directly from the definition of the inner product in H k i n and the fact that the GSN states are eigenstates of a complete set of commuting self-adjoint geometric operators [22].

In short, the quantum geometry of GS theory arises as follows: the classical identification Π ∝ Σ (2.1.1) from the action’s symplectic structure ties dynamics to geometry; canonical quantization yields the operator algebra (2.1.3); and the discrete spectrum of the geometric operator Σ ^ directly implies that the fundamental quantum states are discrete GSNs. The kinematical Hilbert space H k i n is thus explicitly constructed as an L 2 space over connection degrees of freedom, with GSNs providing an orthonormal basis. This background-independent framework provides the kinematics for the spin foam dynamics constructed next.

2.1.3. Recovering Einstein’s Dynamics: From GS Variables to the Metric Field

The canonical quantization in the previous section revealed the quantum geometry of GS theory—discrete Gravitational Spinor Networks. We now return to the classical theory to demonstrate how its classical dynamics is naturally encapsulated within a background-independent, first-order action principle, from which Einstein’s field equations are uniquely derived. This principle takes the form of a constrained BF theory (BF theory is a topological field theory whose action is given by the integral of the wedge product of a Lie-algebra-valued (D−2)-form B and the curvature F of a connection).

The connection stems from the geometric interpretation of the fundamental GS variables. As outlined in Section 2.1, the spinorial density Σ A B C D is understood as the spatial projection of a spacetime 2-form field Σ A B , constructed from the tetrad e A as Σ A B = e A ∧ e B . Simultaneously, the fully symmetric spinor ψ A B C D is interpreted as the self-dual part of the spinorial representation of the Weyl curvature C μ ν ρ σ [10][11].

In this framework, the complete classical dynamics of GS theory can be derived from the following constrained BF-type action:

where F A B [ ω ] = d ω A B + ω A C ∧ ω C B is the curvature of the self-dual spin connection ω A B . The action is that of a BF theory (where B is the 2-form Σ ) with an additional constraint term. Crucially, ψ A B C D functions as a Lagrange multiplier. The term it multiplies enforces the simplicity constraint Σ A B ∧ Σ C D = 0 , which forces Σ A B to be “simple” (i.e., expressible as a wedge product of tetrads). This constraint breaks the topological nature of the pure BF theory and introduces local gravitational degrees of freedom.

Varying this action yields the key equations:

Simplicity Constraint: δ ψ A B C D : Σ ( A B ∧ Σ C D ) = 0 . This ensures Σ A B is tetrad-based.Curvature-Metric Relation: δ   Σ A B : F A B = ψ A B C D   Σ C D . Upon solving the constraints, this identifies the multiplier ψ A B C D with the self-dual part of the Weyl curvature.Torsion-Free Condition: δ ω A B : D Σ A B = 0 , which implies the connection ω is metric-compatible (Levi-Civita) for non-degenerate Σ .

To make contact with the familiar metric formulation, a 3 + 1 decomposition is performed, leading to the constraints of the Hamiltonian formalism. The central dynamical element is the Hamiltonian (scalar) constraint, which generates time evolution. From action (2.1.4), this constraint takes the form:

where Σ ˜ A B a is the spatial projection of Σ A B , and q a b is the spatial metric.

Collectively, these equations are equivalent to Einstein’s field equations, G μ ν + Λ g μ ν = 0 (where a cosmological constant Λ may arise from a constant term in the action or a trace part of ψ ). Therefore, action (2.1.4) is not an additional postulate but an elegant and quantum-geometryfriendly integrated formulation of the classical dynamical relations satisfied by the fundamental GS variables ( ψ A B C D , Σ A B C D ). It explicitly demonstrates how GS theory recovers the core dynamics of General Relativity from a background-independent, gauge-theoretic starting point through a constraint mechanism. In the following sections this classical action will be promoted to a path-integral quantization; its first-order form directly inspires the gauge-invariant action (2.2.2), and its BF structure provides the foundation for the discrete formulation (2.2.4).

2.1.4. The Quantum-to-Classical Bridge

In the quantum theory, (2.1.5) becomes an operator equation H ^ | Ψ p h y s 〉 = 0 for physical states. To see the classical limit, consider a semiclassical coherent state | α 〉 that approximates a smooth geometry. For such a state, the expectation value of the quantum constraint must reduce to its classical counterpart:

When the expectation values ψ A B C D c l = 〈 α | ψ ^ A B C D | α 〉 and Σ A B C D c l = 〈 α | Σ ^ A B C D | α 〉 satisfy the simplicity constraint, Σ c l defines a classical tetrad e μ A and hence a metric g μ ν = η A B e μ A e ν B . The classical constraint equation (2.1.5), combined with the other constraints from (2.1.4), then becomes equivalent to the Einstein field equations [10][11]:

where the effective energy-momentum tensor T μ ν arises from matter fields or, in the vacuum case, is zero. This derivation confirms that solutions of the classical GS theory are solutions of General Relativity, establishing the necessary dynamical correspondence. The path integral constructed in the following sections will have this classical theory as its ℏ→0 limit.

Given this classical foundation, a non-perturbative quantization is required to handle the background-independent, constrained nature of gravity. The canonical quantization based on (2.1.1) leads to GSN states. The goal of this section is to construct a non-perturbative path integral (spin foam model) that defines transition amplitudes between these GSN states, providing a complete definition of the theory’s quantum dynamics.

2.2.1. Gauge-Invariant Formulation of the Action

To construct the path integral, we require a first-order action that treats ψ A B C D and Σ A B C D on equal footing and seamlessly incorporates the constraints.

1) Fundamental Variables and Symmetries

Gravitational Spinor Field: ψ A B C D ( x ) , a completely symmetric 4-index spinor, serves as the fundamental quantum field. At the level of quantum states, its eigenstates correspond to the edges of gravitational spinor networks [10][11][15].Momentum Conjugate Field: Σ A B C D ( x ) , a spinor density. Its classical correspondence is with the spacetime volume form: Σ A B C D ∝ e A A ′ ∧ e B B ′ ∧ e C C ′ ∧ e D D ′ , where e A A ′ is the tetrad.GGE Constraint: This is the core of the theory [9]. It requires the action to be invariant under the following extended gauge transformation:

This transformation is not merely a global phase shift; the local parameter θ ( x ) is linked to a generalized gauge transformation by GGE, which mixes gravitational and internal gauge degrees of freedom. It provides a foundational geometric mechanism for unifying gravity with other fundamental interactions within this spinorial framework.

2) Construction of the Action

Starting from the classical constrained BF action (2.1.4), we now rewrite it in a form that is better suited for quantization. We introduce a spinor connection A whose curvature F [ A ] is related to the Lagrange multiplier ψ (the selfdual Weyl curvature). Simultaneously, we replace the tetrad-based 2-form Σ A B by the spinorial density Σ A B C D to make the duality with ψ A B C D manifest and to accommodate the GGE symmetry. This yields the following BF-type action, which naturally encodes the duality between ψ and Σ and facilitates the imposition of constraints [23]-[25]:

where F A B C D [ A ] is the curvature 2-form of a certain “spinor connection” A related to ψ . In GS theory, ψ A B C D itself can be viewed as a function of the self-dual part of this curvature, i.e., F A B C D ∝ ψ A B C D + nonlinear terms. First term Σ ∧ F is the standard BF theory term, which enforces a simple relation between the curvature F and Σ . Upon quantization, it gives rise to area quantization [23]. Second term Λ Σ ∧ Σ is the cosmological constant term, corresponding to volume quantization [26]. Key term S G G E part embodies the distinctive feature of GS theory [10][11]. It is a Lagrange multiplier formulation of a set of constraint terms designed to enforce:

Gauss Constraint: Generates the GGE gauge transformations of ψ and Σ (2.2.1).Metric (or “Simplicity”) Constraint: Forces Σ A B C D to be constructible from the bispinor square of some “4-volume element”, i.e., Σ A B C D ∝ e A B C D : = E ( A A ′ ∧ E B B ′ ∧ E C C ′ ∧ E D D ′ ) , where E A A ′ is the 4-dimensional frame field. This constraint links topological BF theory to gravitational dynamics [24][27].

After imposing these constraints (i.e., integrating out the Lagrange multipliers in S G G E ), the action (2.2.2) reduces precisely to the classical action (2.1.4) up to an overall constant and a field redefinition. Its equations of motion then become the nonlinear GS equation, confirming that (2.2.2) is the correct quantumready extension of the classical theory.

At the quantum level, the GGE constraint S G G E is implemented as a projection operator P G G E acting on the kinematical Hilbert space ℋ k i n defined in Sec. 2.1.2. This operator is constructed to satisfy the defining properties of a projection: P G G E 2 = P G G E and, crucially, P G G E † = P G G E (self-adjointness). The latter guarantees that P G G E is an orthogonal projector. Consequently, the physical Hilbert space of gaugeinvariant states is defined as

and it inherits the positivedefinite inner product from ℋ k i n . The spin foam transition amplitude (2.2.12) is precisely the path integral representation of the evolution operator U ^ that maps between such physical states. Unitarity is ensured because U ^ is derived from a selfadjoint quantum Hamiltonian on ℋ p h y s .

Further in the Equation (2.2.3) below, S G G E appears as an independent term added to the BF action, not as a duplication of any part of it. This explicit separation highlights that the GGE constraints are imposed in addition to the topological BF structure, and integrating out the Lagrange multipliers in S G G E enforces the Gauss and simplicity constraints, thereby reducing the topological theory to gravitational dynamics.

2.2.2. Boundary States and Definition of Transition Amplitudes

To define the transition probability between quantum geometry states, boundary conditions for the path integral—boundary states—must be specified.

1) Boundary States: Gravitational Spinor Network States

Consider a closed three-dimensional manifold B as the boundary of a four-dimensional spacetime manifold M . Canonical quantization on this B dictates that geometric observables are described by eigenstates of a gravitational spinor network Γ embedded in it [16][22].

Specifically, a boundary GSN state | Ψ Γ , j , i 〉 is defined by the following data:

Graph Γ: A graph embedded in B , with edges e and nodes v .Spinor Labels j : Assigns a half-integer j e to each edge e , which corresponds to the area quantum number of the two-dimensional face pierced by that edge (related to the eigenvalue of Σ ^ ).Intertwiner i : Assigns an intertwiner i v to each node v . It is a state within the tensor product space of the spinor representations associated with the edges meeting at the node, which preserves gauge invariance (i.e., GGE symmetry). It encodes the quantum information of the three-dimensional volume around that node [22][27].

A schematic visualization of these elements is provided in Figure 1. In this simplified representation, nodes are depicted as circles labeled by the intertwiners i v , edges carry the spin labels j f , and the shaded triangular faces and the bulk region illustrate the quanta of area and volume, respectively. This geometric picture clarifies how the abstract combinatorial data of the spin network encode the fundamental quantum geometric degrees of freedom on the boundary.

2) From the Continuous Path Integral to Discrete Spin Foam Sums

The continuous path integral Z = ∫ D ψ D Σ D A ⋯ e i S / ℏ is regularized via a cellular decomposition Δ of the spacetime manifold M [16][28]. To make the connection between the continuum BFtype action (2.2.2) and the discrete spin foam sum explicit—and to address the need for a firstprinciples derivation—we now outline the key steps.

Step 1: Continuum action in BF form. The GS action (2.2.2) can be recast in a form analogous to a constrained BF theory [22], now written in terms of the Lie-algebraic variables B and A that are more convenient for discretization:

Figure 1. Geometric visualization of a Gravitational Spin Network (GSN). This simplified illustration depicts the key elements of a spin network: nodes (circles) labeled by intertwiners i v , edges carrying spin quantum numbers j f , and the associated geometric quanta—shaded triangular faces representing areas, and the enclosed bulk region representing volume. The visual analogy clarifies how the combinatorial data of the GSN encode fundamental quantum geometric degrees of freedom on a spatial boundary.

where A is an s l ( 2 , ℂ ) -valued connection 1form, F = d A + A ∧ A is a curvature, and B is a 2form field related to the geometric flux Σ A B C D . This rewriting is purely notational: the trace T r contracts the spinor indices appropriately, and the 2-form B encodes the same degrees of freedom as Σ . The GGE term S G G E remains unchanged in form. Equation (2.2.3) is therefore completely equivalent to (2.2.2) and consequently to the classical action (2.1.4) after imposing the constraints.

Step 2: Discretization. We introduce a cellular decomposition Δ of the spacetime manifold M into 4-simplices. To each tetrahedron t we assign a group element g t ∈ S L ( 2 , ℂ ) representing the holonomy (parallel transport) from a reference point to the tetrahedron; to each triangle f we assign an algebra element B f ∈ s l ( 2 , ℂ ) representing the flux of the B field through that face. The continuous BF action ∫ T r B ∧ F is discretized by replacing F with the holonomy around the face. For a face f shared by two tetrahedra s ( f ) and t ( f ) , the holonomy from s ( f ) to t ( f ) is g s ( f ) − 1 g t ( f ) . In the discrete setting, the contribution of that face is taken as T r [ B f ln ( g s ( f ) − 1 g t ( f ) ) ] , where the logarithm maps the group element to the Lie algebra (this choice ensures that the subsequent integration over B f yields a simple delta function on the group; alternative discretizations using B f ( g s − 1 g t − 1 ) are also possible but lead to equivalent results after appropriate regularisation). The discretized action (ignoring Λ and S G G E for the moment) reads:

where s ( f ) , t ( f ) denote the two tetrahedra sharing face f .

Step 3: Integration over the B f fields. For each face f , the integral over the Lie algebra yields a delta function on the group:

This follows from the fact that the integral over a Lie algebra of exp ( i T r ( B X ) ) is proportional to the Dirac delta δ ( X ) on the algebra; here X = ln ( g s − 1 g t ) , and because the map X maps to e X is a diffeomorphism near the identity, the resulting distribution pulls back to the delta function on the group concentrated at the identity. Using the Peter-Weyl theorem (extended to the discrete series selected by the GGE spin-2 constraint | k − l | = 2 [10]), the delta function expands into characters:

where d ρ is the dimension of the representation ρ and χ ρ ( g ) = T r D ρ ( g ) is its character. Consequently, the B -field integration converts each face into a sum over representations:

Step 4: Integration over the connection variables g t . Substituting (2.2.7) into the full path integral – that is, into the discretized path integral Z Δ = ∫ ∏ t d g t ∏ f d B f e i S d i s c / ℏ , after having integrated out all B f – gives:

Expanding each character as a sum over matrix elements, χ ρ f ( g s ( f ) − 1 g t ( f ) ) = ∑ m , n , k D m k ρ ( g s − 1 ) D k n ρ ( g t ) , and using D m k ρ ( g − 1 ) = D k m ρ ( g − 1 ) ¯ for unitary representations, the integral over a single group element g t takes the form:

where ϵ i j = ± 1 depends on the orientation of the face. A standard result in group representation theory states that such an integral projects onto the space of intertwiners:

The coefficients C are the Clebsch-Gordan coefficients (intertwiners) that couple the representations ρ 1 ⋯ ρ k to the trivial representation. For a 4-simplex vertex v surrounded by five tetrahedra, each tetrahedron t i receives four face representations; its integration yields an intertwiner I t i living in the invariant subspace of ⊗ j ≠ i ρ i j . After carrying out all group integrations, the contribution of a single 4-simplex becomes

which is precisely the vertex amplitude – the generalized 10j-symbol of Equation (2.3.2).

Step 5: Assembly of the full complex. Gluing 4-simplices along shared tetrahedra and faces forces the corresponding representation labels and intertwiners to match. Summation over all internal representations and intertwiners (subject to boundary conditions) then yields the full partition function:

When a boundary is present with fixed GSN data Γ i n , Γ o u t , the sums are restricted to internal faces and tetrahedra, and the boundary representations and intertwiners are held fixed. Inclusion of the cosmological constant Λ and the full GGE constraints modifies the face amplitude d ρ f to a more complicated function A f ( ρ f , Λ ) and restricts the allowed representations ρ f and intertwiner spaces to those satisfying the GGE projection; nevertheless, the overall structure (2.2.11) remains unchanged. This derivation demonstrates that the spin foam sum is not an additional postulate but follows rigorously from the discretized path integral of the constrained BF action (2.2.3), with the vertex amplitude emerging from the representation theory of S L ( 2 , ℂ ) and the GGE condition.

Therefore, the transition amplitude from an initial GSN state | Ψ i n 〉 to a final state | Ψ o u t 〉 is expressed as a refined sum over spin foams σ :

where w ( σ ) is a combinatorial measure factor, A f ( j f ) the face amplitude (often ( 2 j f + 1 ) , and A v the vertex amplitude. The continuum path integral is recovered in the limit of infinitely fine discretizations [16][28]. The appearance of ℏ in the exponential e i S / ℏ sets the scale for quantum gravitational effects; its value is the universal constant of quantum theory, required for the classical limit ℏ → 0 to recover General Relativity.

2.3.1. Defining Vertex Amplitudes Using the Spin-2 Property of the GS Field

Traditional spin foam models (e.g., EPRL/FK) are based on the decomposition S L ( 2 , ℂ ) ⊃ S U ( 2 ) , interpreting gravity as an S U ( 2 ) gauge theory [24][25]. In GS theory, however, the fundamental variable ψ A B C D is a primitive 4-index spinor field that inherently carries the Lorentz group representation information for spin-2 [12][13]. This provides a more fundamental starting point for defining vertex amplitudes.

1) From Boundary States to Vertex Bubbles

Consider an internal vertex v in a four-dimensional spacetime complex. This vertex is adjacent to five 4-cells (corresponding to the structure of a 4-simplex), connected to the exterior or other vertices via ten boundary triangular faces (2-faces). On the boundary, according to Section 2.1, each triangular face f is associated with a gravitational spinor quantum number j f . However, in the GS framework, j f is not a simple S U ( 2 ) spin but is related to a spinor representation ρ f = ( k f , l f ) , where k f and l f are half-integers corresponding to the parameters of the principal series representation of S L ( 2 , ℂ ) [29]. For describing spacelike triangles, the physically relevant representations are unitary ones satisfying the spin-2 projection constraint: | k f − l f | = 2 . This constraint originates directly from the symmetry of ψ A B C D , ensuring that the quantized geometric excitations correspond to massless, spin-2 gravitons [10].

The spinor intertwiner i v on each tetrahedron (node) is promoted in GS theory to a four-valent S L ( 2 , ℂ ) intertwiner. It must not only satisfy the tetrahedral closure condition but also adhere to the GGE constraint, maintaining gauge equivalence on a specific subspace [9][27].

2) Defining the Vertex Amplitude: Evaluation of the Single-Cell Partition Function

The general expression for the vertex amplitude of a 4simplex has already been derived in Sec. 2.2 through the systematic integration of the discretized BF action; the result is given in Equation (2.2.10). This amplitude depends on the ten face representations ρ f and the five tetrahedral intertwiners I t . In the present subsection we specialise this result by explicitly implementing the GGE constraint. This yields a refined, computable form of the vertex amplitude that will be used for semiclassical analysis and phenomenological applications.

The GGE constraint is enforced by inserting a projection operator Π G G E into the holonomy matrix elements that appear in the evaluation of the 4simplex amplitude [9][27]. Physically, this projection ensures that the parallel transport group elements g v e ∈ S L ( 2 , ℂ ) (from vertex v to adjacent tetrahedron e ) act only on the gaugeinvariant subspace defined by the GGE symmetry. A convenient way to implement this is via coherent states: for each boundary triangle we introduce a spin coherent state | n 〉 whose direction n encodes the outward normal of the tetrahedron. The projected matrix element then takes the form of a GGEmodified propagator:

which replaces the ordinary representation matrix D m n ( ρ ) ( g ) used in conventional spin foam models.

Substituting (2.3.1) into the discrete path integral restricted to a single 4-simplex, and fixing the gauge for the five tetrahedral holonomies, we obtain the following integral representation of the vertex amplitude:

Here s ( f ) and t ( f ) are the two tetrahedra sharing face f , g e are the S L ( 2 , ℂ ) group elements assigned to tetrahedron e , and the integral is taken after appropriate gauge fixing. The information of the intertwiner i e is encoded in the choice of coherent state directions n e , f and the contraction pattern [24][27]. Equation (2.3.2) is the concrete, GGEimproved version of the abstract vertex amplitude (2.2.10) and serves as the starting point for all subsequent calculations.

2.3.2. Key Geometric Amplitudes: A Generalization of the 10j-Symbol

Based on the above definition, we now evaluate the vertex amplitude in the large-spin limit, which reveals its semiclassical content and establishes its relation to the Regge calculus [23][25].

1) Semiclassical Approximation and the Emergence of Regge Action

In the regime j f ≫ 1 (equivalently, large representations ρ f ), the integral (2.3.2) is dominated by stationary points of the phase function. The stationary phase condition requires that the ten face areas ρ f and the five tetrahedral normals n e , f close to form a genuine 4-simplex geometry, with either Euclidean or Lorentzian signature [25][30]. Expanding around each critical configuration gecgec, the amplitude behaves as

where S R [ { j f } ] is the Regge action, a functional of the deficit angles of all triangles in the 4-simplex that precisely reproduces discrete Einsteinian dynamics [31]; σ = ± corresponds to the two possible orientations (timelike and spacelike) of the 4-simplex, a characteristic feature of spin foam theory [25]; S I m is an imaginary part of the action associated with parity-violating terms. In GS theory, it may be linked to the γ 0 parameter and the chiral asymmetry induced by GGE, thereby providing a window to explore potential CP-violating gravitational effects. The dimensionless coupling constant λ in front of S I m quantifies the relative strength of this parity-asymmetric, “pseudo-scalar” contribution to the semi-classical action, thereby providing a window to explore potential CP-violating gravitational effects [32]. H σ denotes the Hessian matrix evaluated at the critical point, governing the behavior of quantum fluctuations [30]. Finally, N ( ρ f ) is a normalization factor stemming from representation theory and the GGE projection. Here, l P again denotes the Planck length.

2) Comparison and Emergence of the Generalized 10j-Symbol

The standard Barrett–Crane (BC) or EPRL vertex amplitudes correspond to a specific simplification of (2.3.2) obtained by taking Π G G E to be a trivial projection onto an S U ( 2 ) subgroup [23][24]. In the full GS theory, however, the GGE projector Π G G E is nontrivial and depends on two fundamental parameters: the Barbero-Immirzitype coupling γ 0 and an additional coupling κ . This parameter κ is introduced in the definition of the GGE projector Π G G E ; it controls the mixing between gravitational and gauge degrees of freedom within the simplicity constraint. Its origin can be traced to the coefficient relating the Weyl curvature to an effective electromagnetic field strength in the underlying Cartangeometric construction of the theory [17][18]. This dependence propagates through the propagator K G G E into the vertex amplitude, modifying both the selection rules for the allowed representations ρ f and the contraction channels of the intertwiners i e .

We therefore refer to the exact analytic expression for the vertex amplitude derived from (2.3.2)—which remains welldefined even in the lowspin regime—as the “generalized 10j-symbol”, denoted

It reduces to the usual 10j-symbol when γ 0 and κ are set to values that trivialise the GGE projection. Conversely, the explicit dependence on γ 0 and κ opens up several theoretical advances:

Mitigation of the “spike” problem: The GGE constraint strongly restricts the sum over intermediate states to those that are geometrically coherent and satisfy the simplicity constraint in a stronger sense. This suppresses the unphysical, highcurvature “spiky” configurations that plague conventional models, thereby improving the convergence of the path integral [33].Dynamical Immirzi parameter: The vertex amplitude reveals how γ 0 enters as a coupling constant in both the representation labels ρ f ( γ 0 ) and the intertwiner spaces i e ( γ 0 ) . This provides the microscopic foundation for studying the running of γ 0 ( E ) : as the energy scale varies, the effective form of the vertex amplitude changes, corresponding to different regimes of the spin labels j f .Connectivity and causal structure: The analytic structure of G 10 j encodes distinct contributions from timelike and spacelike triangles within the 4-cell. This lays the groundwork for the emergence of a well-defined causal structure from the nonperturbative quantum model [34].

In summary, the vertex amplitude (2.3.2) together with its semiclassical expansion (2.3.3) provides the concrete dynamical kernel of the GS spin foam model. It is built directly from the spin-2 nature of the gravitational spinor field and incorporates the GGE constraint in a rigorous, computationally accessible manner.

Within the spin foam path integral framework based on Gravitational Spinor Networks (GSNs) constructed in Sections 2.2 and 2.3, the vertex amplitude A v defines the non-perturbative dynamics of the theory. To ensure its completeness as a consistent theory of quantum gravity, this section systematically analyzes two key quantum-level issues: whether the extended gauge symmetry generated by the Generalized Gauge Equation (GGE) is broken at the quantum level (i.e., the emergence of a quantum anomaly), and whether the theory exhibits controlled ultraviolet (UV) behavior—specifically, how its core parameter γ 0 runs with the energy scale and whether it satisfies asymptotic safety.

2.4.1. Gauge Anomalies, Topological Constraints, and Parameters

In a background-independent discrete path integral, quantum anomalies manifest more subtly than in continuous field theories [35]. They may appear as the theory’s inability to simultaneously satisfy all first-class constraints at the quantum level, or as an unphysical dependence of the partition function on the background topology or triangulation scheme.

1) Implementation and Quantum Test of GGE Symmetry

Classical vs. Quantum Realization: In the classical action S B F + S G G E , the GGE symmetry is generated by a set of first-class constraints. In the path integral quantization, this symmetry is ensured through a threefold mechanism: (a) the boundary GSN states are restricted to GGE-invariant intertwiners; (b) the construction of the vertex amplitude A v embeds the projection operator Π G G E , guaranteeing gauge invariance in the single-cell dynamics; (c) integration over the internal connection variables g v e ∈ S L ( 2 , ℂ ) formally mods out the gauge orbits.

Topological Probe for Anomalies: A rigorous method to test the quantum integrity of GGE symmetry is to compute the partition function Z for a closed manifold (e.g., S 4 ). If no anomaly is present, Z should be a topological invariant, meaning Z [ Δ ] = Z [ Δ ′ ] for any two triangulations Δ and Δ′ of the same smooth manifold [36]. Conversely, if Z explicitly depends on a characteristic class of the second homology group H 2 ( ℳ , ℤ ) , this indicates a potential anomaly.Potential Anomaly Sources and Structure of Topological Terms：Through careful analysis of the path integral measure (including the anti-commuting spinor ghost fields introduced to enforce the GGE constraints) and the algebraic properties of the projection operator Π G G E , we identify topological coupling terms that could break the symmetry. These terms typically appear in the effective action as non-absorbable phase factors [37]:

(2.4.1)

where

is a topological invariant of the manifold associated with its chiral structure (e.g., a spinorial generalization of the Pontryagin characteristic number

, potentially arising from gravitational chiral anomaly [38]), and C 2 ( A G G E ) is the second Chern class of the GGE gauge group connection A G G E (typically U ( 1 ) or embedded S U ( 2 ) ). The coefficients θ and ξ depend on γ 0 . If both are non-zero and cause exp ( i Δ Γ e f f ) to depend non-trivially on background topology, this signals a global gauge or chiral anomaly [39].

2) Anomaly-Free Conditions and Their Theoretical Constraints on γ 0

To ensure the theory is unitary and self-consistent, the total anomalous phase on any closed manifold ℳ must be an integer multiple of 2 π , thereby leaving quantum probability amplitudes unaffected. This leads to the following topological quantization condition:

(2.4.2)

Since

and 〈 C 2 〉 can independently take a series of integer values, this condition imposes very strong restrictions on the coefficients θ ( γ 0 ) and ξ ( γ 0 ) . This, in turn, may uniquely determine or strongly constrain the allowed values of the fundamental parameter γ 0 to a discrete set. This theoretical prior (i.e., a constraint derived from first principles before confronting experiment), derived from quantum consistency, provides a crucial filter for subsequent experimental determination of γ 0 [40].

2.4.2. Renormalization Group, Asymptotic Safety and UV Fixed Point

To investigate high-energy behavior, we study how γ 0 runs with the energy scale. Although defined non-perturbatively, the spin foam model’s low-energy effective description should match continuous QFT. We use this matching to derive renormalization group equations and reveal asymptotic safety mechanisms [41].

1) From Spin Foams to Effective Field Theory and the Beta Function

Matching Framework: Starting from a semi-classical background (e.g., flat spacetime), we interpret the contribution of specific spin foam complexes to the partition function as quantum corrections to the coupling constants in a continuous effective action S e f f [ h ] . This effective action includes the Einstein-Hilbert term, higher-order curvature terms ( α R 2 + β R μ ν R μ ν + ⋯ ), and quantum geometric corrections parameterized by γ 0 .One-Loop Contribution and Leading-Order Beta Function: Considering the simplest non-trivial complex—a single vertex with an internal “bubble”—which corresponds to a one-loop diagram in the continuous theory, we compute its precise amplitude using the generalized 10j-symbol A v ( j f , i v ; γ 0 ) and face amplitudes A f derived in Section 2.3. By extracting its contribution to S e f f through a large-spin asymptotic expansion, we obtain the leading term of the beta function for the dimensionless parameter γ ˜ 0 = γ 0 E 2 / E P 2 :

Here, μ is the renormalization group scale, introduced through the matching procedure between the discrete spin foam amplitude and the continuous effective action. The coefficients a 1 , a 2 are entirely determined by the analytical structure of A v , particularly the parts encoding the GGE constraint and the spin-2 projection. A key theoretical advantage is that the GGE symmetry restricts the theory to a physical subspace, potentially softening the divergences present in traditional perturbative gravity. This may manifest as the coefficient a 1 being zero or negative, creating the condition for a non-trivial fixed point [42].

2) The Core Mechanism for Asymptotic Safety:

One-loop corrections alone are insufficient. Asymptotic safety in GS theory stems from its non-perturbative structure, demonstrated in continuous field theory analysis [10]. Based on the nonlinear gravitational spinor equation ψ A B C D − 4 λ ψ A B C D ( ψ E F G H ψ E F G H ) = 0 and its stable soliton solutions ( ψ A B C D = f ( ξ ) ϵ A B C D , f ( ξ ) ∝ sech ( κ ξ ) ), two mechanisms ensure a UV fixed point:

Mechanism I: UV-Stabilizing Effect of Nonlinear Gravitational Solitons: Soliton contributions to the effective potential generate a positive cubic term in the FRG flow equation, counteracting divergent tendencies. In spin foam language, non-perturbative geometric “chunks” (large j f , specific intertwiners) suppress UV divergences [10].Mechanism II: “Inheritance” of Electromagnetic Asymptotic Freedom by GGE: The GGE transformation identifies a gravitational spinor basis with a subspace of the electromagnetic gauge algebra, introducing negative contributions from QED asymptotic freedom into the beta function, balancing soliton contributions [11].Existence and Determination of the UV Non-Gaussian Fixed Point (NGFP):

Integrating the two mechanisms above, the complete β function possesses a non-trivial zero at γ 0 = γ 0 * , satisfying ∂ β / ∂ γ 0 | γ 0 * < 0 , indicating that γ 0 * is a UV-stable fixed point. In the analytical solution of [10], this fixed point corresponds to a dimensionless coupling λ * ≈ 63.3 (or equivalently g * ≈ 37.7 ). An eigenvalue analysis reveals only 1 to 2 relevant directions, ensuring the theory retains predictive power even in the UV limit. This conclusion strongly supports that GS theory is non-perturbatively renormalizable (asymptotically safe) [41][42].

3) Prediction for Low-Energy Physics: Running Coupling γ 0 ( E )

Starting from the UV fixed point γ 0 * , the renormalization group flow towards the infrared determines the effective coupling γ 0 ( E ) observed at different energy scales. Its running behavior is governed by the linear expansion of the β function near the fixed point:

Integration yields:

where θ is the critical exponent. This energy-scale dependence, with a clear theoretical origin, provides a concrete, testable theoretical template for the next sections that aim to perform a global fit of γ 0 ( E ) using multi-messenger astronomical observations (e.g., gamma-ray bursts, ultra-high-energy cosmic ray spectral cutoffs).

In summary, the analysis of quantum anomalies and renormalization group flows supports the internal consistency of the GSN spin foam model. GGE symmetry appears anomaly-free, imposing stringent constraints on γ 0 . The theory exhibits asymptotic safety from synergy between non-perturbative gravitational soliton stabilization and renormalization properties inherited from asymptotically free electrodynamics via GGE, yielding testable predictions connecting Planck-scale quantum geometry with astrophysical phenomena.

3.1.1. Contrast with LQG: From Connection to Field

Contrasting Fundamental Variables:

LQG: Takes the S U ( 2 ) spin connection A a i and its conjugate electric flux E i a as the fundamental canonical variables. Its quantum geometry is described by spin networks, where edges carry S U ( 2 ) spin representations encoding area quantization, and nodes carry intertwiners encoding volume quantization [6][7].GS Theory: Takes the fully symmetric 4-index spinor field ψ A B C D and its conjugate geometric flux Σ A B C D as the fundamental variables. Its quantum geometry is described by gravitational spinor networks (GSNs), where edges carry spinor quantum numbers related to the area (satisfying the spin-2 projection constraint | k − l | = 2 ), and nodes carry GGE-invariant intertwiners [9][12].Core Difference: LQG originates from the connection dynamics formulation of General Relativity; its fundamental variables are gauge potentials. GS theory treats gravity directly as a spin-2 spinor field; its fundamental variables are the spinor components of the field strength (or curvature). This represents a paradigm shift from “potential” to “field”. The edges of a GSN in GS theory are directly associated with local curvature excitations, whereas the edges in LQG are associated with holonomies (parallel transport) of the connection.

3.1.2. Differences in Constraint Implementation and Dynamics

LQG：After canonical quantization, one must solve the complex quantum constraint equations (Hamiltonian constraint) to extract physical states and dynamics. Its spin foam models (e.g., EPRL/FK) impose causality and simplicity constraints of the spacetime metric by restricting S L ( 2 , ℂ ) representations to specific S U ( 2 ) sub-representations via a specific fusion coefficient. This process involves certain ambiguities, and its connection to the classical limit is sometimes indirect [43].GS Theory: The GGE mechanism deeply integrates gauge symmetry with gravitational dynamics at the classical level. In the path integral, the GGE constraints are directly woven into the definition of the vertex amplitude A v via the projection operator Π G G E (see Section 2.3.1). This makes the imposition of metric (simplicity) constraints more geometric and natural, as it stems directly from the requirement that the spinor field ψ A B C D must be constructible from a real four-volume element. The Regge action is recovered directly and clearly from the semiclassical analysis of the generalized 10j-symbol (Section 2.3.2) [31].The Parameter γ 0 vs. the LQG Immirzi Parameter γ L Q G :LQG: Immirzi parameter γ L Q G is a positive real number introduced into classical connection dynamics. It doesn’t affect classical equations but modifies the quantum spectrum (e.g., area operator eigenvalues = 8 π γ L Q G l P 2 j ( j + 1 ) ) [44]. Its value is conventionally free and needs fixing by matching low-energy conditions like black hole entropy [45].GS Theory: The core parameter γ 0 is intrinsic to theoretical structure, arising naturally from connection between spinor representations and discrete geometric quantum numbers [9]. Importantly, γ 0 is a running effective coupling constant. At low energies where theory approximates connection description, γ 0 at infrared scale E I R can be identified as effective Immirzi parameter:

This implies that the seemingly fundamental constant γ L Q G in LQG is interpreted in GS theory as the “measured value” of a running coupling at infrared energies. Its value is determined by the UV fixed point γ 0 * and the full renormalization group flow, and is in principle no longer free. This offers a novel perspective on resolving the “Immirzi parameter problem” in LQG [44][45].

3.1.3. Contrast with String Theory: Background and Degrees of Freedom

Role of Background:String Theory: Typically formulates perturbative quantization on a fixed classical spacetime background (e.g., Minkowski or Anti-de Sitter space). Spacetime geometry itself is a background field, with the graviton emerging as a specific vibrational mode of closed strings. Non-perturbative definitions (e.g., M-theory) still face significant challenges [11].GS Theory: Belongs, like LQG, to the background-independent framework of quantum gravity. The path integral sums over all possible quantum geometries (labeled by GSNs), with no pre-existing fixed background metric [9]. This constitutes the most fundamental philosophical and methodological distinction from perturbative string theory.Fundamental Degrees of Freedom and Unification Picture:String Theory: Aims to unify all fundamental particles and interactions through the quantum theory of an extended object (the string). Different particles correspond to different vibrational modes and topological configurations of the string [5][8].GS Theory: Focuses on quantizing spacetime geometry itself, with fundamental degrees of freedom being gravitational spinors. Matter and other interactions couple/unify with gravitational field via GGE mechanism [9]. For instance, the gauge structure of the electromagnetic field can be naturally embedded into a subspace of the gravitational spinor algebra through GGE transformations. This represents path based on deep unification of geometric and gauge symmetries, rather than extra dimensions or extended objects.

3.2.1. Structural Simplification and the Classical Limit Problem

The GGE mechanism, by taking the gravitational spinor ψ A B C D (directly corresponding to spacetime curvature) as the fundamental variable, achieves a fundamental simplification of the theoretical structure and provides a historical clarification of the notorious “classical limit problem” in quantum gravity [46].

“Quantum Curvature” Prioritized Over “Quantum Geometry”: Unlike traditional LQG, where the fundamental variable is the holonomy of the connection A a i [7], the basic variable in GS theory is the spinor component of curvature, ψ A B C D . This choice carries profound physical implications: in classical General Relativity, the metric g μ ν and the curvature R μ ν ρ σ are related by differential equations ( R ~ ∂ Γ + Γ Γ , where Γ ~ ∂ g ). In the quantum version of GS, we quantize “curvature” directly, treating the “metric” as a derived effective operator [10][12]:

A concrete construction scheme, stemming from the integral relation g μ ν ~ ∬ R μ ν , can be formulated as:

where C ^ ρ σ is a curvature operator constructed from ψ ^ A B C D , and the integration kernel K μ ν ρ σ satisfies a bi-d’Alembertian equation, □ x 2 K μ ν ρ σ ( x , y ) = δ ( μ ( ρ δ ν ) σ ) δ 4 ( x − y ) + ⋯ . This allows the classical metric to emerge naturally from the condensation of quantum curvature [2].

Clear Classical Correspondence via Coherent States: To describe a semi-classical, macroscopic spacetime, we introduce gravitational spinor coherent states | α 〉 , defined as eigenstates of the operator ψ ^ A B C D :

where α A B C D ( x ) is a classical Weyl spinor field. The expectation value of the effective metric operator in this coherent state directly yields the classical metric:

More crucially, the expectation value of the quantum equations of motion in the coherent state reduces, in the ℏ→0 limit, automatically to the Einstein field equations. Considering the total quantum action S ^ t o t a l = S ^ G S [ ψ ^ , Σ ^ ] + S ^ m [ ϕ ^ , g ^ ] , its equation of motion is δ S ^ t o t a l / δ g ^ μ ν = 0 . Taking the expectation value in the coherent state | α , ϕ 0 〉 and the classical limit, we obtain:

This directly leads to R μ ν − 1 2 g μ ν R + Λ g μ ν = 8 π G T μ ν . Quantum corrections appear naturally as higher-order small terms of the form O ( l P 2 ∇ 4 g ) [4].

The Key to Resolving the “Classical Limit Problem”: In LQG, reconstructing a continuous, smooth metric from discrete spin network states and recovering Einstein equations remains complex [7][16]. In GS theory, this process is intrinsic and direct:Direct Algebraic Relation: The fundamental variable ψ A B C D is curvature itself, directly algebraically related to derivatives of the metric, avoiding the nonlinear complexities of reconstructing the metric from connection holonomies in LQG.Natural Condensation Picture: A macroscopic classical spacetime is interpreted as a condensed phase or coherent state of the quantum gravitational spinor field, with the order parameter 〈 ψ ^ A B C D 〉 ≠ 0 directly corresponding to classical curvature [2].Natural Limiting Behavior of Equations of Motion: Quantum equations of motion smoothly reduce to classical equations in the ℏ→0 limit, without needing additional proofs about constraint operator limits.

Therefore, within GS theory, the transition from quantum gravity to classical General Relativity—the “classical limit problem”—finds a clear, natural, first-principles-derived resolution [2] [46].

3.2.2. Determinacy of the Core Parameter γ 0 : Connection to Grand Unified Theories

In GS theory, γ 0 characterizing quantum geometry discreteness is not free. The GGE framework links it to fundamental interaction coupling constants [9]. The exact analytical expression (3.7) is derived from the synthesis of two cornerstone principles of the theory: the statistical mechanics of black hole horizons and the gauge-gravity unification enforced by the GGE.

Derivation Outline for Equation (3.7):

Constraint from Black Hole Thermodynamics: In the quantum geometric description, the entropy of a black hole is computed by counting the number of GGE-invariant GSN states compatible with a horizon of area A . For a large horizon area ≫ l P 2 , the leading-order result yields the Bekenstein-Hawking law S = A / ( 4 l P 2 ) provided the parameter γ 0 takes a specific value. A detailed state counting, considering the dominant contribution from the smallest non-trivial spin j = 1 / 2 edges, gives the condition:

This is the value required for the quantum geometric entropy to match the classical result in the semi-classical limit.

Constraint from GGE and Renormalization Group Flow: The GGE identifies the gravitational spinor algebra with a subspace of a Grand Unified Theory (GUT) gauge algebra [9]. This identification imposes a relation between the fundamental couplings. Specifically, the effective gravitational coupling γ 0 runs with energy scale E . Its renormalization group (RG) flow, influenced by the gauge couplings, can be computed within the GGE framework. Solving the one-loop RG equation from the GUT scale munifiedmunified to the Planck scale m P , with the boundary condition γ 0 ( m u n i f i e d ) ~ α u n i f i e d , yields a solution of the form:

The specific structure of the quantum corrections within the GGE framework leads to the linear term − α u n i f i e d π in the parentheses.

Synthesis and Uniqueness: The low-energy, semi-classical limit of the theory (black hole thermodynamics) fixes the infrared value of the running coupling, γ 0 I R ≡ γ 0 B H . The high-energy, unified gauge structure determines the ultraviolet behavior and the functional form of γ 0 ( E ) . The unique function that satisfies both the infrared boundary condition (3.7a) and the ultraviolet RG behavior (3.7b) is precisely:

The numerical coefficient ln 2 / π is fixed by the black hole entropy calculation, while the factor ( 1 − α u n i f i e d π ) arises from the one-loop quantum corrections in the gauge-gravity unified theory [47].

The physical implications of Equation (3.7) are profound [47][48]:

Determinacy: It uniquely expresses a fundamental parameter of quantum gravity γ 0 as a function of known (or future measurable) particle physics parameters ( α u n i f i e d , m u n i f i e d ) and fundamental constants ( ln 2 , π , m P ).Unification: It reveals an intrinsic link between the discreteness of quantum spacetime ( γ 0 ) and the unified theory (GUT) describing all non-gravitational interactions, embodying the GGE vision of “gauge-gravity unification”.Testability: This is a quantitative prediction. Taking typical values α u n i f i e d = 1 / 40 = 0.025 , m u n i f i e d = 10 16 GeV, we obtain:

This O ( 0.01 ) value is comparable to LQG values chosen to match black hole entropy (~0.2375) [45], but now with well-defined origin.

Running Behavior of γ 0 and the Immirzi Parameter: Equation (3.7) gives the value of γ 0 at the Grand Unification energy scale m u n i f i e d . As discussed in Section 2.3, γ 0 is a running effective coupling. At lower energy scales (e.g., current cosmological scales), due to quantum corrections, γ 0 evolves to another value, γ 0 I R . When GS theory approximates a connection dynamics at low energies, this infrared value γ 0 I R plays the role of the Immirzi parameter γ L Q G in LQG [44]. Therefore, what is a free parameter requiring external input in LQG is interpreted in GS theory as the value of the running coupling γ 0 ( E ) at a specific low-energy scale, traceable back to Grand Unification physics and the microscopic statistics of black hole entropy.

3.2.3. Dynamical Origin Model for the Cosmological Constant Λ

Based on dimensional analysis and renormalization group arguments, the effective cosmological constant Λ is related to the running γ 0 [41]:

Using Equation (3.7) as the high-energy boundary condition and considering the logarithmic running of γ 0 ( E ) from munifiedmunified down to the Hubble constant H 0 , one can naturally generate a tiny, non-zero Λ consistent with observations [49].

In conclusion, the GGE mechanism brings fundamental breakthroughs: 1) Conceptual and Technical Simplification: Resolves the “classical limit problem” in background-independent quantum gravity [46]; 2) Parameter Determinacy and Unification: Ties γ 0 to particle physics Grand Unification via Equation (3.7) [10][11][47]; 3) Explanation of Profound Puzzles: Provides a dynamical sub-model for cosmological constant problem based on running couplings [41] [49].

This section aims to find self-consistent, static soliton solutions of the nonlinear GS equation under spherical symmetry. Crucially, the nonlinear coupling strength in the equation is not a fixed constant but is dynamically determined by the matter distribution, including the soliton itself, via the response mechanism. This leads to a highly nonlinear self-consistent problem, whose solutions possess a unique radial structure, forming the core foundation for the subsequent description of dark matter [11].

4.1.1. Theoretical Framework and Self-Consistent Equations

We begin with the core equation incorporating the response mechanism [10][11]:

where B A B C D is a spinor tensor that summarizes all nonlinear contributions except for the | ∇ ψ | 2 term, including higher-order terms of ψ and possible derivative coupling. In the general case, B A B C D is a complex function of ψ and its conjugate, the specific form of which is determined by the details of the potential energy U ( ψ ) and quantum corrections, and

To find spherically symmetric static solutions, we adopt a simplified but physically transparent scalar field approximation. This approximation is justified by the symmetry reduction: for localized, static energy configurations under spherical symmetry ( O ( 3 ) symmetry), the tensor degrees of freedom of the 4index spinor can be effectively integrated out or frozen, leaving a dominant real scalar amplitude ϕ ( r ) to characterize the energy density profile. Specifically, under O ( 3 ) symmetry, the most general form of the 4index spinor ψ A B C D that respects spherical symmetry and staticity reduces to a single real scalar amplitude multiplied by the invariant epsilon tensor ϵ A B C D . Any nontrivial tensorial component would necessarily break either rotational invariance or timereversal symmetry and is therefore forbidden for a spherically symmetric, static solution. Moreover, a systematic mode analysis of the linearized GS equation around such a symmetric background reveals that the tensor modes decouple from the scalar mode and possess a mass gap on the order of the inverse core radius 1 / r c . Consequently, for the low-energy, largescale soliton configuration, these tensor modes are not excited dynamically and can be safely integrated out in the effective field theory sense. The resulting effective action for the scalar degree of freedom then takes the form assumed in (4.1.2). Thus we assume the dominant degree of freedom is a real scalar field ϕ related to the gravitational potential Φ ( r ) , such that ψ A B C D ~ ϕ ( r ) ϵ A B C D and | ∇ ψ | 2 ∝ ( ∇ ϕ ) 2 [12][13]. Furthermore, under spherical symmetry, B A B C D effectively reduces to ψ itself (or its linear form), so that the nonlinear term in (4.1.1) becomes ψ | ∇ ψ | 2 ~ ϕ | ∇ ϕ | 2 . We relate the energy density of the soliton itself, ρ ψ ( r ) , to the Hamiltonian density of the scalar field, using a simple model containing kinetic and potential terms:

where μ is a fundamental mass scale of the field and λ s is a self-coupling constant. In this initial exploration, we consider the soliton as an isolated dark matter candidate in the universe, so the total matter density ρ ( r ) = ρ ψ ( r ) , with a background baryonic density ρ b = 0 .

Under spherical symmetry and static conditions, Equation (4.1.1) reduces to a coupled system of nonlinear integro-differential equations for ϕ ( r ) and λ e f f ( r ) :

Field Equation: 1 r 2 d d r ( r 2 d ϕ d r ) − μ 2 ϕ − λ s ϕ 3 = 8 λ e f f ( r ) ( d ϕ d r ) 2 ϕ

Response Equation: λ e f f ( r ) = λ 0 + ∫ 0 ∞ 4 π r ′ 2 d r ′   K ( r , r ′ ) ρ ψ ( r ′ )

Here, K ( r , r ′ ) is the spherically symmetric response kernel:

Where K ( s ) = g 2 e − m s / ( 4 π s ) . The parameter g is a dimensionless coupling strength, and m − 1 is the characteristic length of the kernel (corresponding to the finite range of the response mechanism). Boundary conditions require ϕ ′ ( 0 ) = 0 and ϕ ( ∞ ) = 0 to ensure regularity at the center and localization at infinity.

4.1.2. Numerical Solution Strategy: Iterative Spectral Method

We employ an iterative spectral method to solve this self-consistent system. The procedure is as follows:

1) Initialization: Start with a trial solution, e.g., a Gaussian form ϕ ( 0 ) ( r ) = ϕ 0 e − r 2 σ 2 , and compute the corresponding ρ ψ ( 0 ) ( r ) and initial λ e f f ( 0 ) ( r ) .

2) Spectral Expansion: In the n -th iteration, expand the unknown function ϕ ( n ) ( r ) on the interval [ 0 , R max ] using rational Chebyshev basis functions T k ( x ( r ) ) , where x ( r ) = ( r − L ) / ( r + L ) maps the semi-infinite interval to [ − 1 , 1 ] . This allows for high-precision handling of asymptotic behavior.

3) Iterative Solution:

Field Equation Step: Substitute the current estimate λ e f f ( n − 1 ) ( r ) into the field equation of (4.1.3). Using the collocation method, convert the differential equation into a system of nonlinear algebraic equations for the expansion coefficients, and solve it using the Newton-Raphson method to obtain an updated ϕ ( n ) ( r ) .Response Update Step: From the new ϕ ( n ) ( r ) , compute the updated density ρ ψ ( n ) ( r ) via the density definition. Then, efficiently calculate the integral ∫ K ( r , r ′ ) ρ ψ ( n ) ( r ′ ) d 3 r ′ using a fast Hankel transform algorithm to obtain an updated λ e f f ( n ) ( r ) .