Topological Origin of the Higgs Field: From String-Net Condensation to Geometric Degrees of Freedom

1. Introduction: Revisiting the Higgs Field from the Perspective of Geometric Degrees of Freedom

1.1. The Unsolved Puzzles of the Higgs Field

The Standard Model introduces the Higgs field $\varphi$ as an $SU(2)$ complex doublet scalar field with a Mexican-hat potential:

$$V(\varphi )={\mu }^2|\varphi {|}^2+\lambda |\varphi {|}^4,{\mu }^2<0$$

(1)

This assumption is mathematically self-consistent and has been confirmed by the 2012 LHC experiments [1,2]. Nevertheless, it leaves three fundamental puzzles unresolved [3,4]:

First, why does the Higgs field exist? The Standard Model does not answer where the Higgs field comes from; it is simply "placed" in the theory.

Second, why do the parameters take these values? The quantities ${\mu }^2$,$\lambda$, and the vacuum expectation value $v\approx 246$ GeV are empirical inputs rather than theoretical predictions. The Yukawa couplings—which determine the fermion mass hierarchy—span five orders of magnitude [5], yet there is no explanation for why this hierarchy takes the form it does [6].

Third, why is symmetry broken? [7] The Standard Model describes how symmetry is broken, but does not explain why—the physical origin of symmetry breaking, the "driving force" that propels the universe from the symmetric phase into the broken phase, remains completely unaddressed [8].

Throughout this paper, the term “Higgs parameters” refers specifically to the parameters of the Higgs potential itself—namely, the vacuum expectation value $v$, the quartic coupling $\lambda$, and the Higgs mass $m_H$. The fermion mass spectrum, which is governed by the Yukawa couplings, involves additional bulk-boundary dynamics and is discussed separately in Sec. 6.2 and Appendix B.

These three problems are not new. Yet over the past half-century, attempts to answer them have largely remained within the realm of phenomenological model-building. This paper argues that the fundamental reason lies in the mistaken treatment of the Higgs field as a fundamental entity, rather than as a quantum emergent condensate originating from a deeper structure.

1.2. String-Net Condensation and Metric Emergence: A New Conceptual Framework

Recent advances in topological order and modular tensor categories have provided new conceptual tools for revisiting these issues [9,10,11,12,13].

Concurrently, in quantum condensed matter physics, the observation of chiral gravitons in the fractional quantum Hall effect (Nature Physics, 2026) has directly demonstrated that quantum metric fluctuations are observable physical phenomena [14]. The STAR Collaboration's heavy-ion collision experiments (Science, 2026) have found that baryon number is carried by Y-shaped gluon junctions rather than valence quarks, demonstrating that topological structures can carry conserved charges [15]. Exciton BEC experiments (Nature, 2026) have revealed that multi-component condensates possess switchable internal degrees of freedom [16].

These discoveries collectively point toward a deeper picture: the Standard Model is not a final theory, but rather the low-energy effective theory of a string-net condensate. In this picture, the metric tensor is not fundamental but emerges from the bilinear condensation of the frame field:

$$g_{\mu \nu }(x)={\eta }_{ab}⟨{\widehat{e}}_{\mu }^a(x){\widehat{e}}_{\nu }^b(x)⟩$$

(2)

In previous work, we have derived several independent physical results from the $SU(3{)}_3\otimes SU(2{)}_2$ modular tensor category: a parameter-free prediction of the Cabibbo angle $\lambda =e^{-14/(3\pi )}\approx 0.225$, and the topological origin of neutrino masses and mixing [17]. We have also established a complete emergence mechanism from the string-net condensate to continuous spacetime (Yang et al. 2026a) [18], demonstrating the emergence of the metric tensor as a bilinear frame-field condensate, the emergence of the Einstein equations as an entanglement entropy balance condition, and the emergence of the generalized Gibbs free energy $G_{ST}=H_c-T_{eff}{S}_c$ as the driving principle of cosmic evolution [18]. In the present work, we apply this framework to the electroweak scale, demonstrating that the Higgs field is the scalar mode of the very same metric tensor.

1.3. Selection Criteria for the Category and the Candidate Space

We choose the modular tensor category $\mathcal{C}=\mathrm{S}\mathrm{U}(3{)}_3\otimes \mathrm{S}\mathrm{U}(2{)}_2$ to label the topological order of the bulk string-net condensate. This choice is not based on a single piece of evidence, but rather on the cross-validation provided by the four independent screening criteria listed in Table 1. These criteria differ in screening strength, but their combination significantly compresses the candidate space, making this category the most natural candidate at present. Criteria (A)–(C) are independent selection conditions that compress the candidate space to $SU(3{)}_3\otimes SU(2{)}_2$ without relying on Eq. (13). Criterion (D) serves as a post-hoc consistency check applied after the category has been selected: we test whether the quantum dimension data of this category can yield a reasonable electroweak scale through Eq. (13). If the category selected by (A)–(C) fails (D), the framework would be called into question; in fact, it passes, indicating internal consistency of the framework.

Before proceeding further, an important mathematical clarification is in order. Throughout this paper, $SU(N{)}_k$ always denotes the level-$k$ Wess–Zumino–Witten (WZW) fusion category—an algebraic label of a (2+1)-dimensional topological quantum field theory that encodes the fusion rules, quantum dimensions, and topological spins of anyons in the string-net condensate—not a group-theoretic "extension" of the $SU(N)$ gauge group. More precisely, the correspondence between fusion category $SU(3{)}_3$ and the gauge group $SU(3{)}_c$ is based on Lie algebra isomorphism: the root system of the fusion category shares the same structure as the generators of the $SU(3)$ Lie algebra; hence, when the bulk topological order produces low-energy excitations on the boundary, these boundary states naturally couple to $SU(3{)}_c$ gauge fluxes. Similarly, the Ising anyon structure of $SU(2{)}_2$ couples to $SU(2{)}_L$. Thus, $\mathcal{C}$ is not an "extension" of the gauge group, but rather an algebraic holographic label of the gauge group structure in the underlying topological order—a correspondence analogous to that between the symmetry algebra of the boundary conformal field theory and the bulk gauge symmetry in AdS/CFT. The level values $\mathrm{k}=3$and $\mathrm{k}=2$ are determined by topological constraints; their numerical coincidence with certain gauge group representation dimensions (e.g., $k=3$coinciding with the number of colors $N_c=3$) is an additional constraint on the candidate category in this framework, rather than a consequence of "extension."

Table 1. Selection criteria for category choice and candidates.

Criterion	Screening strength	Specific content	Exclusion effect
Independent criterion (A): Algebraic pairing of gauge group structure	Qualitative screening (hard threshold)	The underlying fusion category must contain the Lie algebra factors $SU(3)$ and $SU(2)$ to ensure that the low-energy boundary can reproduce the Standard Model gauge group $SU(3{)}_c\otimes SU(2{)}_L$	Excludes all categories without the $SU(3{)}_c\otimes SU(2{)}_L$ Lie algebraic structure, such as purely Abelian $U(1)$ series, $G_2$ series, etc.
Independent criterion (B): Three-generation boundary structure	Algebraic constraint	The number of simple objects in the boundary modular tensor category produced by condensing algebra must be 3, corresponding to the three generations of leptons. This requires the bulk category to have sufficiently rich fusion rules	Excludes $\mathrm{S}\mathrm{U}(3{)}_1$ (boundary rank 1), $\mathrm{S}\mathrm{U}(2{)}_3$ (boundary rank 2), $\mathrm{S}\mathrm{U}(3{)}_1\otimes \mathrm{S}\mathrm{U}(2{)}_2$ (boundary rank 2), etc.
Independent criterion (C): Quantitative verification of Cabibbo angle	Strong quantitative test	From the topological temperature $\beta ={\pi }^2/14$ and conformal weight spacing $\mathsf{\Delta }h=1/3$ of this category, one successfully predicts the Cabibbo angle $e^{-14/(3\pi )}\approx 0.225$, consistent with the experimental value 0.2265 to within 0.04%	Excludes categories with different topological temperatures or conformal weight spacings, such as $SU(3{)}_1$ (different topological temperature), $SU(3{)}_4$ (different conformal weight spacing), etc.
Post-hoc consistency check (D): Matching of quantum dimension magnitudes	Post-hoc (verification)	The calculation of the bulk-boundary coupling coefficient in this framework requires the specific combination of quantum dimensions ${\mathrm{d}}_{\mathrm{a}}=2$ (object $(\mathrm{1,0})\otimes \mathbf{1}$ and ${\mathrm{d}}_{\mathrm{b}}=\sqrt{2}$ (object $\mathbf{1}\otimes \mathsf{\sigma }$). Substitution into Eq. (15) precisely yields $\mathrm{v}/{\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}=1.6818$; with ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\approx 146$ GeV, this gives $\mathrm{v}\approx 246$ GeV. Other quantum dimension combinations (e.g., ${\mathrm{d}}_{\mathrm{a}}=\sqrt{3}$) deviate in order of magnitude	Excludes $\mathrm{S}\mathrm{U}(3{)}_2$ (fundamental representation quantum dimension $\sqrt{3}$), $\mathrm{S}\mathrm{U}(4{)}_2$ (different quantum dimension spectrum), etc.

Table 1. Selection criteria for category choice and candidates.

Criterion	Screening strength	Specific content	Exclusion effect
Independent criterion (A): Algebraic pairing of gauge group structure	Qualitative screening (hard threshold)	The underlying fusion category must contain the Lie algebra factors $SU(3)$ and $SU(2)$ to ensure that the low-energy boundary can reproduce the Standard Model gauge group $SU(3{)}_c\otimes SU(2{)}_L$	Excludes all categories without the $SU(3{)}_c\otimes SU(2{)}_L$ Lie algebraic structure, such as purely Abelian $U(1)$ series, $G_2$ series, etc.
Independent criterion (B): Three-generation boundary structure	Algebraic constraint	The number of simple objects in the boundary modular tensor category produced by condensing algebra must be 3, corresponding to the three generations of leptons. This requires the bulk category to have sufficiently rich fusion rules	Excludes $\mathrm{S}\mathrm{U}(3{)}_1$ (boundary rank 1), $\mathrm{S}\mathrm{U}(2{)}_3$ (boundary rank 2), $\mathrm{S}\mathrm{U}(3{)}_1\otimes \mathrm{S}\mathrm{U}(2{)}_2$ (boundary rank 2), etc.
Independent criterion (C): Quantitative verification of Cabibbo angle	Strong quantitative test	From the topological temperature $\beta ={\pi }^2/14$ and conformal weight spacing $\mathsf{\Delta }h=1/3$ of this category, one successfully predicts the Cabibbo angle $e^{-14/(3\pi )}\approx 0.225$, consistent with the experimental value 0.2265 to within 0.04%	Excludes categories with different topological temperatures or conformal weight spacings, such as $SU(3{)}_1$ (different topological temperature), $SU(3{)}_4$ (different conformal weight spacing), etc.
Post-hoc consistency check (D): Matching of quantum dimension magnitudes	Post-hoc (verification)	The calculation of the bulk-boundary coupling coefficient in this framework requires the specific combination of quantum dimensions ${\mathrm{d}}_{\mathrm{a}}=2$ (object $(\mathrm{1,0})\otimes \mathbf{1}$ and ${\mathrm{d}}_{\mathrm{b}}=\sqrt{2}$ (object $\mathbf{1}\otimes \mathsf{\sigma }$). Substitution into Eq. (15) precisely yields $\mathrm{v}/{\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}=1.6818$; with ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\approx 146$ GeV, this gives $\mathrm{v}\approx 246$ GeV. Other quantum dimension combinations (e.g., ${\mathrm{d}}_{\mathrm{a}}=\sqrt{3}$) deviate in order of magnitude	Excludes $\mathrm{S}\mathrm{U}(3{)}_2$ (fundamental representation quantum dimension $\sqrt{3}$), $\mathrm{S}\mathrm{U}(4{)}_2$ (different quantum dimension spectrum), etc.

To illustrate the actual exclusion power of the above screening criteria, consider several "near-miss" candidate categories:

- $SU(3{)}_1\otimes SU(2{)}_2$: Its topological temperature $\beta$ differs from that of $SU(3{)}_3$, and the calculated Cabibbo angle deviates from the experimental value by more than 10%, failing criterion (C).

- $SU(3{)}_3\otimes SU(2{)}_1$: $SU(2{)}_1$ is an Abelian category with boundary rank 1, incapable of producing the three-generation lepton structure, failing criterion (B).

- $SU(3{)}_2\otimes SU(2{)}_2$: The quantum dimension of the object $(\mathrm{1,0})$ is $\sqrt{3}\approx 1.732$ rather than the required${ d}_a=2$; substituting into the bulk-boundary coupling formula gives $v/{\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$ deviating from the correct order of magnitude by about 10%, failing criterion (D).

- $SU(4{)}_2\otimes SU(2{)}_2$: Although one can construct a boundary of rank 3, the quantum dimension spectrum of $SU(4{)}_2$ is $\{1,\sqrt{2},\sqrt{3},\cdots \}$, which does not match the bulk-boundary coupling coefficients required by this framework, failing criterion (D); moreover, its Cabibbo-angle prediction has not been verified against criterion (C).

- $SU(2{)}_6\otimes SU(2{)}_2$: Lacks an $SU(3)$ factor, failing to provide the gauge group pairing for strong interactions, failing criterion (A).

After this fourfold screening, the candidate category space is greatly compressed. We must emphasize: we do not claim that $\mathcal{C}=SU(3{)}_3\otimes SU(2{)}_2$ is mathematically unique. However, among all known modular tensor categories to date, it is the only one that simultaneously satisfies all four criteria (A)–(D). Other potential alternatives either fail one criterion definitively or require additional, unnatural assumptions. Until another category satisfying all criteria is discovered, we adopt this as our working hypothesis. This pattern of "multiple independent constraints converging on a single category" is generally regarded in physics as a strong indicator—much as the choice of $SU(2{)}_L\otimes U(1{)}_Y$ in the electroweak unification is not mathematically unique, but is determined by the intersection of multiple experimental constraints.

A physical remark is in order: Criteria (C) and (D) address two physically independent domains in the Standard Model—flavor physics (the Cabibbo angle) and the electroweak scale (the magnitude of $v$). Both are determined simultaneously by the topological data—the $S$-matrix and quantum dimensions—of the same category, and both agree with experiment. This cross-domain numerical consistency is evidence of the predictive power of our framework, rather than post-hoc fitting.

1.4. The Central Claim of This Paper

Based on the above framework, we propose a precise central claim:

The Higgs field $\varphi (\mathit{x})$ is not an independent fundamental scalar field, but rather the scalar (dilatational) excitation mode of the metric tensor of the string-net condensate at the electroweak scale:

$$\varphi (x)\sim \mathrm{T}\mathrm{r}[\delta g_{\mu \nu }(x)]=g^{\mu \nu }\delta g_{\mu \nu }$$

(3)

This claim transforms the Higgs field from an "externally imposed fundamental entity" into a "degree of freedom of spacetime geometry itself." It implies:

- The existence of the Higgs field is a natural consequence of geometry, requiring no additional assumptions;

- The Higgs potential is a Landau expansion of the cosmological enthalpy around the order parameter, rather than an arbitrarily chosen functional form;

- The parameters of the Higgs mechanism are determined by the topological data of the string-net condensate, rather than being free inputs;

- Symmetry breaking is driven by the competition between configurational entropy and cosmological enthalpy, rather than by a mysterious spontaneous process.

Methodologically, this paper concretizes the quantum entropy–enthalpy competition theory by adopting a generalized free-energy-difference criterion. We argue that the direction of evolution of the system—including whether symmetry is broken and at what temperature this occurs—is determined by the entropy–enthalpy competition between the symmetric and broken phases, specifically by the free-energy difference $\mathsf{\Delta }\mathcal{F}$. When $\mathsf{\Delta }\mathcal{F}>0$, the symmetric phase is stable; when $\mathsf{\Delta }\mathcal{F}<0$, the broken phase is stable; the critical temperature $T_c$ is precisely defined by $\mathsf{\Delta }\mathcal{F}(T_c)=0$. This method reformulates the Higgs phase transition as a computable free-energy minimization problem.

The structure of this paper is as follows: Section 2 establishes the mathematical framework of the metric tensor as an emergent order parameter of the string-net condensate. Section 3 derives the origin of the Higgs potential as a Landau expansion of the cosmological enthalpy. Section 4 demonstrates how Higgs parameters can be computed from the topological data of the modular tensor category. Section 5 proposes testable predictions and establishes analogical support from three 2026 experiments. Section 6 discusses the relationship between topological data and cosmic evolution in mass generation, noting that the complete mass spectrum of elementary particles remains a complex open problem. Section 7 discusses the relationship between our framework and the Standard Model, as well as its boundaries. Section 8 concludes.

2. The Metric Tensor as an Emergent Order Parameter of the String-Net Condensate

2.1. The Frame Field in the String-Net Condensate

The fundamental degrees of freedom of the string-net condensate are the connectivity structures of strings [19]. In this substrate, there exists an intrinsic tendency toward condensation into ordered topological structures—strings tend to form coherent topological defect networks, including vortex lines, braiding structures, and correlated configurations spanning macroscopic distances [20,21]. This condensation produces, near the critical point, a new ordered phase whose order parameter is the frame field ${\mathrm{e}}_{\mathsf{\mu }}^{\mathrm{a}}(\mathrm{x})$ [18].

The frame field ${\mathrm{e}}_{\mathsf{\mu }}^{\mathrm{a}}(\mathrm{x})$ describes the "orientational" structure of the condensate at each point in space. It is a macroscopic variable emerging from the collective behavior of the string-net condensate, not an externally imposed fundamental field. In the string-net condensate, the quantized version ${\widehat{e}}_{\mu }^a(x)$ satisfies [18]:

- Transformation property: Under local Lorentz rotations, ${\widehat{e}}_{\mu }^a(x)$ transforms as a vector.

- Nonzero vacuum expectation value: In the condensed phase, $⟨{\widehat{e}}_{\mu }^a(x)⟩\ne 0$.

- Correlations: Near the critical point, its two-point correlation function takes conformal form.

The vacuum condensation of the frame field spontaneously breaks the internal symmetry of the string-net condensate, producing a new ordered phase—the geometric phase. In this phase, continuous spacetime and its metric structure begin to emerge.

2.2. The Metric Tensor as a Bilinear Condensate

In the geometric phase, the metric tensor is defined as the bilinear form of the vacuum expectation values of the frame field [18]:

$$g_{\mu \nu }(x)={\eta }_{ab}⟨{\widehat{e}}_{\mu }^a(x){\widehat{e}}_{\nu }^b(x)⟩$$

where ${\eta }_{ab}$ is the internal flat metric, determined by the quantum dimensions and fusion rules of the string-net condensate.

The physical meaning of this construction is that spatial distance is not fundamental, but rather a statistical correlation of the frame-field condensate. The distance between two points is, in essence, the macroscopic manifestation of the frame-field correlation function. When the frame field is completely decoupled (high-temperature limit), there is no metric and no space; when the frame field condenses, the metric emerges and space appears.

2.3. The Three Excitation Modes of the Metric Tensor

Excitations of the metric tensor $g_{\mu \nu }$ can be classified into three modes [22], corresponding to different particles:

Spin-2 mode (graviton) [23]: The traceless transverse fluctuation of the metric. In the fractional quantum Hall effect, this mode has been observed as the chiral graviton (Nature Physics, 2026). This experiment demonstrates that quantum metric fluctuations are not mathematical constructs but observable physical effects.

Spin-1 mode (gauge bosons) [24]: The transverse vector fluctuation of the metric. In the string-net condensate, these vector fluctuations are no longer purely gauge artifacts. This suggests that topological terms in the low-energy effective action may modify the structure of gauge constraints, potentially allowing vector modes to carry physical degrees of freedom. This is a plausible mechanism by which the string-net condensate could give rise to spin-1 gauge bosons, though a rigorous demonstration remains to be established.In the string-net condensate, these modes couple to $U(1)$or $SU(2)$ gauge groups, producing electromagnetic and weak forces.

In classical general relativity, vector fluctuations of the metric are typically non-physical because they can be eliminated by diffeomorphism gauge transformations. However, in the string-net condensate, this conclusion no longer holds. The reason is that the string-net condensate provides a preferred reference frame—namely, the "lattice" structure of the string-net ground state—which breaks the full diffeomorphism symmetry, reducing it to the gauge symmetry ($U(1)$ or $SU(2)$) in the low-energy effective theory. This microscopic preferred frame is fluctuating and scale-dependent; its effects are averaged out in the low-energy limit, so that Lorentz symmetry emerges as an effective symmetry. In this context, vector modes that would otherwise be gauge-eliminated become physical: they correspond to collective excitations of transverse topological defects in the string net, projecting in the bulk-boundary correspondence to gauge bosons on the boundary. This is consistent with the physical picture of spin-1 modes (such as magnetorotons) in the fractional quantum Hall effect: in FQH systems, fluctuations of the quantum metric contain spin-1 components that become physical excitations on the boundary via the bulk-boundary correspondence of topological order [25].

More specifically, the low-energy effective action of the string-net condensate contains topological terms (such as Chern-Simons or BF terms) that modify the structure of gauge constraints, so that vector modes are no longer pure gauge redundancies but carry physical degrees of freedom.

Spin-0 mode (Higgs boson) [1,2]: The scalar (dilatational) fluctuation of the metric:

$$\varphi (x)={\displaystyle \frac{1}{\sqrt{d}}}\mathrm{T}\mathrm{r}[\delta g_{\mu \nu }(x)]={\displaystyle \frac{1}{\sqrt{d}}}{g}^{\mu \nu }\delta g_{\mu \nu }$$

(4)

where $d$ is the spacetime dimension ($d=4$ in this paper). This scalar mode is precisely the Higgs field. It carries no spin and is the trace mode of the metric tensor.

We unify these known, disparate pictures by pointing out that they are all different excitation modes of the same geometric entity—the metric tensor emergent from the string-net condensate. This classification implies that the Higgs boson, the graviton, and the gauge bosons are not independent particles, but rather different vibrational modes of the same geometric entity—the metric tensor. The differences between them are not essential but modal.

The above discussion of vector modes becoming physical is a central judgment of this paper, supported by the breaking of diffeomorphism symmetry, the modification of gauge constraints by topological terms, and the experimental analogy from FQH systems. A rigorous demonstration remains for future work.

2.4. Degree-of-Freedom Correspondence for the Scalar Mode

To address concerns regarding the disparity in the number of degrees of freedom between the scalar mode and the Higgs field, we clarify the correspondence in this subsection.

The Standard Model Higgs field is an $SU(2)$ complex doublet with 4 real components in the symmetric phase. In the broken phase, 3 components are absorbed by the W and Z bosons, leaving one physical Higgs particle.

The scalar mode $\varphi =g^{\mu \nu }\delta g_{\mu \nu }$of the metric tensor has only one independent component geometrically, corresponding to the remaining physical Higgs particle after symmetry breaking. The full Higgs doublet structure (4 real components) originates from the $SU(2)$ symmetry and its spontaneous breaking internal to the string-net condensate: in the symmetric phase, this scalar mode together with three Goldstone modes constitutes a doublet; when the competition between configurational entropy and cosmological enthalpy drives condensation, the three Goldstone modes are absorbed by the $W^{\pm }$ and $Z$ bosons via bulk-boundary coupling, leaving one physical scalar excitation. The correspondence between the Standard Model Higgs field and its counterpart in the string-net condensate is systematically represented in Table 2.

This correspondence provides degree-of-freedom support for the central claim that "the scalar mode of the metric is the Higgs field." In the fractional quantum Hall effect [14], quantum metric fluctuations produce spin-2 chiral gravitons—demonstrating that the concept of "quantum metric fluctuations producing particle excitations" is not only legitimate but also physically realizable under certain conditions. If the spin-2 mode of the quantum metric indeed produces observable gravitons, then it is a natural extension of the same physical law that the spin-0 mode of the quantum metric produces the Higgs boson.

Having interpreted the Higgs field as the scalar mode of the metric tensor, a natural question arises: why does this geometric degree of freedom "condense" at the electroweak scale—that is, acquire a nonzero vacuum expectation value? In the Standard Model, the answer is "because ${\mu }^2<0$," but this is merely a description of the shape of the potential, not an explanation of the physical driving force. In our framework, the answer is given by the free-energy-difference criterion: as the temperature decreases, the free-energy difference $\mathsf{\Delta }\mathcal{F}(T)$ between the symmetric phase $\varphi =0$ and the broken phase $\varphi =v$ changes sign; when $\mathsf{\Delta }\mathcal{F}<0$, the geometric degree of freedom condenses spontaneously. Thus, the condensation of the Higgs field is not a mathematical property of the potential shape, but rather a quantum thermodynamic choice of the geometric degree of freedom driven by temperature. This perspective transforms the Higgs mechanism from a problem of "potential shape" into one of "free-energy minimization" of a quantum system, providing the methodological foundation for the Landau expansion in Section 3 and the numerical calculations in Section 4.

3. The Higgs Potential as a Landau Expansion of Cosmological Enthalpy

3.1. Competition Between Configurational Entropy and Cosmological Enthalpy

In the string-net condensate, there exist two competing physical tendencies, which we have elaborated and justified in previous work [18]:

Configurational entropy $S_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$: The string net possesses a vast number of microscopic degrees of freedom in configuration space, driving the system toward disorder and high symmetry.

Cosmological enthalpy$H_c$: The string net has an intrinsic tendency to condense into ordered topological structures, driving the system toward reduced symmetry and ordered condensate formation.

Their competition is expressed through the generalized Gibbs free energy:

$$G_{ST}=H_c-T_{eff}\cdot S_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$$

(5)

where $T_{eff}$ is the effective temperature of the universe. The direction of system evolution is determined by $\delta G_{ST}<0$.

3.2. Landau Expansion Near the Critical Point

Near the electroweak phase transition critical point, the order parameter $\varphi$ (i.e., the Higgs field) is small. We can expand $G_{ST}$ around $\varphi =0$:

$$G_{ST}[\varphi ]=G_{ST}[0]+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\delta }^2{G}_{ST}}{\delta {\varphi }^2}}{|}_0{\varphi }^2+{\displaystyle \frac{1}{4!}}{\displaystyle \frac{{\delta }^4{G}_{ST}}{\delta {\varphi }^4}}{|}_0{\varphi }^4+\cdots$$

(6)

Define:

$${\mu }^2={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\delta }^2{G}_{ST}}{\delta {\varphi }^2}}{|}_0,\lambda ={\displaystyle \frac{1}{4!}}{\displaystyle \frac{{\delta }^4{G}_{ST}}{\delta {\varphi }^4}}{|}_0$$

(7)

Then:

$$G_{ST}\left[\varphi \right]=G_{ST}\left[0\right]+{\mu }^2{\varphi }^2+\lambda {\varphi }^4+\cdots$$

(8)

This is precisely the Standard Model Higgs potential. The key distinction is that ${\mu }^2$and $\lambda$ are no longer input parameters but rather the outcomes of the competition between configurational entropy and cosmological enthalpy.

3.3. Thermodynamic Driving Force of Symmetry Breaking

At high temperatures (large $T_{eff}$), the configurational entropy term dominates, and $⟨\varphi ⟩=0$. As the universe cools below the critical temperature ($T_{eff}<T_c$), the cosmological enthalpy term dominates, and $⟨\varphi ⟩=v\ne 0$.

The critical temperature $T_c$ is determined by the balance condition between configurational entropy and cosmological enthalpy:

$$T_c={\displaystyle \frac{H_c}{S_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}}}$$

(9)

The specific forms of ${\mathrm{S}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ and ${\mathrm{H}}_{\mathrm{c}}$ remain to be derived from the microscopic string-net model; here we use only their phenomenological Landau-expanded forms (see [18] for details).

Therefore, symmetry breaking is not a mysterious "spontaneous" process, but rather a phase transition with a well-defined quantum thermodynamic driving force.

As a simple demonstrative computational example, one may adopt the following toy forms for the configurational entropy and the cosmological enthalpy:

$$S_{\mathrm{c}onf}(T)\sim \ln \mathcal{N}(T),\mathcal{N}(T)\propto {\left({\displaystyle \frac{T}{T_0}}\right)}^{\nu }$$

where $T_0$ is the characteristic temperature scale of the string-net degrees of freedom, $\nu$ is a positive exponent related to the effective number of string-net degrees of freedom, and $H_0$ is the characteristic condensation energy. The free-energy difference is then

$$\mathsf{\Delta }\mathcal{F}(T)=H_c(T)-T{S}_{\mathrm{c}onf}(T)\sim -H_0\left(1-{\displaystyle \frac{T}{T_0}}\right)-\nu T\ln \left({\displaystyle \frac{T}{T_0}}\right)$$

The critical temperature $T_c$, defined by $\mathsf{\Delta }\mathcal{F}(T_c)=0$, satisfies

$$H_0\left(1-{\displaystyle \frac{T_c}{T_0}}\right)+\nu T_c\ln \left({\displaystyle \frac{T_c}{T_0}}\right)=0$$

Assuming the electroweak phase transition occurs near this characteristic scale, i.e., ${T_c~T}_0$, the critical temperature is approximately

$$T_c\simeq {\displaystyle \frac{H_0}{H_0+\nu T_0}}{T}_0$$

In the string-net condensate, the reference scale $T_0$ is naturally identified with the string tension scale ${\mathsf{\Lambda }}_{\mathrm{s}tring}$, while $H_0$ is set by the condensation enthalpy scale. Thus $T_c$ is determined by the competition between the entropy exponent $\nu$ and the enthalpy scale $H_0$, providing a simple but concrete realization of the general free-energy criterion of Section 3.

The above example is not intended as a microscopic derivation, but it demonstrates that the critical temperature and the phase structure are calculable once the microscopic forms of $S_{\mathrm{c}onf}$ and $H_c$ are specified.

4. Computing Higgs Parameters from Modular Tensor Category Data

4.1. Topological Data of the Modular Tensor Category (MTC)

In our framework, the bulk string-net condensate is described by the modular tensor category $\mathcal{C}=SU(3{)}_3\otimes SU(2{)}_2$. The $S$-matrix, quantum dimensions, and conformal weights of this category have been computed and verified [17].

We require the following data:

- Bulk object $a=(\mathrm{1,0})\otimes \mathbf{1}$: quantum dimension $d_a=2$,

- Boundary object $b=\mathbf{1}\otimes \sigma$: quantum dimension $d_b=\sqrt{2}$,

- Reference object $c=\mathbf{1}\otimes \mathbf{1}$: quantum dimension $d_c=1$,

- $S$-matrix elements of the tensor-product category: $S_{ab}=\sqrt{2}/6$, $S_{ac}=1/6$.

4.2. Definition and Physical Origin of the Bulk-Boundary Coupling Coefficient

In the string-net condensate, the bulk topological order and boundary excitations are not independent but are correlated through the bulk-boundary correspondence. The strength of this correlation, i.e., the bulk-boundary coupling coefficient $C$, determines how the condensation amplitude of the bulk order parameter (the Higgs field) projects onto boundary physics.

4.2.1. Physical Picture of Bulk-Boundary Coupling

The bulk object $a=(\mathrm{1,0})\otimes \mathbf{1}$ is a fundamental excitation carrying the $SU(3{)}_3$ topological charge in the string-net condensate; the boundary object $b=\mathbf{1}\otimes \sigma$ is an $SU(2{)}_2$ Ising anyon on the boundary (corresponding to lepton-type excitations). When a bulk excitation propagates to the boundary, it can be transformed into a boundary excitation via a quantum topological process—"splitting" or "tunneling." This process is analogous to:

- In the fractional quantum Hall effect, bulk quasi-holes split into chiral edge fermions at the boundary;

- In AdS/CFT, the projection value of a bulk field on the boundary is determined by the bulk-boundary coupling constant.

The bulk-boundary coupling coefficient $C$is precisely the physical quantity that quantifies the transfer amplitude of this "bulk excitation $\to$ boundary excitation" process.

4.2.2. Topological Definition of the Coupling Coefficient

In the modular tensor category, the fusion/splitting process between anyons is encoded jointly by the $\mathit{S}$-matrix and quantum dimensions. Specifically:

- $\mathit{S}$-matrix element ${\mathrm{S}}_{\mathrm{a}\mathrm{b}}$: Measures the topological overlap integral between the bulk object $a$ and the boundary object $b$. In anyon theory, ${\mathrm{S}}_{\mathrm{a}\mathrm{b}}$ is proportional to the amplitude for two anyons to fuse into the vacuum. Hence, $S_{ab}/S_{ac}$ reflects the relative coupling strength of the bulk object $a$ to the boundary object $b$ relative to the reference object $c$ (the vacuum).

- Quantum dimension ratio $\sqrt{d_b/d_c}$: The quantum dimension $d_x$ corresponds to the Hilbert space dimension of object $x$ (i.e., the ground-state degeneracy). When a bulk excitation couples to the boundary, one must account for normalization of the boundary state density. The ratio $\sqrt{d_b/d_c}$ originates from the normalization factor of the boundary propagator, ensuring the coupling coefficient is dimensionally correct and gauge-invariant.

Thus, the bulk-boundary coupling coefficient is defined as:

$$C={\displaystyle \frac{S_{ab}}{S_{ac}}}\sqrt{{\displaystyle \frac{d_b}{d_c}}}$$

(10)

where $c=\mathbf{1}\otimes \mathbf{1}$ is the vacuum object of the tensor-product category, serving as the reference. This definition is not entirely new in mathematical physics; it shares the same structure as the overlap integral between the boundary state and the bulk field in boundary conformal field theory (BCFT) (cf. the generalization of the Affleck–Ludwig boundary entropy formula [26]).

Physical intuition: If $S_{ab}$ is large, the bulk object $a$ is topologically "close" to the boundary object $b$, implying strong coupling; if $d_b$is large, the boundary object $b$ has more internal degrees of freedom available to participate in the coupling, enhancing the effective coupling. The product of these two effects yields the total bulk-boundary coupling strength. It is worth noting that the above definition of the bulk-boundary coupling coefficient shares the same mathematical structure as the Ishibashi state construction of the Affleck–Ludwig boundary entropy ($g$-function) in BCFT [26]. In BCFT, the Ishibashi expansion coefficient of the boundary state $|B⟩$is proportional to $S_{ab}/\sqrt{S_{ac}}$[27], and the $g$-function is defined as the projection of the boundary state onto the vacuum, $g=⟨0|B⟩$ [28]. After normalization, one obtains $|C{|}^2=g^2\cdot d_b/d_c$. Since the $g$-function satisfies the $g$-theorem—monotonic decrease along the boundary renormalization-group flow [28]—this provides a clear field-theoretic identity for our bulk-boundary coupling coefficient: $|C{|}^2$ is not a freely adjustable parameter, but a topological invariant satisfying a monotonicity constraint. A complete derivation is provided in Appendix C.1.

4.2.3. Numerical Calculation

From the modular tensor category data listed in Section 4.1, we have:

- $S$-matrix elements of the tensor-product category $\mathcal{C}=SU(3{)}_3\otimes SU(2{)}_2$:

$$S_{ab}={\displaystyle \frac{\sqrt{2}}{6}},S_{ac}={\displaystyle \frac{1}{6}}$$

These numerical values are verified by SageMath in Appendix A.

- Quantum dimensions:

$$d_a=2,d_b=\sqrt{2},d_c=1$$

Substituting into Eq. (10):

$$C={\displaystyle \frac{\sqrt{2}/6}{1/6}}\sqrt{{\displaystyle \frac{\sqrt{2}}{1}}}=\sqrt{2}\cdot 2^{1/4}=2^{3/4}$$

(11)

Therefore:

$$|C{|}^2=2^{3/2}=2\sqrt{2}\approx 2.8284$$

(12)

Physical meaning of this value: It quantifies the effective coupling strength between the fundamental $SU(3{)}_3$ topological charge in the bulk and the $SU(2{)}_2$ Ising anyon on the boundary. In the absence of bulk-boundary coupling, $|C{|}^2=1$; here, $|C{|}^2\approx 2.83$ indicates that boundary effects significantly enhance the condensation amplitude of the bulk order parameter by a factor of about 2.83. This is precisely why the vacuum expectation value $v$ can be larger than the bare string-net energy scale ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$ by an $O(1)$ factor.

4.3. Topological Expression for the Vacuum Expectation Value

4.3.1. Relation Between the Vacuum Expectation Value $\mathit{v}$ and the Characteristic String-Net Energy Scale ${\mathsf{\Lambda }}_{\mathbf{s}\mathbf{t}\mathbf{r}\mathbf{i}\mathbf{n}\mathbf{g}}$

Methodologically, drawing on the historical precedents of the Bethe ansatz and the Luttinger ansatz, we position Eq. (13) below as a phenomenological ansatz—it is not the end point of a first-principles derivation, but rather a tentative form based on physical intuition and symmetry constraints, with a rigorous derivation left for future work.

Within this framework, the relation between the vacuum expectation value $v$ and the characteristic string-net energy scale ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$ is:

$$v={\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\cdot {\displaystyle \frac{|C{|}^2}{\sqrt{d_a{d}_b}}}$$

(13)

The structure of this phenomenological ansatz is naturally determined by two lines of reasoning: dimensional analysis and the combination of the bulk-boundary coupling coefficient with quantum-dimension normalization factors. It contains no freely adjustable functional form, leaving only an $O(1)$ coefficient $\alpha$ at the overall scale. Hence it possesses the characteristics of an ansatz: simple form, structurally constrained by symmetry, and awaiting rigorous derivation and testing. Dimensional analysis shows that the structure is determined entirely by the unique combination of topological data—$|C{|}^2$ is the unique scalar invariant of the bulk-boundary coupling, and $\sqrt{d_a{d}_b}$ is the unique natural normalization factor. In this sense, Eq. (13) is the simplest and most natural form through which topological data can enter the vacuum expectation value. If dimensional analysis provides formal uniqueness for Eq. (13), then the BCFT $g$-theorem provides support at the first-principles level: since $|C{|}^2$is monotonically related to the $g$-function (see Appendix C.1), the order-parameter condensation amplitude $v$ determined by $|C{|}^2$ must be a definite function of the MTC topological data.

The coupling coefficient from Section 4.2 directly enters the vacuum expectation value formula (13). The physical logical chain is:

$$\mathrm{T}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}(S,d)⟶\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}|C{|}^2⟶\mathrm{V}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{u}\mathrm{m}v$$

The physical meaning of each step is clear: the topological data determine the bulk-boundary coupling strength, which in turn, via dimensional analysis (combined with the string-net scale ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$), determines the condensation amplitude of the order parameter. In this section we present the final expression for $v$.

The physical motivation for Eq. (13) is as follows: In the string-net condensate, the vacuum expectation value is the condensation amplitude of the order parameter. The condensation amplitude should be determined by the coupling strength between the bulk and boundary states—stronger coupling implies deeper condensation and a larger vacuum expectation value. The quantum dimensions $d_a$and $d_b$serve as natural normalization factors that convert the coupling strength into energy dimensions. Thus, $v\propto |C{|}^2/\sqrt{d_a{d}_b}$ is the most natural form from dimensional analysis and physical intuition. The exact form of this formula awaits rigorous derivation from the complete OPE theory of bulk-boundary coupling. We present it here as a demonstration of the framework's calculability, not as a final exact prediction.

Substituting the numerical values:

$$\sqrt{d_a{d}_b}=\sqrt{2\cdot \sqrt{2}}=2^{3/4}\approx 1.6818$$

(14)

$${\displaystyle \frac{v}{{\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}}}={\displaystyle \frac{2\sqrt{2}}{2^{3/4}}}=2^{3/4}\approx 1.6818$$

(15)

Thus:

$$v\approx 1.6818\cdot {\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$$

(16)

The structure of Eq. (13) is not arbitrary. Within the MTC framework, the bulk-boundary coupling is characterized by a single scalar invariant—the squared modulus $|C{|}^2$of the coupling coefficient—because the relative phase is unobservable in the vacuum expectation value. The normalization must involve the quantum dimensions of the participating objects; dimensional analysis requires a factor of energy dimension $-1$, and the unique combination of $d_a$ and $d_b$ with that dimension is $\sqrt{d_a{d}_b}$. No other gauge-invariant, dimensionless combination of the MTC data can enter at this order. Thus Eq. (13) is the simplest and most natural ansatz consistent with the symmetries and degrees of freedom of the framework, up to an overall $O(1)$ coefficient. This is precisely the same logic that underlies the Ginzburg-Landau expansion: the form is dictated by symmetry and dimensionality, while the coefficients are to be determined by experiment or by a more microscopic theory.

4.3.2. Determination of the String-Net Tension Scale ${\mathsf{\Lambda }}_{\mathbf{s}\mathbf{t}\mathbf{r}\mathbf{i}\mathbf{n}\mathbf{g}}$

To determine the string-net tension scale ${\mathsf{\Lambda }}_{\mathbf{s}\mathbf{t}\mathbf{r}\mathbf{i}\mathbf{n}\mathbf{g}}$, we utilize the cooling history of the early universe from the high-temperature plasma state to the low-temperature condensed state. Above the electroweak scale, the universe is in a high-temperature plasma state, and the string-net degrees of freedom are violently excited by thermal fluctuations, unable to form coherent geometric condensates. As the universe expands and cools, when the temperature drops near the electroweak phase transition temperature $T_{\mathrm{E}\mathrm{W}}$, topological defects in the string net are for the first time able to form coherent configurations spanning macroscopic scales. This process is described by the Kibble–Zurek mechanism [29]: the typical energy scale of defect network formation is set by the critical temperature $T_c$, which in this case is precisely the electroweak phase transition temperature $T_{\mathrm{E}\mathrm{W}}$. Therefore, the tension scale at which the string-net condensate forms is not an independently adjustable parameter, but rather the characteristic energy scale set by the cosmic cooling history:

$${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\sim T_{\mathrm{E}\mathrm{W}}$$

The electroweak phase transition temperature of the Standard Model has been independently determined by lattice simulations to be $T_{\mathrm{E}W}\approx 159\pm 10$ GeV [30,31]. Substituting this independent constraint, taking $T_{\mathrm{E}\mathrm{W}}\approx 149-169$ GeV, and using the ratio $v/{\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\approx 1.68$ from the MTC data, we obtain

$$v\approx 1.68\times (149-169)\mathrm{G}\mathrm{e}\mathrm{V}\approx 250-284\mathrm{G}\mathrm{e}\mathrm{V}$$

The experimental value $v\approx 246$ GeV lies within this order-of-magnitude interval, demonstrating consistency between the topological prediction, the cosmic thermal history, and the measured electroweak scale. We stress that this is a consistency check rather than a precise numerical prediction, because both the $O(1)$ coefficient in the Kibble–Zurek relation and the exact proportionality factor in Eq. (13) remain to be determined from first principles.

The input value $T_{EW}\approx 159\pm 10$ GeV is taken from lattice simulations of the Standard Model electroweak theory [30,31], which are performed within the conventional Higgs framework. Our purpose here is not to claim that $T_{EW}$ is derived independently of all Higgs-related physics, but rather to show that—taking the Standard Model's own prediction of the crossover temperature as a phenomenological input—the topological relation $v/{\mathsf{\Lambda }}_{string}\approx 1.68$ obtained from the MTC data yields an estimate of $v$ that falls in the same range as the experimental value. This demonstrates that the topological framework is numerically consistent with the known thermal history of the electroweak theory, without requiring any adjustment of the MTC data to fit $v$.The non-trivial aspect of our framework is that the MTC data alone fix the dimensionless ratio $v/{\mathsf{\Lambda }}_{\mathrm{s}tring}$, so that once the electroweak scale is specified by the Standard Model thermal history, the vacuum expectation value is determined up to an $O(1)$ factor.

In this picture, the string-net condensate may undergo multiple phase transitions at different scales. The full metric structure (and hence continuous spacetime) is expected to emerge at a much higher scale, possibly near the Planck scale, whereas the scalar mode of the metric—the Higgs field—condenses later at the electroweak scale. The tension scale ${\mathsf{\Lambda }}_{\mathrm{s}tring}$ appearing in Eq. (13) should therefore be understood as the effective tension scale associated with this electroweak-scale condensation, not as the fundamental scale of spacetime emergence.

4.3.3. Self-Consistency Argument for Eq. (13)

We again emphasize that Eq. (13) is a phenomenological ansatz whose exact form awaits rigorous derivation. This ansatz is not untestable—once the complete OPE theory of bulk-boundary coupling is established, its exact form will be determined. Appendix C presents the theoretical basis and verification schemes for Eq. (13), including the supporting argument from the BCFT $g$-function, a semi-classical derivation from the effective field theory perspective, and a tensor-network numerical verification pathway.

4.4. Topological Expression for the Higgs Mass

The mass of the Higgs boson is given by the second derivative of the potential at the vacuum expectation value:

$$m_h^2={\left.{\displaystyle \frac{d^{2}V}{d{\varphi }^2}}\right|}_{\varphi =v}=2\lambda v^2$$

Here $\lambda$ comes from the fourth-order expansion coefficient of $G_{ST}$, which is determined by the topological data of the string-net condensate. Thus ${\mathrm{m}}_h$ is not a free parameter; it can be obtained from topological data in conjunction with the Landau coefficients, and its specific numerical value depends on the complete form of ${\mathrm{G}}_{\mathrm{S}\mathrm{T}}$.

As described above, we have extracted from the $S$-matrix of the $SU(3{)}_3\otimes SU(2{)}_2$ modular tensor category the bulk-boundary coupling coefficient $|C{|}^2=2\sqrt{2}$, and from this obtained the relation $v=1.6818\cdot {\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$ between the vacuum expectation value and the string-tension scale. These numerical results are not merely simple outputs of the topological data; they play a decisive role in the free-energy-difference criterion.

At finite temperature, the free-energy difference between the symmetric and broken phases can be expanded as:

$$\mathsf{\Delta }\mathcal{F}(T)=\mathcal{F}(0,T)-\mathcal{F}(v,T)=-{\displaystyle \frac{1}{2}}{\mu }^2(T)v^2+{\displaystyle \frac{1}{4}}\lambda v^4+\cdots$$

(17)

where the temperature dependence of ${\mu }^2(T)$is determined by the competition between the configurational entropy and the cosmological enthalpy of the string-net condensate. The quantity $|C{|}^2$ computed in this section is precisely the key topological factor that determines the magnitude of the sign change of ${\mu }^2(T)$ at the critical temperature. Without these MTC data, the magnitude of the free-energy difference cannot be determined, and hence the phase transition temperature cannot be predicted. Therefore, the topological calculation of this section provides irreducible, first-principles numerical input for the free-energy criterion of Section 3—this is the core feature that distinguishes our framework from the phenomenological Standard Model.

5. Testable Predictions

5.1. Cross-Scale Experimental Support and Conceptual Consistency

Chiral graviton in the fractional quantum Hall effect (Nature Physics, 2026) [14]

This experiment observed multiple chiral gravitons in FQH states at filling factors $\nu =2/7$ and $\nu =1/4$, demonstrating that quantum metric fluctuations are observable physical phenomena. Our framework interprets the Higgs field as the scalar mode of the metric tensor—fully corresponding to the role of the graviton as a metric-tensor mode in FQH.

Of course, the FQH system is (2+1)-dimensional, while our spacetime is (3+1)-dimensional, so there are topological differences. We invoke the FQH experiment here only as a conceptual illustration that "quantum metric fluctuations can be observed in physical systems," not as a direct dimensional correspondence or theoretical verification.The (2+1)-dimensional FQH system is topologically distinct from our (3+1)-dimensional spacetime; the analogy is illustrative rather than demonstrative.

Y-shaped gluon junctions in STAR heavy-ion collisions (Science, 2026) [15]

This experiment found that baryon number is carried by Y-shaped gluon junctions rather than valence quarks, demonstrating that topological structures can carry conserved charges. In our framework, lepton number is similarly carried by boundary topological structures.

Multi-component condensates in exciton BEC(Nature, 2026) [16]

This experiment found that exciton BECs contain switchable internal configurations, demonstrating that condensates possess microscopic degrees of freedom. In our framework, the condensation enthalpy similarly possesses microscopic degrees of freedom.

5.2. Testable Predictions

Prediction 1: Additional scalar excitation modes of the Higgs boson

As the scalar mode of the metric tensor, the Higgs boson may possess additional excited states originating from higher-order terms of the scalar metric mode, manifesting as:

- Anomalous resonance structures in certain rare decay channels,

- Small deviations from Standard Model Higgs couplings,

- Detectable deviations at high-energy colliders (HL-LHC, FCC-ee).

Prediction 2: Vanishing of masses at extremely high temperatures

When $T\gg T_c$, the configurational entropy dominates, and all fermions should revert to massless states.

Prediction 3: Universality of internal structures in quantum condensates

Different types of quantum condensates should possess analogous, controllable hidden structures. The 2026 exciton BEC experiment already supports this prediction.

6. Relationship Between Our Framework and the Standard Model, and Its Boundaries

6.1. Compatibility Rather Than Competition

As noted in Section 1.3, the $SU(3{)}_3$ and $SU(2{)}_2$ appearing in this paper denote WZW fusion categories, which are algebraic pairings to the gauge group structure of the Standard Model, rather than group-theoretic extensions. This distinction establishes the topological-order foundation of the framework, rather than a grand unification of gauge groups.

Our framework is not in competition with the Standard Model. The Standard Model is precisely effective within its domain of applicability, and its computational results are verified to experimental precision. What we attempt to answer are the "why" questions that the Standard Model leaves unanswered: (1) Why is there a Higgs field? (2) Why do the parameters take these values? (3) Why is symmetry broken?

These questions lie beyond the scope of the Standard Model, because the Standard Model takes the Higgs field as a fundamental input rather than a derived object. The value of our framework lies in filling this explanatory gap—it does not alter the computational results of the Standard Model, but rather reveals the underlying structure.

6.2. Topological Skeleton vs. Complete Mass Spectrum

It must be emphasized that the topological framework established in this paper primarily provides a zeroth-order skeleton of the mass hierarchy and coupling structure, rather than a complete dynamical explanation of the mass spectrum. The static modular tensor category data—quantum dimensions, $S$-matrix elements, $F$-symbols—essentially describe the ground-state symmetries and fusion rules of the topological order. They can determine which particle types can exist, the overall ordering of the coupling strengths of different fermions to the condensate, and basic properties such as chirality and algebraic structure. However, the specific numerical values of particle masses, particularly the fine mass ratios among the three generations, are likely not determined by these static topological data alone.

Physically, the decay of one generation of elementary particles into the next, the evolution of the early-universe high-temperature plasma state to the current low-temperature matter phase, and non-equilibrium effects during cosmic expansion all influence the finally observed mass spectrum. In other words, elementary particle masses themselves may be cumulative quantities with cosmic history, rather than instantaneous projections of the initial topological order. Neglecting this evolutionary history and attributing particle masses entirely to combinations of a few static matrix elements would underestimate the complexity of the problem.

It is worth reflecting that this problem is not unique to our framework. The great physicist Weinberg, well into his eighties, continued to work on the problem of lepton mass ratios, ultimately without success [32]. This suggests that, within the framework of the Standard Model alone—whether through symmetry principles or renormalization-group methods—deriving the precise mass ratios among the three generations of fermions may itself be a problem requiring new physics input beyond the Standard Model. The Standard Model can describe the mechanism of mass generation (the Higgs mechanism), but it cannot explain the specific numerical values of masses—this is an inherent limitation of the Standard Model as an effective theory.

At a deeper level, the modular tensor category (MTC) itself has structural limitations in describing bulk-boundary coupling. MTC describes the equilibrium topological properties of a closed system; it does not contain the following three classes of dynamical physical effects:

(1) Channel-degree variation. As the system temperature or external conditions change, the effective number of channels through which the system exchanges with its environment changes accordingly. At high temperatures, the channel degree is maximal and the system is fully coupled to its environment; at low temperatures, the system enters the condensed phase and the channel degree tends to close. The quantum dimension $d_i$of MTC is static, independent of temperature, and cannot describe this evolution.

(2) Decoherence. Bulk excitations interact with other degrees of freedom of the string-net condensate during propagation to the boundary, leading to decoherence. The decoherence rate depends on the channel degree—the larger the channel degree, the faster the decoherence. MTC is unitary; all fusion and splitting processes are reversible and contain no dissipation or decoherence mechanisms.

(3) Radiative flux. Bulk-boundary coupling is not an instantaneous process. Bulk excitations require finite time to propagate to the boundary, and the radiative flux propagates outward at finite speed, carrying away part of the energy and quantum information. MTC contains no metric or velocity concepts and cannot describe this transport process.

These three are not independent but are different facets of a unified dynamical process: decreasing temperature drives the channel degree to decrease, a decreasing channel degree suppresses decoherence, and the efficiency of bulk-boundary radiative flux is also determined by the channel degree. A complete theory of bulk-boundary coupling must describe all three aspects simultaneously—MTC can currently handle only the initial end of the chain (providing a set of static coupling strengths), but cannot describe the subsequent dynamical evolution.

Therefore, we adopt the following prudent stance: the modular tensor category and its boundary structure provide the topological skeleton and symmetry constraints for elementary particle mass generation, while the complete calculation of the mass spectrum requires the introduction of cosmic evolution equations or equivalent non-equilibrium dynamics. The topological data answer structural questions such as "why there are only three generations," "why neutrinos are much lighter than quarks," and "why the coupling strengths exhibit the ordering bulk excitation > boundary current > pure boundary current" as determined by the condensing algebra. The more refined question of "why the mass ratios of the three generations take the specific values they do," together with the complete dynamics of channel-degree variation, decoherence, and radiative flux in bulk-boundary coupling, requires supplementation by future theoretical frameworks.

This distinction does not imply the failure of our topological framework, but rather clarifies its domain of applicability. Indeed, the work of Coleman and Weinberg has shown that radiative corrections can themselves be a dynamical source of symmetry breaking [33], yet even they did not derive the precise mass ratios of the three fermion generations. Our previous attempts from the perspective of "cosmic evolutionary sedimentary phases" [34,35] have already suggested that the formation of mass hierarchies may be closely related to phase transitions and sedimentation processes at different stages of cosmic history. The present work can be viewed as a topological upgrade of this idea: the topological order provides the initial conditions and constraint space for the evolutionary process, while the evolution equations describe how the system gradually generates the specific mass spectrum under these constraints.

In future work, we plan to pursue three directions: (1) continue to seek a rigorous derivation of bulk-boundary coupling from boundary effective field theory or tensor-network models to test the correctness of the topological skeleton; (2) attempt to construct simplified evolution models from the early high-temperature state to the current low-temperature phase, to investigate the modification of the mass spectrum by cosmic evolution; (3) introduce open quantum system dynamics (such as the Lindblad equation) to supplement the static description of MTC, incorporating channel-degree variation, decoherence, and radiative flux into the framework.

6.3. Boundaries of Our Framework

Table 3. Boundaries of our framework.

Addressed	Not yet addressed
What is the Higgs field (geometric degree of freedom)	Precise mass ratios of the three fermion generations
Where the Higgs potential comes from (Landau expansion of cosmological enthalpy)	Excitation mode assignments for dark matter/dark energy
Why symmetry is broken (entropy–enthalpy competition)	Precise calculation of ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$
Why parameters take these values (determined by MTC data)	Rigorous derivation of Eq. (13); complete dynamics of bulk-boundary coupling (channel-degree variation, decoherence, radiative flux)—MTC can only handle static topological structures

Table 3. Boundaries of our framework.

Addressed	Not yet addressed
What is the Higgs field (geometric degree of freedom)	Precise mass ratios of the three fermion generations
Where the Higgs potential comes from (Landau expansion of cosmological enthalpy)	Excitation mode assignments for dark matter/dark energy
Why symmetry is broken (entropy–enthalpy competition)	Precise calculation of ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$
Why parameters take these values (determined by MTC data)	Rigorous derivation of Eq. (13); complete dynamics of bulk-boundary coupling (channel-degree variation, decoherence, radiative flux)—MTC can only handle static topological structures

6.4. Distinction from Existing Geometric Scalar Theories

In existing geometric scalar theories (such as $f(R)$ gravity and Brans–Dicke theory), the scalar mode typically originates from an extension of the Einstein–Hilbert action, with its coupling determined by ad hoc parameters. The scalar fields in f(R) and Brans–Dicke are extra introduced degrees of freedom (via the choice of the $f(R)$ function or the coupling parameter ω), whereas in our framework the Higgs field is the trace mode of the metric tensor itself, requiring no additional degrees of freedom. This is an essential mechanistic distinction.

The distinction of our framework lies in the fact that the coupling of the scalar mode is naturally determined by the topological data of the modular tensor category. The Yukawa coupling coefficients are proportional to the $F$-matrix elements, and the coupling strength of fermions to the scalar metric mode is determined by the fusion rules of the boundary Ising category. This is consistent with the Standard Model Higgs mechanism, and differs from other geometric scalar theories.

6.5. Unification of the Free-Energy-Difference Criterion and Framework Boundaries

The generalized free-energy-difference criterion employed in this paper is not limited to the analysis of the Higgs mechanism. In companion works on spacetime emergence [18,36], the same criterion is used to understand cosmic acceleration (driven by negative Komar energy) and local structure formation (driven by gravitational potential energy reduction). This reveals a deep unification: whether it is cosmic acceleration on cosmological scales, gravitational collapse on galactic scales, or Higgs condensation on the electroweak scale, the essence is the system seeking a minimum of the generalized free energy under different conditions. The differences between scales lie only in which terms dominate the free-energy expression.

However, this unification also has its boundaries. Our framework can currently explain only whether the Higgs field exists and why it condenses at this energy scale; it cannot explain why the mass ratios of the three fermion generations take the specific numerical values they do. The precise calculation of mass ratios requires the complete $F$-matrix elements of the bulk $SU(3{)}_3$—which lies beyond the current computational capabilities of modular tensor category tools (see Appendix B for discussion). Thus, what we reveal is the topological skeleton of the Higgs field, not the complete mass spectrum. This demarcation of boundaries is not a failure of the framework, but an honest assessment of the current state of physics, and it points the way toward future calculations.

7. Conclusion

This paper has proposed a topological framework for the origin of the Higgs field. The central conclusions are:

1. The Higgs field is a geometric degree of freedom. The Higgs field is not a fundamental scalar field, but rather the scalar excitation mode of the metric tensor of the string-net condensate. It shares the same geometric origin as the graviton and the gauge bosons.

2. The Higgs potential is a Landau expansion of the cosmological enthalpy. The Mexican-hat potential of the Standard Model is the manifestation, at the electroweak scale, of the competition between configurational entropy and cosmological enthalpy. Symmetry breaking is a phase transition with a well-defined thermodynamic driving force.

3. Under the phenomenological ansatz proposed in this paper, a quantitative connection between the Higgs parameters and the modular tensor category topological data has been preliminarily established. The core relation (13) plays a role analogous to the Bethe ansatz: it captures, in the simplest form, the connection between topological data and the electroweak scale, while its rigorous first-principles derivation awaits future work. Specifically, the bulk-boundary coupling coefficient and the ratio $v/{\mathsf{\Lambda }}_{\mathrm{s}tring}$ are determined by the $S$-matrix elements and quantum dimensions of the modular tensor category. The string tension scale ${\mathsf{\Lambda }}_{\mathrm{s}tring}$ is independently constrained by the electroweak phase transition temperature through the cosmic thermal history. The quartic coupling $\lambda$ of the Higgs potential, however, still remains to be derived from higher-order topological data such as the $F$-matrices.

4. Three experiments from 2026 provide cross-scale analogical support. The chiral graviton in the fractional quantum Hall effect, the Y-shaped gluon junction in STAR heavy-ion collisions, and the multi-component condensate in exciton BEC collectively point toward the ubiquity of topological structures and quantum metric fluctuations in physical systems.

5. The framework is testable. We have made several specific predictions, particularly regarding additional scalar excitation modes of the Higgs boson and the universality of internal structures of quantum condensates.

Our framework is conceptually consistent with several recent quantum condensed-matter experiments, such as the observation of chiral gravitons in the fractional quantum Hall effect, the discovery of topological gluon junctions in hadronic collisions, and the revelation of internal structures in exciton condensates. These experiments demonstrate that quantum metric fluctuations and topological defects have observable physical effects across multiple energy scales, providing indirect analogical support for the geometric picture of our framework.

What we have revealed is the topological skeleton of the Higgs field—it explains the origin of the Higgs field as a geometric degree of freedom, but this picture remains at an exploratory stage. If the Standard Model is indeed the low-energy effective theory of a quantum string-net condensate, then a reasonable theoretical aspiration is that it should not only effectively explain all known phenomena of the Standard Model, but also derive the elementary particle mass spectrum that the Standard Model cannot explain. However, a complete theory of bulk-boundary coupling in the strong-coupling regime has not yet been established; the complete mass-spectrum calculation requires combining the constraints of topological order with the non-equilibrium dynamics of cosmic evolution. This is an open problem of our framework and the direction of future work.