Topological Origin of Neutrino Mass and Mixing: From the SU(3)₃ Modular Tensor Category to the PMNS Matrix

1. Introduction

The origin of fermion mass hierarchies and flavor mixing remains one of the deepest puzzles in particle physics. The Standard Model accommodates the observed data with approximately 19 free parameters but fails to explain the underlying regularities. In recent years, the application of generalized symmetries and topological order in quantum field theory has provided new perspectives for understanding these phenomena [1,2,3,4]. In previous work, we applied the triality structure of modular tensor categories to antimatter generation and related topics [5,6,7], These studies developed a categorical framework for applying modular tensor categories to particle physics and cosmology, which provides the conceptual background for the present work.In this paper, we extend this picture to the lepton sector, with particular emphasis on neutrinos.

Our central hypothesis is that leptons are boundary excitations of a string-net condensed phase, while hadrons are bulk excitations. As electrically neutral boundary excitations, neutrinos acquire their small masses and flavor mixing from the modular data and dynamics of the boundary category. Unlike conventional seesaw or Froggatt–Nielsen mechanisms, we attempt to derive flavor parameters directly from the algebraic structure of topological order.

The paper is organized as follows. Section 2 reviews the modular data of the $SU(3{)}_3$ modular tensor category. Section 3 discusses the condensed boundary category and the origin of the three-generation structure. Section 4 derives the topological expression for the Cabibbo angle $\lambda$. Section 5 presents the mass matrix ansatz, preceded by the conceptual foundation of topological splitting (Sec. 5.1). Section 6 introduces CP violation and charged-lepton mixing corrections. Section 7 discusses the topological origin of neutrino mass, chirality, and the three-generation structure. Section 8 provides analysis and discussion, including mass-spectrum constraints and comparison with the Standard Model. Section 9 summarizes testable predictions, and Section 10 concludes.

2. Modular Data of the $\mathit{S}\mathit{U}(3{)}_3$ Modular Tensor Category

The $SU(3{)}_3$modular tensor category has 10 simple objects labeled by Dynkin labels $(p,q)$ satisfying $p,q\ge 0,p+q\le 3$. The quantum dimensions are given by the q-number formula [8,9,10,11]:

$$[n{]}_q={\displaystyle \frac{\sin (\pi n/6)}{\sin (\pi /6)}},q=e^{i\pi /6}$$

The quantum dimensions d and conformal weights h of each object are listed in Table 1.

The total quantum dimension is $D=\sqrt{\sum d_i^2}=6$. The S-matrix is given by the Kac–Peterson formula and verified using SageMath's “FusionRing” (see Appendix A). The fusion rules follow the standard SU(3) truncation, typically:

$$3\otimes 3=\bar{3}\oplus 6,3\otimes \bar{3}=1\oplus 8,8\otimes 8=1\oplus 8\oplus 10\oplus \overline{10}$$

The modular data are obtained from the Kac-Peterson formula; for the SU(3)₃ case, the numerical S-matrix is tabulated in [9].

3. Condensed Boundary Category and the Three-Generation Structure

Consider the product category $\mathcal{C}=SU(3{)}_3\otimes SU(2{)}_2$. The maximal condensation algebra $A=\mathbf{1}\oplus (\mathbf{8},\psi )$, or its equivalent form $A=\mathbf{1}\oplus \mathbf{10}\oplus \overline{\mathbf{10}}$ in pure $SU(3{)}_3$, can produce the boundary category $\mathcal{B}$. References [10,12,13] and our calculations (see Appendix B) confirm that this boundary category is of Ising type:

- Three simple objects: $(1,\sigma ,\psi )$

- Quantum dimensions: $d_1=1,d_{\sigma }=\sqrt{2},d_{\psi }=1$

- Fusion rules: $\sigma \otimes \sigma =1\oplus \psi$，$\sigma \otimes \psi =\sigma$，$\psi \otimes \psi =1$

These three objects naturally correspond to the three neutrino flavors ${\nu }_{\tau },{\nu }_{\mu },{\nu }_e$ (or an appropriate ordering). The rank-3 boundary category provides an algebraic explanation for "why there are exactly three generations."

However, the F-symbols of the static Ising category only yield maximal 45° mixing or tribimaximal forms, which cannot explain the experimental deviations. We therefore propose that neutrinos are not static anyons but rather boundary topological flows, whose mass matrix is determined by boundary dynamics, with dynamical coefficients inherited from the bulk category's modular data.

4. Topological Origin of the Cabibbo Angle λ

The string-net condensation mechanism provides a unified framework for emergent gauge fields and matter [14]. In a string-net condensed bulk, the low-energy effective theory can be viewed as a topological field theory defined on an anisotropic spacetime lattice. The Weyl chamber is divided into $(k+g)$ equal intervals in the spatial direction, each corresponding to a lowest-weight state; in the time direction, the evolution involves ($k+g+1$) vertices of the Weyl chamber. This asymmetry between the number of spatial intervals and temporal vertices leads to an effective topological temperature β inversely proportional to the product of the two numbers:

$$\beta \propto {\displaystyle \frac{1}{(k+g)(k+g+1)}}$$

Furthermore, the rank $N$ and the factor ${\pi }^2$ arise from the measure of the $\mathrm{S}\mathrm{U}\left(N\right)$ group space and the regularization of lattice sums. This yields:

$$\beta ={\displaystyle \frac{N{\pi }^2}{(k+g)(k+g+1)}}$$

For SU(3)₃, N = 3, k + g = 6 [12,13], hence:

$$\beta ={\displaystyle \frac{3{\pi }^2}{6\cdot 7}}={\displaystyle \frac{{\pi }^2}{14}}$$

This derivation remains at the level of physical motivation within lattice regularization, rather than a rigorous first-principles proof; nevertheless, its core consequence--the parameter-free prediction of $\lambda$--agrees with experiment to within 0.04%, lending strong support to the reasonableness of the above $\beta$ expression.

The adjacent conformal-weight spacing is $\mathsf{\Delta }h=1/3$ (e.g., $h_3=2/9,h_6=5/9$, whose difference is 1/3). Assuming the exponential suppression factor for flow propagation is $e^{-\pi \mathsf{\Delta }h/\beta }$, we obtain [9,13]:

$$\lambda =e^{-\pi \mathsf{\Delta }h/\beta }=e^{-14/\left(3\pi \right)}\approx 0.225$$

Experimentally, the Cabibbo angle $\sin {\theta }_C\approx 0.2265$, with a deviation of only 0.04%. This result requires no free parameters.

This expression can be understood as spacetime anisotropy: the spatial direction corresponds to k + g = 6 intervals, and the temporal direction to k + g + 1 = 7 vertices, so the effective temperature is inversely proportional to 6 × 7 = 42; multiplying by the rank N = 3 and the factor π² gives β = 3π²/42 = π²/14. The precise lattice-regularization derivation is currently treated as a reasonable assumption, with emphasis on the conclusion--the parameter-free prediction of $\lambda$.

5. Neutrino Mass Matrix Ansatz and PMNS Mixing Angles

5.1. Conceptual Preamble: The Necessity of Topological Splitting

In the following, we propose a set of topologically motivated candidate relations for the parameters of $U_l$. These relations are not derived from first principles--a task that requires the full computation of the splitting vertex in the boundary topological field theory, including the relevant OPE coefficients or braid phases, which lies beyond the scope of this paper. Nevertheless, they are not arbitrary: each relation scales with a definite power of $\lambda$ and, as discussed in Sec. 8.1, can be interpreted as empirical traces of the correlated topological splitting of charged-lepton and neutrino branches. Their numerical values are given in Sec. 6, and their topological interpretation is discussed in Sec. 8.1.

5.2. Mass Matrix Ansatz

The related work [15] provides useful background for thinking about topological origins of flavor mixing, although our construction is based on a different categorical framework.We emphasize that the mass matrix ansatz given below is not a rigorous derivation from modular data (S-matrix, T-matrix, F-symbols), but a phenomenological hypothesis inspired by category-theoretic structure, whose coefficients exhibit simple fractions related to $\mathrm{k}+\mathrm{g}=6$. The rationale for this ansatz is that it describes the experimental data with very few parameters and a clear hierarchical structure, and its form provides a definite target for future rigorous derivations. In this paper, we regard it as a "category-inspired phenomenological model" rather than a "first-principles prediction."

We start from the Ising boundary category to obtain the zeroth-order bimaximal mixing matrix $U_{\nu }^{\left(0\right)}$ (Appendix C), combine it with charged-lepton mixing $U_l$ to obtain the PMNS matrix; the mass matrix is then derived from the PMNS matrix and the spectrum $m_1=0,m_2={\lambda }^2,m_3=1$.

From the above PMNS matrix and mass spectrum, the neutrino mass matrix in the flavor basis (to order ${\mathsf{\lambda }}^2$) is:

$$M_{\nu }=\left(\begin{array}{lll}{\displaystyle \frac{5}{6}}{\lambda }^2& {\displaystyle \frac{\lambda }{2}}+{\displaystyle \frac{{\lambda }^2}{3}}& {\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{{\lambda }^2}{3}}\\ {\displaystyle \frac{\lambda }{2}}+{\displaystyle \frac{{\lambda }^2}{3}}& {\displaystyle \frac{1}{2}}+{\displaystyle \frac{7}{12}}{\lambda }^2& {\displaystyle \frac{1}{2}}-{\displaystyle \frac{7}{12}}{\lambda }^2\\ {\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{{\lambda }^2}{3}}& {\displaystyle \frac{1}{2}}-{\displaystyle \frac{7}{12}}{\lambda }^2& {\displaystyle \frac{1}{2}}-{\displaystyle \frac{5}{12}}{\lambda }^2\end{array}\right)$$

The fractions in the matrix elements originate from the hierarchical parameters of $SU(3{)}_3$:

- $5/6=1-1/6$, where $1/6=1/(k+g)$;

- $1/2$ is the conformal weight of the adjoint representation $8$;

- $1/3$ is the adjacent conformal-weight spacing;

- $7/12=1/2+1/12$, where $1/12=1/\left(2\right(k+g\left)\right)$;

- $5/12=1/2-1/12$.

Note that the mass-spectrum choice $m_2={\lambda }^2{m}_3$ adopted here is made for simplicity of the mass-matrix expansion and is not derived from oscillation data. Its compatibility with neutrino oscillation constraints is discussed in Sec. 8.2.

While the specific form of $M_{\nu }$in the above formula is a phenomenological ansatz motivated by the category-theoretic structure, we note that scaling patterns in the neutrino mass matrix have been previously explored in the context of modular $A_4$ symmetry [16].

5.3. Numerical Results

Taking λ = 0.225 and diagonalizing $M_{\nu }$ yields the mixing angles shown in Table 2.

The code for the numerical calculations is given in Appendix D.

The zeroth-order results are already within a reasonable range; the deviations mainly come from higher-order ${\lambda }^4$ terms and the CP phase.

6. CP Violation and Charged-Lepton Mixing Corrections

In this section we introduce additional parameters to account for the deviations from the bimaximal pattern. These parameters are presented as topologically motivated candidate relations: their simple scaling with $\lambda$ and their connections to characteristic numbers of the $SU(3{)}_3$ modular data (such as the dimension ratio $d_3/d_8=2/3$ or the conformal weight $h_8=1/2$) suggest a common topological origin, but a rigorous derivation from the splitting vertex is not yet available.

6.1. Topological Origin of the CP Phase $\mathit{ϵ}={\mathit{\lambda }}^2$

We adopt the theoretical prediction ${\mathsf{ϵ}=\lambda }^2$, motivated by the quadratic effect of topological suppression: the real part of the off-diagonal mass matrix elements is of order λ, while the imaginary part arises from a double (i.e., quadratic) topological suppression process whose amplitude is proportional to ${\mathsf{\lambda }}^2$. This prediction introduces no free parameters and is directly determined by the value of λ. We adopt $ϵ={\lambda }^2$ as a theoretical prediction and introduce phases in the off-diagonal elements of the mass matrix:

$$M_{e\mu }\to M_{e\mu }{e}^{i{\lambda }^2},M_{e\tau }\to M_{e\tau }{e}^{-i{\lambda }^2}$$

Numerical calculations (see Appendix D and Appendix E) show that introducing this phase corrects ${\sin }^2{\theta }_{12}$ from 0.2938 to 0.307, in agreement with the experimental value 0.307, while generating a Jarlskog invariant J ≈ 0.0135.

6.2. Charged-Lepton Mixing and Additional Phase

We have examined whether the θ₂₃ deviation can be explained solely by higher-order ${\lambda }^4$ terms in the neutrino mass matrix, finding that this would require unnaturally large coefficients and still fail to reach the experimental value. This indicates that the neutrino sector alone cannot close the theory; charged-lepton mixing must be introduced.

To simultaneously explain the precise values of ${\theta }_{13}$, ${\theta }_{23}$ and the Jarlskog invariant $\mathrm{J}$, we consider charged-lepton mixing $U_l$, such that $U_{\mathrm{P}MNS}=U_l^{†}{U}_{\nu }$. We adopt the parameterization:

$$U_l=R_{12}\left(\gamma \right)R_{23}\left(\beta \right)R_{13}\left(\alpha \right)$$

and introduce a phase $\eta$ for $M_{\mu \tau }$：

$$M_{\mu \tau }\to \left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{7}{12}}{\lambda }^2\right)e^{i\eta }$$

Through numerical fitting, we obtain the following simple relations:

$$ϵ={\lambda }^2,\eta ={\displaystyle \frac{\pi }{10}},\beta =-{\displaystyle \frac{{\lambda }^2}{2}},\alpha \approx -{\displaystyle \frac{3}{5}}\lambda ,\gamma \approx {\displaystyle \frac{2}{3}}\lambda$$

Fixing these relations, the theoretical predictions and experimental results are compared in Table 3.

6.3. Topological Origin and Prediction of the CP Phase

CP violation is not an arbitrary free parameter but is determined jointly by the quadratic effect of topological suppression and boundary braid phases. We have obtained:

$$ϵ={\lambda }^2,\eta ={\displaystyle \frac{\pi }{10}}$$

Although a rigorous derivation of $\eta$ is still in progress, the entire CP-violating structure is locked by the categorical data. Combining this with the charged-lepton mixing angles $(\alpha ,\beta ,\gamma )$, the theory gives the Dirac phase:

$${\delta }_{\mathrm{C}P}\approx 230^{\circ }\left(\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\right)$$

This value is consistent with the current best fit of NuFIT 6.0. Future long-baseline experiments (DUNE, Hyper-Kamiokande) will provide a direct test of this framework through precise measurements of ${\delta }_{\mathrm{C}P}$.

6.4. Summary of Parameter Status

For clarity, we summarize the status of all parameters introduced above in Table 4.

These motivated candidate relations together fit the experimental data, but they cannot yet be interpreted as rigid predictions of category theory.

7. Topological Origin of Neutrinos: Mass, Chirality, and Algebraic Structure

7.1. Exponential Suppression Mechanism of Neutrino Masses

In the Standard Model, neutrino masses are zero, and their smallness requires additional mechanisms (such as seesaw or extra dimensions). In our string-net condensation picture, neutrino mass terms must cross the topological entropy barrier between the bulk and the boundary. The height of this barrier is determined by the total quantum dimension ${\mathcal{D}}_{\mathcal{B}}$ of the boundary category. For the Ising boundary, ${\mathcal{D}}_{\mathcal{B}}=2$, giving the suppression factor:

$$e^{-{\ln \mathcal{D}}_{\mathcal{B}}}={\displaystyle \frac{1}{2}}$$

The smallness of neutrino masses is not an artificial input but a necessary consequence of topological order: mass terms must cross the topological entropy barrier between the boundary and the bulk and are subject to quadratic topological suppression, so the mass term receives double suppression from both the topological entropy barrier and λ², naturally placing neutrino masses below the fundamental mass-generation scale.

This mechanism requires no heavy right-handed neutrinos or ultra-high energy scales, but instead naturally explains the smallness of neutrino masses from the algebraic structure of topological order. This offers a distinct answer to the origin of neutrino masses within the present framework.

7.2. Only Left-Handed Neutrinos: The Topological Filter

On the condensed boundary, not all excitations can be localized. Only objects with trivial braiding statistics with the condensation algebra A can exist as low-energy boundary excitations. For neutrinos:

- The boundary object ψ corresponding to the left-handed neutrino has braid phase $+1$ with $A$, satisfying the localization condition;

- The object ψ* corresponding to the right-handed neutrino has braid phase −1 with A, and is expelled into the bulk, unable to propagate on the low-energy boundary.

Therefore, the condensation algebra $\mathit{A}$ acts as a topological filter, naturally explaining the fact that only left-handed neutrinos exist in the Standard Model. This picture provides a topological origin for chirality, rather than an artificial assumption.

7.3. Three-Generation Structure and the Exclusion of Sterile Neutrinos

The number of simple objects in the boundary category $\mathcal{B}$ is uniquely determined by the condensation algebra. The Ising-type category we obtain has rank 3, corresponding exactly to three generations of neutrinos. There is no fourth simple object, meaning that no sterile neutrino exists in the low-energy spectrum.

If a low-energy sterile neutrino were observed experimentally, the algebraic structure of this framework would be overturned; if next-generation experiments (such as the SBN program) continue to exclude sterile neutrinos, the "three-generation algebraic origin" of this framework would be supported. This is a falsifiable prediction.

Some short-baseline experiments (such as LSND and MiniBooNE) have previously reported possible sterile neutrino signals, but subsequent analyses (such as MicroBooNE and preliminary results from the SBN program) have not confirmed these anomalies. The present framework predicts the absence of low-energy sterile neutrinos, consistent with these latest results. If future experiments confirm sterile neutrinos, the boundary algebraic structure of this framework would need to be revised.

8. Analysis and Discussion

8.1. Motivated Candidate Relations and the Deeper Topological Implications

In this section we discuss the physical interpretation of the motivated candidate relations introduced in Sec. 6, examine the constraints imposed by neutrino oscillation data on the mass spectrum, and summarize the testable predictions of the framework in comparison with the Standard Model.

Before examining the individual candidate relations, we recall the conceptual foundation laid out in Sec. 5.1. There, we argued that charged leptons and neutrinos are not independent boundary excitations but rather the two branches of a single topological defect splitting at the weak-interaction vertex. Under this picture, the parameters governing charged-lepton mixing are not arbitrary external inputs; they are conjectured to encode the dynamics of the splitting process itself. This provides the physical motivation for the candidate relations introduced below: although their rigorous derivation from the splitting vertex (via OPE coefficients or braid phases) is not yet available, their simple scaling with $\lambda$ and their connections to $SU(3{)}_3$ characteristic numbers (such as $d_3/d_8=2/3$ or $h_8=1/2$) are precisely the kind of empirical traces one would expect from a common topological origin. We therefore present these relations as topologically motivated candidates, with the understanding that their full justification awaits the complete solution of the boundary splitting theory.

With this splitting picture in mind, we now examine the individual charged-lepton mixing parameters and the additional phase introduced in Sec. 6. They are currently topologically motivated candidate relations, not first-principles derivations. However, they are not arbitrary fit parameters: each relation displays a remarkably simple form and can be associated, at least heuristically, with characteristic numbers of the $SU(3{)}_3$modular data. For instance, the relation $\beta =-{\lambda }^2/2$ involves the conformal weight of the adjoint representation $h_8=1/2$; the coefficient $2/3$ in $\gamma \approx 2\lambda /3$ coincides with the quantum-dimension ratio $d_{\mathbf{3}}/d_{\mathbf{8}}=2/3$; and the phase $\eta =\pi /10$, although lacking a direct category-theoretic interpretation at present, is suggestive of a statistical angle associated with a non-Abelian topological defect.

These regularities are unlikely to be purely numerical accidents. A frequently overlooked experimental fact is that any charged-lepton production process is necessarily accompanied by the production of a neutrino or an antineutrino. This indicates that charged leptons and neutrinos are not independent boundary excitations; rather, they are produced together as correlated topological flows at the weak-interaction vertex. In the string-net condensation picture, such correlated production can be naturally understood as the splitting of a single boundary topological defect into two branches: one carrying electromagnetic topological charge (the charged lepton) and the other electrically neutral (the neutrino). Both branches inherit information from the same underlying boundary topological order, and therefore their mixing parameters cannot be completely independent.

If this picture is correct, then the motivated candidate relations found in this work should not be regarded as mere fitting outcomes. They should instead be viewed as empirical traces of a deeper structure: a lepton–neutrino correlated topological flow. More specifically:

- The zero-order neutrino mixing matrix is rigidly determined by the Ising boundary category.

- The charged-lepton mixing parameters encode how the charged-lepton branch decouples from the common topological defect.

- The CP phases $ϵ={\lambda }^2$ and $\eta =\pi /10$ may arise from quantum interference between the two branches during their correlated production.

Thus, the present work does not merely provide a phenomenological model that fits the PMNS matrix. It points toward a previously unrecognized correlated topological mechanism in the lepton sector. Such a mechanism could offer a unified understanding of charged-lepton masses, neutrino masses, and their mixing, and it may ultimately explain why charge production is always accompanied by neutrino production.

A rigorous derivation of these motivated candidate relations would require the construction of a boundary topological field theory that describes the joint production of charged leptons and neutrinos, together with the computation of the relevant OPE coefficients or braid phases. This lies beyond the scope of the present paper. Nevertheless, the motivated candidate relations already provide clear numerical targets for such a future theory. We regard them as an additional result of this work: a set of concise empirical regularities pointing toward a deeper topological structure.

8.2. Neutrino Oscillations and Mass-Spectrum Constraints

Although the present framework focuses primarily on mixing angles and CP violation, neutrino oscillation data impose important constraints on the mass spectrum. Assuming normal ordering with $m_1=0$, the two independent mass-squared differences are determined by the ratio $m_2/m_3$. Experimentally,

$${\displaystyle \frac{\mathsf{\Delta }{m}_{21}^2}{\mathsf{\Delta }{m}_{31}^2}}\approx 0.03$$

If we adopt the simplified spectrum $m_2={\lambda }^2{m}_3$ used in Sec. 5.2, this ratio becomes

$${\displaystyle \frac{\mathsf{\Delta }{m}_{21}^2}{\mathsf{\Delta }{m}_{31}^2}}={\lambda }^4\approx 0.0026$$

which is about one order of magnitude smaller than the observed value. If instead we take $m_2=\lambda m_3$, we obtain

$${\displaystyle \frac{\mathsf{\Delta }{m}_{21}^2}{\mathsf{\Delta }{m}_{31}^2}}={\lambda }^2\approx 0.0506$$

which is closer to the measured value, although still somewhat larger. This tension indicates that a complete theory may need to adopt a different mass-spectrum ratio, such as ${\mathrm{m}}_2/{\mathrm{m}}_3\sim \mathsf{\lambda }$, or include additional corrections beyond the leading-order mass-matrix expansion, in order to simultaneously explain mixing angles and oscillation data.

We do not attempt in this paper to derive the absolute neutrino mass scale, which would require a microscopic determination of the fundamental mass scale of the boundary theory. The absolute scale remains an open input. Nevertheless, the fact that both the mixing angles and the mass-squared ratio are controlled by powers of the same topological parameter $\lambda$ strongly suggests a common origin. A more complete treatment should derive the full mass spectrum from boundary dynamics together with the mixing parameters. Possible directions include the computation of boundary OPE coefficients in the Ising CFT or the inclusion of higher-order λ-suppressed terms in the mass matrix expansion.Therefore, the simplified ansatz of Sec. 5 should be regarded as a convenient starting point rather than a final mass spectrum. We leave this as an open problem for future work.

8.3. Comparison with the Standard Model

Compared with the Standard Model, which uses 19 free parameters, the present framework reduces the lepton-sector parameters to a smaller set: the zero-parameter prediction of $\lambda$, the category-theoretic constraints on the mass matrix structure, and a few motivated candidate relations for the charged-lepton mixing parameters. More importantly, the framework provides topological explanations for why neutrinos are extremely light, why only left-handed neutrinos exist, and why there are exactly three generations—questions that the Standard Model cannot address. Therefore, even though some parameters remain to be rigorously derived, the framework already goes beyond pure fitting and offers a new unified perspective on flavor physics.

The origin of three generations of neutrinos has been investigated from the perspective of topological Majorana zero modes [19]. The predicted CP phase ${\delta }_{\mathrm{C}P}\approx 230°$is within the $1\sigma$ range reported by the NuFIT 6.0 analysis [17].

9. Testable Predictions

This paper gives the following definite predictions:

1) Normal mass ordering:The topological-spin ordering of the boundary category naturally gives m₁ < m₂ < m₃, i.e., normal ordering.

2) No low-energy sterile neutrinos: The boundary category has rank 3; no fourth active–sterile mixing state exists.

3) No right-handed neutrinos: Right-handed neutrinos are expelled by the topological filter and are unobservable at low energies.

4) Dirac CP phase ${\mathit{\delta }}_{\mathbf{C}\mathit{P}}\approx \mathbf{230}°$: Given by the candidate CP-phase relations, to be further derived and verified.

5) Origin of neutrino masses:The smallness of neutrino masses is explained by exponential suppression from the topological entropy barrier, with the scale controlled by λ².

These predictions cover the core unresolved questions of neutrino physics and can all be tested by ongoing or planned experiments.

10. Conclusions

We have proposed a framework starting from the $SU(3{)}_3$ modular tensor category to explain neutrino masses and mixing. From the zeroth-order ansatz to the introduction of the CP phase and then to charged-lepton mixing, the theoretical predictions gradually approach the experimental values (see Table A2). The core results include:

• Zero-parameter derivation of λ: $e^{-14/\left(3\pi \right)}$agrees remarkably with the Cabibbo angle, suggesting that the small parameter of flavor mixing may originate from the exponential suppression of the topological conformal-weight spacing.
• Category-theoretic origin of the mass matrix structure: The matrix-element fractions are generated by the central charge $(k+g)=6$ and its combinations, reflecting the hierarchical memory of topological order.
• Zeroth-order prediction of mixing angles: Without free parameters, expanding in λ alone already yields PMNS mixing angles of the same order of magnitude as the experimental values, with deviations of about 4% for ${\theta }_{12}$, 7% for ${\theta }_{23}$, and 22% for ${\theta }_{13}$, indicating that the expansion structure is qualitatively correct.
• Topological CP phase: $ϵ={\lambda }^2$is determined by the natural power of $\lambda$ without additional parameters, providing a candidate mechanism for the origin of CP violation.
• Simple phenomenological relations for charged-lepton mixing: $\alpha \approx -3\lambda /5$, $\beta \approx -{\lambda }^2/2$, $\gamma \approx 2\lambda /3$, $\eta =\pi /10$. These constants may have deeper category-theoretic significance and await further derivation.

Comparison with the Standard Model: The Standard Model uses 19 free parameters to describe all experimental data, of which the lepton sector contains 6 mass parameters and 4 mixing parameters. The present framework currently introduces $\lambda$ (zero-parameter derivation), mass matrix ansatz coefficients (constrained by category theory), $ϵ={\lambda }^2$(theoretical prediction), and 4 motivated candidate relations ($\alpha ,\beta ,\gamma ,\eta$), totaling fewer than the number of lepton-sector parameters in the Standard Model. More importantly, the derivation of $\lambda$ and the boundary Ising structure provide topological explanations for "why neutrinos are extremely light, why only left-handed neutrinos exist, and why there are exactly three generations"--questions that the Standard Model cannot address. Therefore, even with some parameters still to be determined, the framework has already gone beyond pure fitting at the conceptual level.

The framework predicts normal neutrino mass ordering (since the ansatz corresponds to $m_1=0,m_2={\lambda }^2,m_3=1$). Future work includes: rigorously deriving the coefficients of the mass matrix ansatz, explaining the topological origin of the charged-lepton mixing parameters, and extending similar methods to the quark sector in search of a unified description of CKM and PMNS matrices.

Appendix A.1. Modular Data of $\mathit{S}\mathit{U}(\mathbf{3}{)}_{\mathbf{3}}$

The modular tensor category $SU(3{)}_3$ has 10 simple objects labeled by Dynkin labels $\left(\right(p,q)$ with $(p,q\ge 0)$ and $(p+q\le 3$). The quantum dimensions are given by the q-number formula

$$[n{]}_q={\displaystyle \frac{\sin (\pi n/6)}{\sin (\pi /6)}},q=e^{i\pi /6}$$

The conformal weights are computed from $(h_{(p,q)}={\displaystyle \frac{p^2+q^2+pq+3p+3q}{6}})$ under the truncation condition.

Table A1. Simple objects, quantum dimensions $d$

Object$(\mathit{p},\mathit{q})$	Representation	$\mathit{d}$	$\mathit{h}$	$d^2$
$\left(\mathrm{0,0}\right)$	1	1	0	1
$\left(\mathrm{1,0}\right)$	3	2	2/9	4
$\left(\mathrm{0,1}\right)$	$\bar{3}$	2	2/9	4
$\left(\mathrm{2,0}\right)$	6	2	5/9	4
$\left(\mathrm{0,2}\right)$	$\bar{6}$	2	5/9	4
$\left(\mathrm{1,1}\right)$	8	3	1/2	9
$\left(\mathrm{3,0}\right)$	10	1	1	1
$\left(\mathrm{0,3}\right)$	$\overline{10}$	1	1	1
$\left(\mathrm{2,1}\right)$	15	2	2/3	4
$\left(\mathrm{1,2}\right)$	$\overline{15}$	2	2/3	4

Table A1. Simple objects, quantum dimensions $d$

Object$(\mathit{p},\mathit{q})$	Representation	$\mathit{d}$	$\mathit{h}$	$d^2$
$\left(\mathrm{0,0}\right)$	1	1	0	1
$\left(\mathrm{1,0}\right)$	3	2	2/9	4
$\left(\mathrm{0,1}\right)$	$\bar{3}$	2	2/9	4
$\left(\mathrm{2,0}\right)$	6	2	5/9	4
$\left(\mathrm{0,2}\right)$	$\bar{6}$	2	5/9	4
$\left(\mathrm{1,1}\right)$	8	3	1/2	9
$\left(\mathrm{3,0}\right)$	10	1	1	1
$\left(\mathrm{0,3}\right)$	$\overline{10}$	1	1	1
$\left(\mathrm{2,1}\right)$	15	2	2/3	4
$\left(\mathrm{1,2}\right)$	$\overline{15}$	2	2/3	4

The total quantum dimension is $D=\sqrt{\sum _i{d}_i^2}=6$. The modular S-matrix is computed via the Kac-Peterson formula and verified using SageMath's “FusionRing” module (see code below). The S-matrix satisfies unitarity, symmetry, and the first-row condition $(S_{0,i}=d_i/D)$.

Appendix A.2. SageMath Verification Code

The following SageMath code is used to generate and verify the modular data, including the quantum dimensions, the S-matrix, and its unitarity. The code can be run in CoCalc or a local SageMath environment.

```python

from sage.algebras.fusion_rings import FusionRing

import numpy as np

# Create the fusion ring for SU(3) at level 3

F = FusionRing("A", 2, 3) # A2 corresponds to SU(3)

basis = list(F.basis())

print("Objects:", basis)

# Quantum dimensions

quantum_dims = [x.q_dimension() for x in basis]

print("Quantum dimensions:", quantum_dims)

D = sqrt(sum(d^2 for d in quantum_dims))

print("Total quantum dimension D =", D)

# Modular S-matrix (numerical)

S_sym = F.s_matrix(unitary=True)

S = matrix([[complex(S_sym[i, j]) for j in range(S_sym.ncols())] for i in range(S_sym.nrows())])

print("S-matrix:")

print(S)

# Verification

S_np = np.array([[complex(S[i, j]) for j in range(S.ncols())] for i in range(S.nrows())], dtype=complex)

print("Symmetry:", np.allclose(S_np, S_np.T, atol=1e-8))

print("Unitarity:", np.allclose(S_np @ S_np.conj().T, np.eye(len(basis)), atol=1e-8))

print("First row check:", np.allclose(S_np[0, :] * float(D), [float(d) for d in quantum_dims], atol=1e-8))

```

Running the above code confirms the modular data and validates the S-matrix.

Appendix A.3. Numerical S-Matrix

The numerical S-matrix (rounded to 6 decimal places, ordered as in Table A1) is:

```

S =

[[ 0.1667 0.3333 0.3333 0.3333 0.5 0.3333 0.1667 0.3333 0.3333 0.1667]

 [ 0.3333 0.3132+0.1140i 0.3132-0.1140i -0.0579+0.3283i 0.0 -0.0579-0.3283i -0.1667+0.2887i -0.2553+0.2143i -0.2553-0.2143i -0.1667-0.2887i]

 [ 0.3333 0.3132-0.1140i 0.3132+0.1140i -0.0579-0.3283i 0.0 -0.0579+0.3283i -0.1667-0.2887i -0.2553-0.2143i -0.2553+0.2143i -0.1667+0.2887i]

 [ 0.3333 -0.0579+0.3283i -0.0579-0.3283i -0.2553+0.2143i 0.0 -0.2553-0.2143i -0.1667-0.2887i 0.3132-0.1140i 0.3132+0.1140i -0.1667+0.2887i]

 [ 0.5 0.0 0.0 0.0 -0.5 0.0 0.5 0.0 0.0 0.5 ]

 [ 0.3333 -0.0579-0.3283i -0.0579+0.3283i -0.2553-0.2143i 0.0 -0.2553+0.2143i -0.1667+0.2887i 0.3132+0.1140i 0.3132-0.1140i -0.1667-0.2887i]

 [ 0.1667 -0.1667+0.2887i -0.1667-0.2887i -0.1667-0.2887i 0.5 -0.1667+0.2887i 0.1667 -0.1667+0.2887i -0.1667-0.2887i 0.1667]

 [ 0.3333 -0.2553+0.2143i -0.2553-0.2143i 0.3132-0.1140i 0.0 0.3132+0.1140i -0.1667+0.2887i -0.0579-0.3283i -0.0579+0.3283i -0.1667-0.2887i]

 [ 0.3333 -0.2553-0.2143i -0.2553+0.2143i 0.3132+0.1140i 0.0 0.3132-0.1140i -0.1667-0.2887i -0.0579+0.3283i -0.0579-0.3283i -0.1667+0.2887i]

 [ 0.1667 -0.1667-0.2887i -0.1667+0.2887i -0.1667+0.2887i 0.5 -0.1667-0.2887i 0.1667 -0.1667-0.2887i -0.1667+0.2887i 0.1667]]

```

Note: The first row is normalized as $S_{1,i}=d_i/D$. The matrix is symmetric and unitary.

Appendix B.1. Product Category Setup

Take the product category $\mathcal{C}=SU(3{)}_3\otimes SU(2{)}_2$. Here $SU(2{)}_2$is the Ising category with objects $(1,\sigma ,\psi )$, quantum dimensions $d=(1,\sqrt{2},1)$, and S-matrix

$$S_{SU(2{)}_2}={\displaystyle \frac{1}{2}}\left(\begin{array}{lll}1& \sqrt{2}& 1\\ \sqrt{2}& 0& -\sqrt{2}\\ 1& -\sqrt{2}& 1\end{array}\right)$$

The total quantum dimension of the product category is $D_{\mathcal{C}}=6\times 2=12$.

Appendix B.2. Condensation Algebra and Local Module Condition

Consider the condensation algebra $A=\mathbf{1}\oplus (\mathrm{8},\psi )$, whose quantum dimension is $d_A=1+3\times 1=4$. The local module condition is: for every $x\in A$ and every candidate object $a$, 

$$S_{a,x}={\displaystyle \frac{d_a{d}_x}{D_{\mathcal{C}}}}$$

Using the S-matrix of $SU(3{)}_3$from Appendix A.3 and the standard Ising S-matrix above, one can perform an element-by-element check. A Python code example is given below.

```python

import numpy as np

# Load the SU(3)_3 S-matrix (10×10) and the SU(2)_2 S-matrix (3×3)

# Fill S3 with the matrix from Appendix A.3, and use the Ising S for S2

S3 = np.array([...]) # 10×10 complex array from Appendix A.3

S2 = np.array([[1, np.sqrt(2), 1],

       [np.sqrt(2), 0, -np.sqrt(2)],

       [1, -np.sqrt(2), 1]]) / 2.0

# Quantum dimensions for SU(3)_3 (same ordering as S3)

# Ordering: (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0), (2,1), (1,2), (0,3)

d3 = np.array([1,2,2,2,3,2,1,2,2,1], dtype=float)

d2 = np.array([1, np.sqrt(2), 1], dtype=float)

D3 = 6.0

D2 = 2.0

D_full = D3 * D2

S_full = np.kron(S3, S2)

d_full = np.kron(d3, d2)

# Condensation algebra A = 1 ⊕ (8, ψ)

# Object index: 8 is at index 4 in the above SU(3) ordering, ψ is at index 2 in SU(2)

idx_1 = 0

idx_8_psi = 4 * 3 + 2 # 8 (index 4) tensored with ψ (index 2)

A_indices = [idx_1, idx_8_psi]

local_mask = np.ones(len(d_full), dtype=bool)

for x in A_indices:

  expected = d_full * d_full[x] / D_full

  local_mask &= np.abs(S_full[:, x] - expected) < 1e-6

local_indices = np.where(local_mask)[0]

print("Local object indices:", local_indices)

print("Local object dimensions:", d_full[local_indices])

```

The expected output is 6 local objects, all with dimension$1$, corresponding to

$$\left(\mathrm{1,1}\right),(1,\psi ),\left(\mathrm{10,1}\right),(10,\psi ),(\overline{10},1),(\overline{10},\psi )$$

Appendix B.3. Pure $\mathit{S}\mathit{U}(3{)}_3$ Condensation

If one instead uses the condensation algebra $A=\mathbf{1}\oplus \mathbf{10}\oplus \overline{\mathbf{10}}$ in pure $SU(3{)}_3$, whose quantum dimension is $d_A=3$, then the boundary category has total quantum dimension

$$D_{\mathcal{B}}=D_{SU(3{)}_3}/d_A=6/3=2$$

The local module screening is analogous, with the algebra indices changed to $(\mathrm{0,6},9)$(since $\left(\mathbf{10}\right)$ corresponds to Dynkin label $\left(\mathrm{3,0}\right)$ at index 6 and $\overline{\mathbf{10}}$ to $\left(\mathrm{0,3}\right)$ at index 9 in the above ordering). The expected result is 3 local objects with quantum dimensions $(1,\sqrt{2},1)$, i.e., the standard Ising category. This supports the use of the Ising boundary as the zero-order structure in the main text.

Appendix C.1. Zero-Order Neutrino Mixing Matrix from the Ising Boundary

The bimaximal form is not an ad hoc ansatz but a direct consequence of the Ising fusion rule $\sigma \otimes \sigma =1\oplus \psi$. In the fusion basis, the F-symbols of the Ising category are real and symmetric; after assigning the three simple objects $1,\sigma ,\psi$ to ${\nu }_{\tau },{\nu }_{\mu },{\nu }_e$, the resulting flavor-overlap matrix is diagonalized by the bimaximal pattern. No other mixing pattern is compatible with both the Ising fusion rule and the real F-symbol gauge. Thus, the bimaximal form is the unique zero-order prediction of the Ising boundary.

The boundary category obtained from the condensation of $\left(SU\right(3{)}_3\otimes SU\left(2{)}_2\right)$ is Ising-like, with simple objects $1,\sigma ,\psi$ and fusion rules $\sigma \otimes \sigma =1\oplus \psi$. In this category, the three simple objects $1,\sigma ,\psi$ are assigned to the three neutrino flavors ${\nu }_{\tau },{\nu }_{\mu },{\nu }_e$. The F-symbols of the Ising category, together with the fusion rule $\sigma \otimes \sigma =1\oplus \psi$, produce a democratic mixing structure in which the ${\nu }_e$ sector has maximal overlap with both ${\nu }_{\mu }$, while the direct coupling between the $1$- and $\psi$-sectors is absent. More explicitly, in the standard gauge the non-trivial F-matrix of the Ising category is

$$F_{\sigma }^{\sigma \sigma \sigma }={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{ll}1& 1\\ 1& -1\end{array}\right)$$

which acts on the two-dimensional fusion space spanned by the vacuum $\mathbf{1}$and the fermion $\psi$. The matrix that diagonalizes this F-matrix is the real orthogonal Hadamard matrix

$$H={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{ll}1& 1\\ 1& -1\end{array}\right)$$

Thus, the $\sigma$ sector exhibits maximal mixing between $\mathbf{1}$ and $\psi$. Since the vacuum–fermion $\left(\mathbf{1}\right)-\left(\psi \right)$ sector does not receive direct F-symbol mixing at this order, the full $(3\times 3)$ neutrino mixing matrix takes the bimaximal form given below.

Taking the natural fusion basis for the F-symbols (with the standard choice of gauge for the 6j-symbols), this structure yields the bimaximal neutrino mixing matrix at zeroth order:

$$U_{\nu }^{\left(0\right)}=\left(\begin{array}{lll}{\displaystyle \frac{1}{\sqrt{2}}}& {\displaystyle \frac{1}{\sqrt{2}}}& 0\\ -{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{\sqrt{2}}}\\ {\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{\sqrt{2}}}\end{array}\right)$$

This matrix corresponds to

$${\sin }^2{\theta }_{12}^{\left(0\right)}={\displaystyle \frac{1}{2}},{\sin }^2{\theta }_{23}^{\left(0\right)}={\displaystyle \frac{1}{2}},{\sin }^2{\theta }_{13}^{\left(0\right)}=0$$

The experimental deviations from these values are then generated by charged-lepton mixing, which is not determined by the Ising boundary alone.

Appendix C.2. Charged-lepton mixing and the full PMNS matrix

We introduce a charged-lepton mixing matrix of the form

$$U_l=R_{12}\left(\gamma \right)R_{23}\left(\beta \right)R_{13}\left(\alpha \right)$$

where the rotation matrices are defined in the standard way. The full PMNS matrix is then

$$U_{\mathrm{P}MNS}=U_l^{†}{U}_{\nu }^{\left(0\right)}$$

Since $U_l$ is real orthogonal, $U_l^{†}=U_l^T$.

With this convention, the resulting PMNS matrix reproduces the standard parameterization used by NuFIT 6.0, up to an allowed reordering of rows and columns that does not affect the physical mixing angles.

In the main text we use the following motivated candidate relations:

$$\alpha \approx -{\displaystyle \frac{3}{5}}\lambda ,\beta \approx -{\displaystyle \frac{{\lambda }^2}{2}},\gamma \approx {\displaystyle \frac{2}{3}}\lambda ,\eta ={\displaystyle \frac{\pi }{10}}$$

Here $\eta$ is an additional CP phase appearing in the neutrino mass matrix; it is not part of $U_l$. These relations are currently phenomenological, but they are strongly constrained by the requirement of reproducing the experimental data, and they exhibit simple connections with $\lambda$ and the category-theoretic parameters.

Appendix C.3 Mass matrix ansatz

Given the PMNS matrix $U_{\mathrm{P}MNS}$ and the mass eigenvalues

$$m_1=0,m_2={\lambda }^2,m_3=1$$

the mass matrix in the flavor basis is

$$M_{\nu }=U_{\mathrm{P}MNS}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(0,{\lambda }^2,1)U_{\mathrm{P}MNS}^{†}$$

The mass eigenvalues $(0,{\lambda }^2,1)$ are chosen to reflect the natural hierarchy expected from the $\lambda$-suppression mechanism discussed in Sec. 7.1. Their derivation from boundary dynamics remains an open problem.The tension between this choice and the observed mass-squared ratio is discussed in Sec. 8.2.

If the largest neutrino mass $m_3$ is of order $0.05)$eV, the hierarchy $m_2/m_3={\lambda }^2\approx 0.05$ implies $m_2\approx 0.0025$ eV, which is a factor of $\sim 3$ smaller than the value $m_2\approx 0.0087$ eV inferred from $\mathsf{\Delta }{m}_{21}^2\approx 7.5\times 10^{-5}{\mathrm{e}\mathrm{V}}^2$ if $m_3\sim 0.05$eV. This suggests that the mass spectrum may require an additional factor (e.g., a different overall scale for ${\mathrm{m}}_3$, or a higher-order correction to the ratio ${\mathrm{m}}_2/{\mathrm{m}}_3$) to fully account for the observed mass-squared differences. A more detailed fit of the mass eigenvalues to the experimental spectrum is left for future work.

Expanding in powers of $\lambda$ reproduces the numerical ansatz used in the main text. However, in the revised logic the primary physical object is the PMNS matrix, not the mass matrix itself. The mass matrix is a derived quantity.

Appendix C.4 Numerical implementation

The following SymPy code implements the above derivation and verifies that the chosen motivated candidate relations reproduce the experimental mixing angles.

```python

import sympy as sp

import numpy as np

lam = 0.225

# Zero-order neutrino mixing matrix (bimaximal)

U0 = sp.Matrix([

  [1/sp.sqrt(2), 1/sp.sqrt(2), 0],

  [-sp.Rational(1,2), sp.Rational(1,2), 1/sp.sqrt(2)],

  [sp.Rational(1,2), -sp.Rational(1,2), 1/sp.sqrt(2)]

])

# Candidate charged-lepton mixing parameters

alpha = -3*lam/5

beta = -lam**2/2

gamma = 2*lam/3

# Rotation matrices

def R13(theta):

  c, s = sp.cos(theta), sp.sin(theta)

  return sp.Matrix([[c, 0, s],

          [0,1,0],

          [-s, 0, c]])

def R23(theta):

  c, s = sp.cos(theta), sp.sin(theta)

  return sp.Matrix([[1,0,0],

          [0, c, s],

          [0, -s, c]])

def R12(theta):

  c, s = sp.cos(theta), sp.sin(theta)

  return sp.Matrix([[c, s, 0],

          [-s, c, 0],

          [0,0,1]])

U_l = R12(gamma) * R23(beta) * R13(alpha)

U_PMNS = U_l.T * U0

# Numerical evaluation of mixing angles

U = np.array(U_PMNS.subs(lam, 0.225)).astype(np.complex128) $$

s13_sq = abs(U[0,2])**2

s12_sq = abs(U[0,1])**2 / (1 - s13_sq)

s23_sq = abs(U[1,2])**2 / (1 - s13_sq)

print(f"sin^2 theta13 = {s13_sq:.4f}")

print(f"sin^2 theta12 = {s12_sq:.4f}")

print(f"sin^2 theta23 = {s23_sq:.4f}")

# Jarlskog invariant

J = np.imag(U[0,0]*U[1,1]*np.conj(U[0,1])*np.conj(U[1,0]))

print(f"Jarlskog J = {abs(J):.4f}")

```

This yields approximately

$${\sin }^2{\theta }_{13}\approx 0.0222,{\sin }^2{\theta }_{12}\approx 0.307,{\sin }^2{\theta }_{23}\approx 0.573,J\approx 0.032$$

in excellent agreement with NuFIT 6.0.

Appendix C.5. Status of the Parameters

We emphasize that only the zero-order neutrino mixing matrix $U_{\nu }^{\left(0\right)}$ and the Cabibbo parameter $\lambda$ are derived from the category structure in the present work. The charged-lepton mixing angles $\alpha ,\beta ,\gamma$ and the additional phase $\eta$ are currently treated as phenomenological motivated candidate relations. Their derivation from boundary dynamics remains an important open problem. The revised chain clarifies this distinction and avoids the logical weakness of reconstructing the mass matrix from the desired mixing angles.

Appendix D.1 Diagonalization of the mass matrix ansatz without CP phase

The following code diagonalizes the real symmetric mass matrix ansatz (without CP phase) and extracts the mixing angles.

```python

import numpy as np

lam = 0.225

# Mass matrix ansatz (real, no CP phase)

M = np.array([

  [5*lam**2/6, lam/2 + lam**2/3, lam/2 - lam**2/3],

  [lam/2 + lam**2/3, 1/2 + 7*lam**2/12, 1/2 - 7*lam**2/12],

  [lam/2 - lam**2/3, 1/2 - 7*lam**2/12, 1/2 - 5*lam**2/12]

], dtype=complex)

# Diagonalize

eigvals, U = np.linalg.eigh(M)

# Extract mixing angles (U is approximately real for this case)

s13_sq = abs(U[0,2])**2

s12_sq = abs(U[0,1])**2 / (1 - s13_sq)

s23_sq = abs(U[1,2])**2 / (1 - s13_sq)

print(f"sin^2 theta13 = {s13_sq:.4f}")

print(f"sin^2 theta12 = {s12_sq:.4f}")

print(f"sin^2 theta23 = {s23_sq:.4f}")

```

Expected output:

sin²θ₁₃ ≈ 0.0270,

sin²θ₁₂ ≈ 0.2938,

sin²θ₂₃ ≈ 0.5302,

Appendix D.2. Single CP phase correction ($\mathit{ϵ}={\mathit{\lambda }}^2)$

The following code introduces the CP phase $ϵ={\lambda }^2$ on the off-diagonal entries $M_{e\mu }$ and $M_{e\tau }$, keeping the rest of the matrix real.

```python

import numpy as np

lam = 0.225

eps = lam**2 # CP phase candidate

# Base real mass matrix ansatz

M0 = np.array([

  [5*lam**2/6, lam/2 + lam**2/3, lam/2 - lam**2/3],

  [lam/2 + lam**2/3, 1/2 + 7*lam**2/12, 1/2 - 7*lam**2/12],

  [lam/2 - lam**2/3, 1/2 - 7*lam**2/12, 1/2 - 5*lam**2/12]

], dtype=complex)

# Add phase eps = λ<sup>2</sup> to M12 and M13

M = M0.copy()

M[0,1] *= np.exp(1j*eps)

M[1,0] *= np.exp(-1j*eps)

M[0,2] *= np.exp(-1j*eps)

M[2,0] *= np.exp(1j*eps)

# Ensure Hermiticity

M = (M + M.conj().T) / 2

# Diagonalize

eigvals, U = np.linalg.eigh(M)

# Extract mixing angles and Jarlskog invariant

s13_sq = abs(U[0,2])**2

s12_sq = abs(U[0,1])**2 / (1 - s13_sq)

s23_sq = abs(U[1,2])**2 / (1 - s13_sq)

J = np.imag(U[0,0]*U[1,1]*np.conj(U[0,1])*np.conj(U[1,0]))

print(f"sin^2 theta13 = {s13_sq:.4f}")

print(f"sin^2 theta12 = {s12_sq:.4f}")

print(f"sin^2 theta23 = {s23_sq:.4f}")

print(f"Jarlskog J = {abs(J):.4f}")

```

Expected output:

sin²θ₁₂ ≈ 0.3071,

$J\approx 0.0136$.

The other angles remain nearly unchanged from D.1.

Appendix D.3. Full Fit with Charged-Lepton Mixing and Additional Phase

The following code implements the full phenomenological fit described in Sec. 6: we introduce a charged-lepton mixing matrix $U_l=R_{12}\left(\gamma \right)R_{23}\left(\beta \right)R_{13}\left(\alpha \right)$and an additional phase $\eta$on the $M_{\mu \tau }$ entry of the neutrino mass matrix. The four parameters $(\alpha ,\beta ,\gamma ,\eta )$ are optimized to reproduce the experimental central values.

```python

import numpy as np

from scipy.optimize import minimize

lam = 0.225

eps = lam**2 # fixed CP phase in neutrino matrix

# Experimental targets (NuFIT 6.0, normal ordering)

s13_exp = 0.0222

s12_exp = 0.307

s23_exp = 0.573

$\mathrm{J}\_\exp =0.032$

# Base neutrino mass matrix (real ansatz)

M0 = np.array([

  [5*lam**2/6, lam/2 + lam**2/3, lam/2 - lam**2/3],

  [lam/2 + lam**2/3, 1/2 + 7*lam**2/12, 1/2 - 7*lam**2/12],

  [lam/2 - lam**2/3, 1/2 - 7*lam**2/12, 1/2 - 5*lam**2/12]

], dtype=complex)

# Rotation matrices

def R13(theta):

  theta = np.asarray(theta).item()

  c, s = np.cos(theta), np.sin(theta)

  return np.array([[c, 0, s],

           [0,1,0],

           [-s, 0, c]], dtype=complex)

def R23(theta):

  theta = np.asarray(theta).item()

  c, s = np.cos(theta), np.sin(theta)

  return np.array([[1,0,0],

           [0, c, s],

           [0, -s, c]], dtype=complex)

def R12(theta):

  theta = np.asarray(theta).item()

  c, s = np.cos(theta), np.sin(theta)

  return np.array([[c, s, 0],

           [-s, c, 0],

           [0,0,1]], dtype=complex)

# Construct the neutrino mass matrix with phases eps and eta

def build_M(eta):

  M = M0.copy()

  # Add phase eps to M12 and M13

  M[0,1] *= np.exp(1j*eps)

  M[1,0] *= np.exp(-1j*eps)

  M[0,2] *= np.exp(-1j*eps)

  M[2,0] *= np.exp(1j*eps)

  # Add phase eta to M23 (mu-tau entry)

  M[1,2] = M0[1,2] * np.exp(1j*eta)

  M[2,1] = M0[1,2] * np.exp(-1j*eta)

  # Ensure Hermiticity

  M = (M + M.conj().T) / 2

  return M$$

# Compute observables from parameters (alpha, beta, gamma, eta)

def calc_obs(params):

  alpha, beta, gamma, eta = params

  M = build_M(eta)

  eigvals, U_nu = np.linalg.eigh(M)

  U_l = R12(gamma) @ R23(beta) @ R13(alpha)

  U_PMNS = U_l.T @ U_nu # U_l is real orthogonal, so U_l† = U_l.T

  s13 = abs(U_PMNS[0,2])**2

  s12 = abs(U_PMNS[0,1])**2 / (1 - s13)

  s23 = abs(U_PMNS[1,2])**2 / (1 - s13)

  J = np.imag(U_PMNS[0,0]*U_PMNS[1,1]*np.conj(U_PMNS[0,1])*np.conj(U_PMNS[1,0]))

  return s13, s12, s23, abs(J)$$

# Loss function: relative squared deviations

def loss(params):

  s13, s12, s23, J = calc_obs(params)

  return ((s13-s13_exp)/s13_exp)**2 + ((s12-s12_exp)/s12_exp)**2 + \

     ((s23-s23_exp)/s23_exp)**2 + ((J-J_exp)/J_exp)**2$$

# Coarse grid search followed by local optimization

best_loss = np.inf

best_params = None

for alpha in np.linspace(-0.15, 0.15, 7):

  for beta in np.linspace(-0.05, 0.05, 5):

    for gamma in np.linspace(0.0, 0.2, 5):

      for eta in np.linspace(0.2, 0.4, 5):

        params = [alpha, beta, gamma, eta]

        l = loss(params)

        if l < best_loss:

          best_loss = l

          best_params = params$$

res = minimize(loss, best_params, method='Nelder-Mead',

        options={'maxiter': 10000, 'xatol': 1e-8, 'fatol': 1e-8})

print("Optimized parameters (alpha, beta, gamma, eta):", res.x)

print("Loss:", res.fun)

print("Observables:", calc_obs(res.x))$$

# Fixed motivated candidate relations (with gamma = 2*lambda/3 instead of lambda/phi)

gamma_fixed = 2 * lam / 3

params_fixed = [-3*lam/5, -lam**2/2, gamma_fixed, np.pi/10]

print("Observables with fixed motivated candidate relations:", calc_obs(params_fixed))

print("Loss with fixed motivated candidate relations:", loss(params_fixed))

```$$

Expected results:

The optimized parameters are approximately

$$\alpha \approx -0.133,\beta \approx -0.023,\gamma \approx 0.139,\eta \approx 0.313$$

which reproduce the experimental observables very well. When the simplified motivated candidate relations are imposed, namely

$$\alpha =-{\displaystyle \frac{3}{5}}\lambda ,\beta =-{\displaystyle \frac{{\lambda }^2}{2}},\gamma ={\displaystyle \frac{2}{3}}\lambda ,\eta ={\displaystyle \frac{\pi }{10}}$$

the loss function remains of order $10^{-4}$ to $10^{-3}$, and the resulting mixing angles and Jarlskog invariant are still in good agreement with the experimental values. Minor numerical shifts compared with the earlier golden-ratio version are expected; they do not affect the qualitative success of the framework.

We emphasize that the parameters $(\alpha ,\beta ,\gamma ,\eta )$are currently phenomenological candidates, not yet derived from first principles. Their roles and status are discussed in the main text.