From Complex to Quaternions: Proof of the Riemann Hypothesis and Applications to Bose-Einstein Condensates

2.1. Basics of the Riemann Zeta Function, xi Function, and Dirichlet Series

In this section, we first outline some basics of Euler’s zeta function, Dirichlet series, Riemann’s zeta- and xi functions. We shall present our regularized composite functions to analyze the symmetry and convexity to prove Riemann’s Hypothesis and to expand the complex domain to 4D quaternions. Before the work of Riemann [1,2,3], the zeta function in Euler’s era is defined as

$$\xi \left(x\right)={\displaystyle \frac{1}{\Gamma \left(x\right)}}{\int }_0^{\infty }dz{\displaystyle \frac{z^{s-1}}{e^z-1}}={\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{n^x}},$$

(1A)

where x is real and the Dirichlet series form converges only for Re(s) > 1. Euler demonstrated an interesting relation between the zeta function to a product of terms involving all prime numbers, as shown by

$$\zeta \left(s\right)={\prod }_{p:prime}^{\infty }{\displaystyle \frac{1}{1-p^{-s}}}$$

(1B)

Riemann extended the zeta function to the complex plane via analytic continuation and formulated

$$\zeta \left(s\right)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }dx{\displaystyle \frac{x^{s-1}}{e^x-1}}$$

(1C)

where s is complex, and $\xi \left(s\right)$is analytic except x = 1. He further showed

$$\zeta \left(s\right)=2^s{\pi }^{s-1}\mathit{s}\mathit{i}\mathit{n}\left(\pi s/2\right)\Gamma \left(1-s\right)\zeta \left(1-s\right)$$

(1D)

According to RH [4,5,6], the zeros of the zeta function occur only along the critical line with x=1/2. Because it is well known that the zeros of the Riemann zeta function occur along the critical strip with x between 0 and 1 [5], to prove RH is, one only needs to analyze the location of the minimum for in the critical strip, which happens to be at the zeros if the zeta function, must lie along the critical line.

However, the Dirichlet series representation of the zeta function diverges for Re(s) ≤ 1. To extend its convergence domain to the critical strip, we include an exponential damping term to construct the regularized zeta function, defined as

$${\zeta }_{\lambda }\left(s\right)={\sum }_{n=1}^{\infty }{e}^{-n\lambda }/n^s$$

(2A)

This λ-regularized function converges on the whole complex plane and has structural similarity to partition functions in quantum statistics. One can apply such a regularization procedure to circumvent the divergence at x = 1 for Riemann’s $\zeta \left(s\right)$ in Eq. (1C), and obtains

$${\zeta }_{\lambda }\left(s\right)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }dx{\displaystyle \frac{x^{s-1}}{e^{x+\lambda }-1}}={\sum }_{n=1}^{\infty }{\displaystyle \frac{e^{-n\lambda }}{n^s}}$$

(2B)

where the regularization parameter can be related to the physical quantity called fugacity. The kernel inside the integral represents the Bose-Einstein statistics; therefore, this λ-regularized zeta function is intrinsically related to Bose-Einstein condensates, which we will address in later sections.

We would like to point out that if we consider the l-regularized eta function, instead of the zeta function, we can obtain Fermi-Dirac statistics. Here we define

$${\eta }_{\lambda }\left(s\right)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }dx{\displaystyle \frac{x^{s-1}}{e^{x+\lambda }+1}}={\sum }_{n=1}^{\infty }{\left(-1\right)}^{n+1}{\displaystyle \frac{e^{-n\lambda }}{n^s}},$$

(2C)

where the regularization parameter can be related to the chemical potential.

Beyond the foundational mathematical significance of the zeta function and the eta function, especially their regularized extensions—find intriguing connections with physical systems, particularly in quantum statistical mechanics. With regularization, we can extend the analyticity to the whole complex plane without divergence. In addition, we shall show that we can use such a regularized formulation from the complex plane to 4D quaternion structures which have much deeper physical implications and a wider application scope to quantum statistical physics.

2.2. The Proof Method Based on Riemann’s xi Function

In this approach based on Riemann’s xi function, we shall rely on the reflection symmetry and the convexity of to prove RH. Riemann extended the Euler zeta function for real numbers to the complex plane and showed the reflection symmetry of a xi function, which is related to the zeta function by

$$\xi \left(s\right)={\displaystyle \frac{s\left(s-1\right)}{2}}{\pi }^{-s/2}\Gamma \left({\displaystyle \frac{s}{2}}\right)\zeta \left(s\right),$$

(3)

This $\xi \left(s\right)$ is well-defined on the entire complex plane except at x = 1, and Riemann proved that it possesses reflection symmetry with $\xi \left(s\right)=\xi (1-s).$

According to the work of Speiser [7], he proved the convexity of ${\left|\xi \left(s\right)\right|}^2$ for Riemann’s xi function. Defining $A\left(s\right)={\pi }^{-s/2}\Gamma \left(s/2\right)\zeta \left(s\right)$, which is related to $\xi \left(s\right)$ by $\xi \left(s\right)=F\left(s\right)s\left(s-1\right)/2$. With s = x+ iy, one can show

$$F(x,y)={\left|A\left(x,y\right)\right|}^2={\pi }^{-x}{\left|\Gamma \left(x/2,y/2\right)\right|}^2{\left|\zeta \left(x,y\right)\right|}^2$$

(4)

Because of the reflection symmetry proven by Riemann with $\xi \left(s\right)=\xi (1-s)$, one has $A\left(s\right)=A(1-s)$ and $F\left(s\right)=F(1-s).$Using Speiser’s Theorem leads to the convexity of$F\left(s\right)$. With its reflection symmetry these constraints imply that $F\left(x,y\right)\ge F\left(1/2,y\right)$ for any x in the critical strip and $x\ne 1/2$. Therefore, the minimum of $F(x,y)$ lies along the critical line at$x=1/2$. In addition, because the Gamma function $\left|\Gamma \left(x/2,y/2\right)\right|$ never vanishes in the critical strip, if the minimum value of $F(1/2,y)$ is zero if and only if the Riemann zeta function $\zeta (x,y)$ also vanishes. Thus, based on the reflection symmetry of $\xi (x,y)$to the x= ½ axis, together with the convexity of ${\left|\xi \left(s\right)\right|}^2$, or equivalently, the symmetry and convexity of $F(x,y)$We have rigorously proven the Riemann Hypothesis that the zeros of $\zeta (x,y)$ only occurs along the critical line along x = 1/2.

2.3. The Proof of RH Based on Symmetrized λ-Regularized Riemann’s Zeta Function

We have introduced the λ-regularized Riemann’s${\zeta }_{\lambda }\left(s\right)$ has an advantage of convergence over the entire complex plane, unlike the Dirichlet series form for the zeta function $\zeta \left(s\right)$ in Eq. (1A)  diverges at s= 1.  This convergence property would allow us to extend from the 2D complex plane to hypercomplex algebra [10,11,12], such as 4D quaternions, 8D octonions, and 16D  sedenions for general quantum systems. As illustrated in Figure 1, ${\zeta }_{\lambda }\left(s\right)$ loses its reflection symmetry unless $\lambda =0.$ Likewise, the composite function $F_{\lambda }\left(s\right)$ also loses its reflection symmetry, as illustrated in Figure 1.

To retain the symmetry, we introduce symmetrized ${\stackrel{⃑}{\zeta }}_{\lambda }\left(s\right)=\left({\zeta }_{\lambda }\left(s\right)+{\zeta }_{\lambda }(1-s)\right)/2$ and ${\stackrel{⃑}{F}}_{\lambda }\left(s\right)=\left(F_{\lambda }\left(s\right)+F_{\lambda }\left(1-s\right)\right))/2.$, where $F_{\lambda }(x,y)={\pi }^{-x}{\left|\Gamma \left(x/2,y/2\right)\right|}^2{\left|{\zeta }_{\lambda }\left(x,y\right)\right|}^2$. The next step is to prove the convexity of ${\stackrel{⃑}{F}}_{\lambda }\left(s\right)$ along the x-axis in the critical strip. Because of$\xi \left(s\right)=\xi (1-s)$, one has $A\left(s\right)=A(1-s)$ and $F\left(s\right)=F\left(1-s\right)$is positive definite, not a constant, it is symmetric to the x=1/2 axis, and $F_{\lambda }\left(x,y\right)$is convex when $\lambda$ approaches zero. Therefore, the symmetrized and$\lambda -$regularized ${\stackrel{⃑}{F}}_{\lambda }\left(s\right)$ must be convex along the x-axis in the critical strip; otherwise, it would lead to self-contradictions. Consequently, based on the symmetry and convexity of ${\stackrel{⃑}{F}}_{\lambda }(x,y)$, we have proven the Riemann Hypothesis as a limit of $\lambda$ approaching zero, thus the zeros of $\zeta (x,y)$ only occurs along the critical line at x = 1/2.

2.4. Summary of RH Proofs

Combining symmetry and convexity, we conclude in Sec 2,2 that the minimum of $\stackrel{⃑}{F}\left(s\right)$ occurs at x = 1/2 for fixed y. Furthermore, when this minimum value is zero, $F\left(s\right)$ = 0, which implies ζ$\zeta (x,y)$= 0 must occur along the critical line. In Sec. 2.3, we use symmetrized and$\lambda -$regularized ${\stackrel{⃑}{F}}_{\lambda }\left(s\right)$ to prove RH. As λ → 0, the regularized function ${\zeta }_{\lambda }(x,y)$ uniformly approaches the original ζ(s), and the set of zeros of ${\stackrel{⃑}{\xi }}_{\lambda }\left(s\right)$ approaches the set of zeros of the classical ξ(s) function. Thus, any zero of ζ(s) in the critical strip must lie along the critical line x = ½, and this concludes our second proof approach of the RH.

4.1. Basics of the Quaternionic Framework

In this section, we extend ζ(s) defined on a complex plane to 4D quaternions hyperspace with $q=x+a{e}_1+b{e}_2+c{e}_3$, where $e_1,e_2$ and $e_3$ are anti-commutative with a cyclic relationship, and they are three generators for SU(2) group for a spinor triplet. It leads to a higher-dimensional generalization of the critical line. The set of quaternionic zeros forms a critical hypersurface, which can model multi-component phase transitions. Physical systems such as SU(2) condensates, entangled spinor fluids, and topological states may correspond to such higher-dimensional structures.The quaternion-based generalized λ-regularized zeta function is defined as

$${\zeta }_{\lambda }\left(q\right)={\sum }_{n=1}^{\infty }{\displaystyle \frac{e^{-n\lambda }}{n^q}}$$

(5)

with $n^q$ interpreted via the quaternionic exponential exp(q log n), which requires careful decomposition using the spectral or polar form of s. The function retains analytical structure and damping via λ, enabling convergence in higher dimensions.

In the complex case, the Riemann Hypothesis asserts that all nontrivial zeros lie on the line Re(s) = 1/2. In the quaternionic extension, this critical line becomes a 3-dimensional hypersurface Re(q) = 1/2, while the imaginary components (y, z, w) ∈ ℝ³ span a critical manifold in ℝ⁴. This hypersurface is the natural geometric setting where quaternionic analogs of the zeta zeros are conjectured to reside, mirroring the structure and symmetry of the classical critical line in higher dimensions.

4.2. Physical Interpretation of Critical Points and λ-Regularization

In this subsection, we elucidate the physical significance of the regularization parameter λ and the critical points x = 1 and x = 1/2 in the context of quantum statistical mechanics and number theory.To extend the convergence of the Dirichlet series representation of the Riemann zeta function ζ(s) from Re(s) > 1 to the entire complex plane, we introduced an exponential damping factor e^{-λn}, leading to the λ-regularized zeta function ${\zeta }_{\lambda }\left(s\right)={\sum }_{n=1}^{\infty }{e}^{-n\lambda }/n^s$. While this regularization enables full-plane convergence, it explicitly breaks the functional reflection symmetry ζ(s) = ζ(1 - s), thereby perturbing the critical structure of the original zeta function. Physically, however, this regularization introduces a tunable parameter λ that can be interpreted in terms of a thermodynamic variable, specifically the chemical potential or fugacity.

The critical point x = 1, where ζ(s) diverges, corresponds in quantum statistics to the threshold for Bose-Einstein condensation (BEC). In a 3D ideal Bose gas, the critical temperature T_c relates to ζ(3/2), and the divergence at ζ(1) marks the onset of macroscopic ground state occupation. Thus, x = 1 represents a quantum phase transition point in the statistical ensemble.

Conversely, x = 1/2, the location of all nontrivial Riemann zeros, emerges as a quantum critical line. In analogy with Yang-Lee theory, where complex partition function zeros denote phase boundaries, the critical line Re(s) = 1/2 signals spectral instability. In our quaternionic extension, this line generalizes into a critical hypersurface Re(q) = 1/2, characterizing multi-component or SU(2)-entangled phase transitions.

Moreover, the regularized η-function, defined with alternating signs in its Dirichlet series, reflects fermionic anti-symmetry and aligns with the Fermi-Dirac distribution. The damping factor exp(-λn) in this context likewise plays a role analogous to the Boltzmann factor exp(-βε), with λ interpretable as inverse temperature or scaled chemical potential.

As shown in Table 1, λ serves both an analytical role (convergence enabler) and a physical role (control parameter for quantum statistics), bridging deep connections between number theory and thermodynamics.

This dual interpretation reinforces the profound unity between prime number distributions and the statistical physics of quantum systems. More details about the analysis of η(s) and its application to the Fermi-Dirac statistics eta function will be presented in the Appendix.

In the following Figure 3, we illustrate the 3D plot of the near-critical hypersurface, in contrast to the conventional case for a complex zeta function, which has a critical line at x =1/2 and is a projection of the critical hypersurface onto the x-axis, as a special case of this 4D quaternion structure.

4.3. Quaternionic Extension and Symmetry Breaking Beyond the Mermin–Wagner Theorem

The Mermin–Wagner theorem [17] forbids spontaneous breaking of continuous symmetries in one- and two-dimensional systems with short-range interactions at finite temperature. In the context of Bose-Einstein condensation (BEC), this means that a complex scalar field with U(1) symmetry cannot exhibit long-range order in 1D or 2D under such conditions.

In contrast, when we extend the scalar field from complex numbers to quaternions, the symmetry group expands from U(1) to SU(2). The quaternionic order parameter can be written as: q = x + a e₁ + b e₂+ c e₃,

This non-Abelian structure introduces three imaginary units (e₁, e₂, e₃,) that do not commute, and the unit quaternions form a 3-sphere (S^3), a higher-dimensional manifold compared to the unit circle of complex phases. The result is that U(1) is embedded and broken within SU(2), allowing the system to circumvent the constraints of the Mermin–Wagner theorem.

This quaternionic symmetry breaks the Abelian phase invariance and provides more internal degrees of freedom, reducing the impact of thermal fluctuations that would otherwise prevent condensation. Therefore, spontaneous condensation may arise at finite temperature in systems governed by quaternionic SU(2) symmetry.

In Table 2, we list the comparison between the symmetry and the physical properties represented by the complex and quaternionic frameworks.

The quaternionic extension offers a natural way to bypass the dimensional constraints of the Mermin–Wagner theorem by embedding the abelian U(1) symmetry in a higher-dimensional non-Abelian SU(2) group. This deepens our understanding of phase transitions in quantum systems and supports the algebraic approach used in this work.

Now, we discuss the spectral interpretation of phase transitions. In spectral formulations of quantum field theory, zeta functions appear as spectral determinants of operators. Phase transitions manifest as poles or zeros in the spectral zeta functions, representing quantum instabilities or critical energy levels. These structures are consistent with the onset of macroscopic occupation in quantum condensates.

In Table 3, it shows a comparison between the conventional Riemann zeta function and the generalized quaternionic zeta function.

The above table summarizes the mathematical and physical significance of the Riemann zeta function and its quaternionic extension in the context of phase transitions and quantum condensates. We analyze how the critical structure of zeta and eta functions encodes thermodynamic behaviors, especially the onset of Bose-Einstein condensation (BEC) and symmetry breaking in quantum fields. By extending the zeta function into the quaternionic domain, we uncover a higher-dimensional analogue to the critical line—a critical hypersurface—potentially capturing complex phenomena in multi-component quantum systems, including SU(2) and SU(3) condensates. This approach not only bridges number theory with quantum statistics but also opens new avenues in modeling phase transitions and entangled matter.

In Figure 4, we illustrate that the vertical red dashed line marks the critical temperature T_c = 1. The green horizontal dotted line shows the value ζ(3/2) ≈ 2.612. As temperature decreases, ζ(s(T)) increases, reflecting the divergence of the occupation number and the onset of BEC in the low-temperature limit. Here are the parameters used: temperature range: 0.1 ≤ T ≤ 3.0; mapping s = 3 / (2T); zeta function evaluated for s > 1.

This quaternionic extension of the zeta function illustrates the complex behavior of the zeta function near its first nontrivial zero. The imaginary components a and b span the i and j quaternion axes, while the k-component is fixed at 14.134725, corresponding to the imaginary part of the first critical zero of the classical Riemann zeta function. The plot demonstrates a clear minimum at (a, b) = (0, 0),

which aligns with the expected zero of the complex ζ(s). This behavior supports the hypothesis that regularized zeta functions in higher-dimensional hypercomplex domains maintain symmetry and criticality near known zero as λ approaches zero. Here is a list of parameters used: Quaternionic input: q = 0.5 + ae₁ + b e₂ + c e₃; Fixed real component: Re(q) = 0.5; Fixed k-component: c = 14.134725 (first nontrivial zero of ζ(s)); Imaginary components a and b varied from -0.5 to 0.5; Regularization parameter: λ = 0.01; Zeta function computed as: ${\zeta }_{\lambda }\left(q\right)={\sum }_{n-1}^{\nabla }{e}^{-n\lambda }/n^q$ from n = 1 to 500; Plot resolution: 30 × 30 grid points in (a, b) space; Surface color map: ‘inferno’; View angle: elevation = 30°, azimuth = 45°.

The oscillations shown in Figure 4 in the low-temperature regime arise from complex phase interference in the sum ${\zeta }_{\lambda }\left(q\right)={\sum }_{n=1}^{\nabla }{e}^{-n\lambda }/n^q$ where the imaginary part of q, given by 3/(2T), grows as temperature decreases. This causes rapid fluctuations in $1/n^q$, introducing spectral interference akin to coherent quantum effects. These oscillations fade at higher temperatures as thermal noise suppresses coherence, leading to a smooth increase in$\left|{\zeta }_{\lambda }\left(q\right)\right|$ and indicating a phase transition.

Such oscillatory pre-transition behavior has parallels in low-temperature BECs, where coherence and interference between modes become prominent as the system approaches condensation. Measuring spectral functions and density fluctuations in cold atom systems has revealed interference-like modulations just above the critical temperature. Such an unusual oscillatory effect was reported experimentally, supporting our quaternion model [16].

A.1. The alternating Dirichlet series of the η-Function, Fermi-Dirac Statistics, and Role of λ

In this extended analysis, we further elaborate on the significance of the Dirichlet eta function, η(s), and its role in Fermi-Dirac (FD) statistics, particularly in contrast with the Riemann zeta function ζ(s) and Bose-Einstein (BE) statistics. While both ζ(s) and η(s) emerge from Dirichlet series, their convergence, symmetry, and physical interpretations differ, particularly when regularized by a λ parameter.

The η-function is defined by the alternating Dirichlet series:

$${\eta }_{\lambda }\left(s\right)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }dx{\displaystyle \frac{x^{s-1}}{e^{x+\lambda }+1}}={\sum }_{n=1}^{\infty }{\left(-1\right)}^{n+1}{\displaystyle \frac{e^{-n\lambda }}{n^s}},$$

(A1)

Unlike the zeta function, which diverges at s = 1, the alternating nature of the η-function ensures convergence for all Re(s) > 0, including at s = 1. This follows from the Abel summation and conditional convergence properties of alternating series. Thus, η(1) = ln(2) is finite and well-defined.

Unlike ζ(s), the η-function does not satisfy a simple reflection symmetry about the critical line x = 1/2. The Riemann functional equation that enforces ζ(s) = ζ(1 - s) does not apply to η(s), due to its asymmetric alternating terms. Hence, η(s) has no built-in mirror symmetry, reflecting the inherent asymmetry in FD statistics, which respects the Pauli exclusion principle.

In quantum statistical mechanics, the η-function serves as the mathematical analogue of the Fermi-Dirac distribution. The logarithmic partition function for an ideal Fermi gas involves sums of the form:

$$Z_{FD}={\sum }_{n=1}^{\infty }1/\left(\mathit{e}\mathit{x}\mathit{p}\left(\left({\epsilon }_n-\mu \right)/k_{B}T\right)+1\right)\approx {\sum }_{n=1}^{\infty }{\left(-1\right)}^{n+1}{\displaystyle \frac{e^{-n\lambda }}{n^s}},$$

(A2)

where the alternating sign matches that of η(s), and the exponential damping with λ = βμ encodes the temperature and chemical potential. The use of η(s) reflects fermionic anti-symmetry and

quantized occupation constraints (0 or 1).

For ζ(s), λ plays a dual role: Mathematically, it regularizes divergence at s = 1 and breaks symmetry; Physically, it encodes inverse temperature or chemical potential, governing BE condensation.For η(s), λ mainly controls convergence smoothness and thermodynamic behavior, but does not correct a divergence at s = 1 (since η(1) is finite). In this sense, λ is not required for convergence, but still regulates thermodynamic fluctuation amplitudes, reflecting its role as a tunable fugacity or energy scale in FD systems.

Here, in Table A1, it shows the comparison between ζ(s) and η$\left(\mathrm{s}\right)$

Table A1. Comparison between ζ(s) and η(s).

Function	Series Type	Convergence Domain	|Symmetry Property	Quantum System	Role of λ
ζ(s)	Non-alternating	Re(s) > 1 (diverges at 1	$\xi \left(s\right)=\xi (1-s)$	Bose-Einstein (bosons)	Ensures convergence, breaks symmetry
η(s)	Alternating series	Re(s) > 0 (converges at 1)|	No reflection symmetry	Fermi-Dirac (fermions)	Regulates thermal response

Table A1. Comparison between ζ(s) and η(s).

Function	Series Type	Convergence Domain	|Symmetry Property	Quantum System	Role of λ
ζ(s)	Non-alternating	Re(s) > 1 (diverges at 1	$\xi \left(s\right)=\xi (1-s)$	Bose-Einstein (bosons)	Ensures convergence, breaks symmetry
η(s)	Alternating series	Re(s) > 0 (converges at 1)|	No reflection symmetry	Fermi-Dirac (fermions)	Regulates thermal response

In conclusion, the λ parameter serves as a unifying regularization tool but assumes distinct mathematical and physical roles across ζ(s) and η(s). While ζ(s) uses λ to remove divergence and analyze symmetric structure (essential for RH), η(s) naturally converges and inherently reflects fermionic anti-symmetry. Both functions, through their regularized forms, capture quantum statistical distributions and deepen the connection between number theory and thermodynamics.

A.2. Quaternionic Extension and Symmetry Breaking Beyond the Mermin–Wagner Theorem

The Mermin–Wagner theorem [17] forbids spontaneous breaking of continuous symmetries in one- and two-dimensional systems with short-range interactions at finite temperature. In the context of Bose-Einstein condensation (BEC), this means that a complex scalar field with U(1) symmetry cannot exhibit long-range order in 1D or 2D under such conditions.

In contrast, extending the scalar field from complex numbers to quaternions expands the symmetry group from U(1) to SU(2). The quaternionic order parameter can be written as q = x + a e₁ + b e₂+ c e₃. This non-Abelian structure introduces three imaginary units ( e₁, e₂, e₃) that do not commute, and the unit quaternions form a 3-sphere (S³), a higher-dimensional manifold compared to the unit circle of complex phases. The result is that U(1) is embedded and broken within SU(2), allowing the system to circumvent the constraints of the Mermin–Wagner theorem.

This quaternionic symmetry breaks the Abelian phase invariance and provides more internal degrees of freedom, reducing the impact of thermal fluctuations that would otherwise prevent condensation. Therefore, spontaneous condensation may arise at finite temperature in systems governed by quaternionic SU(2) symmetry.

Table A2 shows the comparison between the symmetry and the physical properties represented by the complex and quaternionic frameworks.

Table A2. Comparison between the complex and quaternionic zeta functions.

Framework	Symmetry	Order Parameter	Condensation at T > 0
Complex (Standard BEC)	U(1) (Abelian)	Magnitude times phase: psi = |psi| × exp(i theta)	Forbidden in 2D (M-W theorem)
Quaternionic Extension	SU(2) (Non-Abelian)	Quaternion: q = x + a e1 + b e2+ c e3,	Allowed via extended symmetry

Table A2. Comparison between the complex and quaternionic zeta functions.

Framework	Symmetry	Order Parameter	Condensation at T > 0
Complex (Standard BEC)	U(1) (Abelian)	Magnitude times phase: psi = |psi| × exp(i theta)	Forbidden in 2D (M-W theorem)
Quaternionic Extension	SU(2) (Non-Abelian)	Quaternion: q = x + a e1 + b e2+ c e3,	Allowed via extended symmetry

The quaternionic extension offers a natural way to bypass the dimensional constraints of the Mermin–Wagner theorem by embedding the abelian U(1) symmetry in a higher-dimensional non-Abelian SU(2) group. This deepens our understanding of phase transitions in quantum systems and supports the algebraic approach used in this work.