Spontaneous Complex Order on a Stable Interface

Research status. An explicit classical field-theory construction with a stable interface. It does not generate fermions, gauge fields, or gravity; its relation to Standard Model CP violation is unproven. Not peer reviewed.

Abstract

A five-dimensional scalar model with real couplings admits a stable domain wall whose localized internal order selects one of two complex-conjugate orientations. The wall saturates an exact energy bound, and the full seven-real-component fluctuation operator is nonnegative. The model illustrates how an internal configuration can be dynamically selected instead of prescribed.

1. Dynamical action

Introduce a real field (phi) and three complex components (chi_i). Let (s=\sum_i|\chi_i|^2), (a=1/r-2), (00). The action is

\(\displaystyle S=\mathcal C\int d^5x\left[\frac12(\partial\phi)^2+\frac12\sum_i|\partial\chi_i|^2-V\right],\qquad \mathcal C>0.\)

Choose the real potential

\(\displaystyle \begin{aligned}V={}&\tfrac12(1-\phi^2-rs)^2+2r^2\phi^2s+\tfrac\lambda2[s-a(1-\phi^2)]^2\\&+\tfrac{\eta_2}{2}(|\chi_2|^2-2|\chi_1|^2)^2+\tfrac{\eta_3}{2}(|\chi_3|^2-3|\chi_1|^2)^2+\tfrac\kappa2|\chi_1+\chi_2+\chi_3|^2.\end{aligned}\)

The homogeneous zero-energy vacua are ((\phi,\chi)=(\pm1,0)). The action is invariant under common phase rotation and simultaneous complex conjugation.

2. Spontaneous complex orientation

The interactions favor squared-amplitude ratios (1:2:3) and a vanishing component sum. Fixing the common phase so the first component is real selects the two unit vectors

\(\displaystyle u_\pm=\begin{pmatrix}1/\sqrt6\\\pm i/\sqrt3\\-1/\sqrt6\mp i/\sqrt3\end{pmatrix}.\)

The conjugation-odd quantity (J_C=\operatorname{Im}(\chi_1^*\chi_2)) distinguishes the orientations. Their relative complex phase follows from real interactions, not an explicitly complex source.

3. Exact wall and energy minimum

For the selected interface sector, the coupled equations admit

\(\displaystyle \boxed{\phi(z)=\tanh[2r(z-z_0)],\quad \chi(z)=\sqrt{\frac1r-2}\operatorname{sech}[2r(z-z_0)]e^{i\alpha}u_\pm.}\)

Introduce (W=\phi-\phi^3/3-r\phi s). The potential splits into (rac12|\nabla W|^2+V_{\mathrm{aux}}) with (V_{\mathrm{aux}}\ge0). Completing squares gives the interface tension

\(\displaystyle T=\frac43+\int dz\left[\frac12|q'-\nabla W|^2+V_{\mathrm{aux}}\right]\ge\frac43.\)

The exact solution saturates this bound: (T=4/3). It is a minimum in its topological wall sector, not proof of cosmological formation.

4. Linear stability

Writing all complex fields as real components gives seven real scalar fluctuations. Their operator has the factorization

\(\displaystyle \mathscr H=Q^\dagger Q+\sum_Ac_Av_Av_A^T,\qquad c_A>0.\)

Hence (mathscr H\ge0). Two normalizable zero modes correspond to translation and common phase rotation. The latter is an ungauged massless scalar, not a demonstrated Higgs particle.

5. Numerical consistency checks

At (r=0.25,\lambda=0.10,\eta_2=0.40,\eta_3=0.70,\kappa=0.60), the original study reports the tension (1.333333334823) against the exact (4/3), lowest positive scalar eigenvalue (0.108218025), continuum threshold (0.25), and core order (\sqrt2/3\). Small changes to the interaction coefficients preserve stable ordered solutions in the tested cases.

An uncondensed wall has an unstable mode; time evolution confirms its initial growth but does not demonstrate universal relaxation to a unique ordered state. Stability of the constructed equilibrium and generic formation of that equilibrium are different claims.

6. Scope

The construction demonstrates a dynamically generated localized complex scalar order. Its seven real field components are not seven extra spatial dimensions. It does not derive electrons, quarks, fermionic exchange statistics, Standard Model charges, the Higgs doublet, or the four interactions. The exact solution uses a special parameter relation whose microscopic protection has not been shown.

Research provenance

Web adaptation of an AI-assisted October 2026 investigation. Related Bloch walls, frustrated multicomponent order, Bogomolny energy bounds, and spontaneous conjugation breaking are established research topics. Priority and publishability have not been certified.

Study 01Fermion spectral calculation →Philosophical essayThe original ontological question →

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