Static Fourier sector and hypercomplex phase structure

Analyze the complementary static Fourier sector of the hypercomplex dissipative partition function by determining its stationary conditions and integration cycle, and characterize its possible phase structure without identifying it a priori with a conventional Landau potential or Bose–Einstein condensate.

Background

The partition function separates the static zero-frequency and zero-momentum contribution from the nonstatic thermal fluctuations through the factor Z_0. In the strict non-condensed thermal domain, this isolated mode contributes only subextensively to intensive thermodynamic quantities, but the authors retain it because its behavior may become nontrivial when the zero-momentum sector is macroscopically occupied or approaches a genuinely degenerate configuration.

The paper does not impose an interpretation of Z_0 as a conventional Landau effective potential or Bose–Einstein condensate. Resolving the stationary conditions and selecting an appropriate integration cycle for the static-sector hypercomplex action is therefore left as an unresolved problem, including the possibility of a distinct phase structure associated with this sector.