Fluctuation-Induced BCC Selection in Z2-Symmetric Isotropic Fluids: An Exact Combinatorial Approach to Structural Selection

Periodic cellular structures frequently emerge when an isotropic medium undergoes a fluctuationdriven ordering transition. In systems with strict Z2 symmetry, all cubic invariants vanish, forcing structural selection to arise solely from quartic four-wave resonances. We develop a fully closed analytical framework showing that the Body-Centered Cubic (BCC) structure is the unique fluctuationselected minimum across the entire isotropic local quartic class.At the single-shell level, we perform an exact enumeration of nontrivial four-wave resonances on the critical momentum shell and obtain the strict geometric hierarchy CBCC > CFCC > CSC. Because the one-loop Brazovskii correction factorizes into a structure-independent shell integral and the geometric weight CS, this hierarchy directly determines the ordering of the renormalized quartic coefficients.We then prove that this BCC dominance is universal under all isotropic extensions of the theory. Finite shell thickness and O(n) multiplets preserve the same factorization. At arbitrary loop order, every diagram decomposes into a structure-independent shell integral multiplied by a polynomial in CS with nonnegative coefficients, ensuring that higher-loop corrections cannot invert the geometric hierarchy. For weak multi-shell coupling, cross-shell bubble integrals are parametrically suppressed, and mixed-shell resonance multiplicities admit universal bounds. Even in the resonant case—where k1/k0 lies in a finite exceptional set permitting mixed-shell resonances—explicit multiplicity estimates guarantee that the BCC>FCC ordering survives.Together, these results establish a complete universality theorem: except for a measure-zero set of finely tuned resonant ratios, the fluctuation-induced quartic sector of isotropic Z2-symmetric media inevitably selects the BCC structure. By Fourier duality, the real-space cell is the truncated octahedron, providing a mathematically rigorous and physically universal foundation for BCC cellular order in isotropic continuous media.