An Exact Algebraic Reduction of the Hexagonal-Face Monostationarity Problem in Dual Phosphorylation

This record documents an exact algebraic reduction of the hexagonal-face case arising in the monostationarity problem for dual phosphorylation. The work starts from the polynomial formulation of the hexagonal-face region considered by Cai, Himmelmann, and Ostermann (2025), in the parameter regime a>0, b<0a>0,\\ b<0. The positive-variable polynomial is reduced by factoring out the monomial x2zx^2z and minimizing the remaining expression with respect to zz. The key reduction is an exact one-variable minimization of the form F(x)=Ax2+Bx+C+Dx+Ex2,F(x)=Ax^2+Bx+C+\\frac{D}{x}+\\frac{E}{x^2}, where the coefficients A,B,C,D,EA,B,C,D,E are explicit functions of the biochemical parameters. The stationary condition reduces to the quartic equation 2Ax4+Bx3−Dx−2E=0.2Ax^4+Bx^3-Dx-2E=0. Because its coefficient sequence has exactly one sign change, Descartes' rule of signs gives at most one positive stationary root. Combined with the limiting behaviour of F(x)F(x), this establishes a unique positive global minimizer x∗x_*. The corresponding boundary is characterized by N=F(x∗).N=F(x_*). Equivalently, the boundary can be obtained algebraically by eliminating xx between the quartic stationary equation and the boundary equation. This record is intended as a dated research record of the derivation and computational verification. The mathematical contribution should be regarded as a candidate exact characterization pending independent verification and comparison with the complete prior literature. Included materials contain the derivation, symbolic calculations, resultant calculation, numerical sanity check, and references.