The Geometry of the Fixed-Point Set of the Einstein–Hilbert Flow with Matter: a Cuspidal Cubic, a Bogdanov–Takens Point and a Periodic Renormalization-Group Orbit

We study the dependence of the ultraviolet fixed point of the derived Einstein–Hilbert flow with minimally coupled matter (optimised cutoff, harmonic gauge) on the matter content $(X,Y)$, treating X and Y as continuous parameters. Three exact results are proved. (i) The fixed-point set is a ruled surface; the saddle–node (fold) curve in the $(X,Y)$ plane, which bounds the existence region found numerically in the author's book on Quantum Distortion Gravity, is a cuspidal cubic, i.e. the semicubical parabola in suitable affine coordinates, with a cusp at the rational point $(X,Y) = (12349/15552, -126613/23328)$ and an explicit rational parametrisation; for any regulator it is the Legendre transform of a function of λ. (ii) At a fixed point the determinant of the stability matrix equals 2F'(u)/(3u² Nω), where F is the fixed-point cubic, so the Poincaré index of every admissible fixed point is the sign of $F'$. (iii) The trace of the stability matrix has a closed form; its zeros on the fold curve are the roots of an explicit quartic, two of which are admissible Bogdanov–Takens points, one at (X,Y) (4.017, -5.345) with (g_*,λ_*) (1.869, 0.0169). From it emanates a Hopf curve with first Lyapunov coefficient positive (subcritical) at all 19 points sampled along it (4.1 ≤ X ≤ 130), so that on the side where the fixed point is ultraviolet attractive it is surrounded by an unstable periodic orbit of the flow, bounded by a homoclinic loop of the saddle. The periodic orbit, the Hopf and homoclinic curves, and the transition to a repulsive focus are established numerically, not proved. The cusp and the Bogdanov–Takens point persist for six regulators tested; the contents assigned to Quantum Distortion Gravity (nf Dirac fermions plus two Proca fields, or two massless vectors) never reach any of these structures. All statements concern the truncated flow and are not claims about the full theory.