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 $\lambda$. (ii) At a fixed point the determinant of the stability matrix equals $2F'(u)/(3u^2 N\omega)$, 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 $(X,Y) \simeq (4.017, -5.345)$ with $(g_*,\lambda_*) \simeq (1.869, 0.0169)$. From it emanates a Hopf curve with first Lyapunov coefficient positive (subcritical) at all 19 points sampled along it ($4.1 \le X \le 130$), 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 ($n_f$ 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.