Hyperbolicity mechanism for period-doubling universality

At the normalized quadratic fixed point $g$, one expanding eigenvalue $\delta >1$ of $D\mathcal{T}(g)$ and a contracting complement give a stable manifold of finite codimension one. A generic one-parameter family crosses that manifold transversely at its cascade accumulation. Its unstable coordinate is initially proportional to parameter distance and multiplies by $\delta$ per renormalization, yielding $s_{\infty }-s_n\sim C{\delta }^{-n}$. Stable components disappear, while the fixed point's orientation-reversing scale $a$ yields signed spatial ratios $a^{-1}=-\alpha$.

// Target: dynamical-systems.bigb