= 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
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