= Solution
Work in a normalized <analytic function> space of even <unimodal maps> with a quadratic critical maximum. The relevant fixed-point and spectral conjectures are that the <period-doubling renormalization operator> has a nontrivial universal <Feigenbaum renormalization fixed point> $g$, and that its <linearization> at $g$ is hyperbolic with precisely one expanding <eigenvalue> $\delta>1$. Its remaining spectral directions contract. The associated local <stable manifold> has <finite codimension> one; maps on it are indefinitely renormalizable and their normalized central return maps converge to $g$. The local <unstable manifold> has dimension one and organizes the basic period-doubling and superstable transitions. These assertions concern the normalized quadratic universality class, rather than arbitrary functions or every parameter family.
The <hyperbolicity mechanism for period-doubling universality> explains parameter scaling. Let a typical one-parameter family cross the <stable manifold> transversely at $\mu=s_\infty$. In local stable and unstable coordinates, write its unstable coordinate as
$$
u(\mu)=c(\mu-s_\infty)+O((\mu-s_\infty)^2),\qquad c\ne0.
$$
Repeated renormalization multiplies this coordinate to leading order by $\delta$ while damping stable components. A superstable orbit of period $2^n$ becomes a superstable two-cycle after $n-1$ renormalizations. Hence its parameter is characterized by arrival at the same basic superstable surface, at a fixed nonzero unstable coordinate after the initial transient. This yields
$$
s_\infty-s_n=C\delta^{-n}(1+o(1)),\qquad C>0,
$$
and therefore
$$
\boxed{\frac{s_n-s_{n-1}}{s_{n+1}-s_n}\longrightarrow\delta.}
$$
The family-dependent coefficient $C$ changes with parametrization, but the ratio does not. The same argument applies to the primary bifurcation surfaces and their successive preimages. <Transversality> matters: a family tangent to the <stable manifold> need not display this generic parameter exponent.
Spatial universality follows from the <fixed point>'s scale $a_g=g(1)<0$. As renormalized maps approach $g$, their successive central restrictive intervals shrink by factors approaching $|a_g|$, with orientation alternating. Thus the signed central spacings have asymptotic ratio $a_g^{-1}$, and
$$
\boxed{\alpha=-a_g^{-1}.}
$$
Stable directions remove the original family's detailed shape; the limiting profile is $g$, the parameter expansion rate is $\delta$, and the spatial contraction rate is $|a_g|$. Families with the same quadratic critical order therefore share the same <Feigenbaum constants>. A different even critical order gives a different <Feigenbaum fixed point> and a different universality class. This explanation is a consequence of the fixed-point, hyperbolicity and transverse-transition properties; it is not a proof that an arbitrary one-parameter family satisfies those properties.
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