= Solution
The <Lanford contraction proof of the Feigenbaum fixed point> replaces an unstable renormalization operator by a contracting Newton correction. Use a <coefficient Banach space for normalized even maps>, represented by $p(x)=1-x^2h(x^2)$ with an absolutely summable coefficient <norm> on $h$. Its complex domain and a small ball are chosen so that every map is admissible and the renormalization composition is analytic there.
A high-accuracy <polynomial> $p_0$ gives an approximate <fixed point>. Use a bounded invertible approximate inverse $J$ for $D\mathcal T(p_0)-I$, and define
$$
A(p)=p-J[\mathcal T(p)-p].
$$
Rigorous estimates certify
$$
\varepsilon=\|A(p_0)-p_0\|,\qquad
\sup_{\|p-p_0\|\leq r}\|I-J[D\mathcal T(p)-I]\|\leq\kappa<1,
\qquad\varepsilon\leq(1-\kappa)r.
$$
The finite coefficient computations use <interval arithmetic> with controlled rounding; analytic estimates bound the infinite tail. Thus the certification applies to the full function space, rather than merely a truncated <polynomial> system.
Part (b) gives a unique <fixed point> of $A$ in the certified ball. Invertibility of $J$ makes it a unique <fixed point> $g$ of $\mathcal T$ there. The ball also preserves the quadratic critical maximum and nontrivial shape, excluding the constant formal normalized solution. Consequently \b[an actual locally unique analytic <Feigenbaum fixed point> exists], with the error bound $\|g-p_0\|\leq\varepsilon/(1-\kappa)$. The numerical approximation locates it; rigorous residual, <derivative> and tail bounds establish its existence and uniqueness. This conclusion is local, not a claim of global uniqueness among all analytic solutions.
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