A Fractal Eigenvector

The paper defines Mn via the 3×3 block recursion and derives the polynomial-eigenvector recursion xn+1(ρ) = [ρ Cn(ρ) xn(ρ); Cn(ρ); xn(ρ)], then shows by direct block substitution that Cn(ρ)e1 + Mn xn(ρ) = ρ xn(ρ) at ρ = ρn+1 and that the last entry is 1, i.e., essentially (Mn − ρ I) xn(ρ) = −Cn(ρ) e1, with Cn+1(ρ) = ρ Cn(ρ)^2 − 1 ensuring (Mn+1 − ρ I) xn+1(ρ) = −Cn+1(ρ) e1 and hence eigenvectorhood when Cn+1(ρ)=0 (equations (4), (10)–(14), (11) in the paper) . The candidate solution reproduces the same construction and identities, proving them for all real ρ by induction and then specializing to ρn+1, which matches the paper’s approach and conclusions.