= Auxiliary-value height argument for the Fredholm series
{title2=$P_N(z,f_l(z))=O(z^{T_N}),\qquad T_N\ge(N+1)^2-1$}
A nonzero <integer> <polynomial> of bidegree at most $N$ can cancel the first $(N+1)^2-1$ Taylor coefficients of the <Fredholm series>. Functional transcendence makes the resulting <analytic function> nonzero. If $\alpha$ and $f_l(\alpha)$ were both algebraic, the relation $f_l(\alpha^{l^r})=f_l(\alpha)-\sum_{j<r}\alpha^{l^j}$ would keep the auxiliary values in one fixed <number field>. Their <Absolute logarithmic Weil height> is at most $N l^r h(\alpha)(1+1/(l-1))+O_N(r+1)$, whereas their chosen absolute value is $O_N(|\alpha|^{T_Nl^r})$. The <Liouville height inequality> forces $T_N=O(N)$, contradicting its quadratic lower bound for sufficiently large $N$.
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