Gel'fond-type transcendence criterion with continuous growth functions

The transcendence criterion, for continuous growth functions.

This is FourExp.transcendence_criterion with two changes. The functions $\sigma_1, \sigma_2$ are also assumed continuous. The conditions

are required only for $x \ge 1$, not for $x > 0$.

Explicitly: let $\alpha \in \mathbb{C}$ and $\varepsilon > 0$. Let $\sigma_1, \sigma_2$ be continuous, strictly increasing and unbounded, and let $a_1, a_2 \ge 1$ satisfy the conditions above for $x \ge 1$. Suppose that for all $N > N_0$ there is a non-zero $P_N \in \mathbb{Z}[X]$ with coefficients at most $e^{\sigma_1(N)}$, degree at most $\sigma_2(N)$, and $|P_N(\alpha)| < \exp(-C\sigma_1(N)\sigma_2(N))$, where $C = \max\{10+\varepsilon, (4+\varepsilon)a_1a_2\}$. Then $\alpha$ is algebraic.

Why this form. Waldschmidt's proof (1971, §3) inverts $\sigma_i$: it defines $z_q = \max\bigl(\sigma_1^{-1}(\tfrac13\log h_q),\ \sigma_2^{-1}((1+\tfrac\varepsilon2)^{-1}\delta_q)\bigr)$ and uses $\sigma_i(\sigma_i^{-1}(y)) = y$, which needs continuity. The paper notes that continuity can be assumed, since only integer values matter. That step is the accepted reduction from FourExp.transcendence_criterion, by piecewise-linear interpolation. The proof only ever uses large $x$, so asking the conditions for $x \ge 1$ loses nothing, and interpolation cannot preserve them on $(0, 1)$ in general.

Intended proof. The 1971 argument, using Gel'fond's height inequality for a divisor, Gel'fond's small-irreducible-factor lemma, and the resultant bound that forces $Q_q \mid P_N$.