= Solution
The <Gelfond–Schneider theorem> says that if $a\ne0,1$ is algebraic and $\beta$ is algebraic irrational, then every value $\exp(\beta\lambda)$ with $e^\lambda=a$ is transcendental. Fix such a nonzero logarithm $\lambda$, and suppose $\gamma=e^{\beta\lambda}$ is algebraic. The <field> $K=\mathbb Q(a,\beta,\gamma)$ then contains all values needed in <Schneider lattice proof of the Gelfond–Schneider theorem>.
Here is an outline including the parameter balance. For a large $L$, choose <polynomial> degrees $M\asymp L\log L$, $N\asymp L/\log L$, with constants large enough that $(M+1)(N+1)>[K:\mathbb Q]L^2$. Impose
$$
P(s+t\beta,a^s\gamma^t)=0\qquad(0\le s,t<L).
$$
Expand these $K$-linear equations in a rational basis and clear denominators. <Siegel lemma> gives a nonzero <integer> <polynomial> $P$ with $\log H(P)=O(L^2/\log L)$. This follows because each coefficient in the grid equations has <arithmetic height> $O(M\log L+NL)$ and the ratio of unknowns to constraints stays bounded away from one. The <exponential polynomial> $F(z)=P(z,e^{\lambda z})$ is not identically zero: distinct exponentials are independent over the <polynomial> ring, as one sees by successively applying powers of $D-j\lambda$ to kill all but one term.
The irrationality of $\beta$ makes the $L^2$ grid points distinct. Their zeros make subsequent grid values extremely small. More precisely, if a whole $S\times S$ grid has been obtained, take an outer radius $R\asymp S\log S$. Growth gives $\log\max_{|z|=R}|F(z)|\le\log H(P)+O(M\log R+NR)$. The product form of Schwarz's lemma contributes $(C/\log S)^{S^2}$ at the next grid points. Therefore
$$
\log|F(s+t\beta)|\le-cS^2\log\log S\qquad(s,t\le S)
$$
for large initial $L$, unless the value is zero. But every such value is algebraic in the same $K$, since $e^{\lambda(s+t\beta)}=a^s\gamma^t$. Its <arithmetic height> is $O(\log H(P)+M\log S+NS)$, so the Liouville lower bound is incompatible with the displayed upper bound for a nonzero value. Induction extends the zero grid indefinitely.
A fixed nonzero <exponential polynomial> has only $O(R)$ zeros in a disc of radius $R$, by its exponential-type growth and <Jensen's formula> centered at a nonzero value. The grids give $S^2$ distinct zeros in a disc of radius $O(S)$, a contradiction. Thus \b[$a^\beta$ is transcendental for every choice of logarithm], completing the <Schneider lattice proof of the Gelfond–Schneider theorem>.
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