= Solution
Fix <fundamental units> $\varepsilon_1,\ldots,\varepsilon_r$ and write $x=\zeta\prod\varepsilon_i^{b_i}$. Rank zero already gives a finite <unit group>. Otherwise put $B=\max(1,|b_i|)$. The logarithmic embedding in Dirichlet's theorem has a weighted sum-zero image and is injective on the free exponent lattice. Norm equivalence and that sum-zero condition give an archimedean embedding $\sigma$ with
$$
|\sigma(x)|\le C e^{-cB}.
$$
This is the <small archimedean value of a large unit>. From the equation, $\sigma(\beta y)=1-\sigma(\alpha x)$ is exponentially close to one. Also $h(y)\le h(x)+O(1)$ by the <arithmetic height> inequalities for $y=(1-\alpha x)/\beta$, so the fundamental-unit exponents of $y$ have size $O(B)$.
For large $B$, take the small-branch logarithm of $\sigma(\beta y)$. With $y=\eta\prod\varepsilon_i^{c_i}$ it is the nonzero form
$$
\Lambda=\log\sigma(\beta\eta)+\sum_i c_i\log\sigma(\varepsilon_i)-2k\log(-1),\qquad |k|=O(B),
$$
using fixed logarithms and $\log(-1)=\pi i$. The <integer> $k$ corrects the branch, and nonzero follows since $\beta y=1$ would force $\alpha x=0$. Its upper bound is $|\Lambda|\le C'e^{-cB}$. The assumed effective logarithmic-form estimate for the fixed <algebraic numbers> gives $\log|\Lambda|\ge-C''\log(2B)$. Thus $cB\le C''\log(2B)+O(1)$, effectively bounding $B$. The finitely many embeddings and torsion choices give uniform constants. Therefore \b[there are only finitely many <unit> solutions, and their exponents are effectively bounded].
For the Thue application, take an irreducible integral <binary form> $F$ of degree $d\ge3$ and $m\ne0$. In a <splitting field>, write $F(X,Y)=a\prod_i(X-\rho_iY)$ and put $\theta_i=a\rho_i$, which are <algebraic integers>. The integral factors $L_i=aX-\theta_iY$ satisfy $\prod_iL_i=a^{d-1}m$. Each <ideal> $(L_i)$ therefore divides one fixed <ideal>, giving finitely many possibilities. Choose a generator $\delta_i$ for each principal possibility, so $L_i=\delta_i u_i$ with <units> $u_i$.
For three distinct roots,
$$
(\theta_2-\theta_3)L_1+(\theta_3-\theta_1)L_2+(\theta_1-\theta_2)L_3=0.
$$
Division by the third term gives a fixed-coefficient <unit> equation in $u_1/u_3,u_2/u_3$. Its solutions are finite, hence there are finitely many values of $L_1/L_2$. This ratio determines the rational direction $X:Y$, since the roots are distinct. In a fixed direction, homogeneity and $F(X,Y)=m\ne0$ leave only finitely many <integer> scales. This proves <Thue theorem>, and the argument also works for a form with at least three distinct projective roots even if reducible.
The hyperelliptic alternative uses a related factorization for the usual nondegenerate case with squarefree $f$ of degree at least three. For $y^2=f(x)$, <ideals> of two different root factors can share primes only above the fixed root differences and the leading coefficient. At other primes their valuations are even. Finite exceptional-prime and ideal-class choices, together with <units> modulo squares, reduce the factors to finitely many expressions $\delta_i z_i^2$. Relations between distinct root factors then give simultaneous norm or Pell-type equations. Writing their solutions in <fundamental units> and applying the same small-value/logarithmic-form estimates bounds the <unit> exponents, hence $x,y$. The common-factor and square-class reductions are essential; a single quadratic norm equation alone can have infinitely many Pell solutions. This outlines the effective hyperelliptic treatment as well as the explicit Thue reduction.
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