Linear forms in logarithms and effective control of S-units in parametric families via an exponent dichotomy

We develop a quantitative theory for S-unit solutions of the equation $\xplusy$ over number fields, with emphasis on the interaction between the Schmidt Subspace Theorem, linear forms in logarithms (in the sense of Matveev and Yu), and parametric families along algebraic subtori of $\Gm^2$. The central goal is to construct an effective global constant $C(K,S,t)$, depending only on the number field $K$, the finite set of places $S$, and the number of terms $t$, which provides uniform lower bounds for linear forms in logarithms of elements of a finitely generated multiplicative group containing the S-unit group and a finite set of fixed coefficients. By combining archimedean and $p$-adic lower bound theorems in a controlled fashion, we obtain a global lower bound principle for linear forms in logarithms whose bases lie in a fixed multiplicative group. This principle is applied to parametric families of S-unit solutions of $\xplusy$ obtained via the Schmidt Subspace Theorem. Along these families, we construct linear forms in logarithms with integer coefficients affine in a parameter $k\in\Z$, and derive, in suitable archimedean embeddings, a purely geometric upper bound for $\log\abs{\LambdaK(k)}$ of at most logarithmic growth in $\abs{k}$. Combining this upper bound with the global lower bounds arising from $C(K,S,t)$ leads to an exponent dichotomy lemma: either the parameter $k$ remains uniformly bounded along the parametric component, or $\LambdaK(k)=0$ for infinitely many $k$, forcing the corresponding solutions to fall into narrower linear subvarieties in the exceptional hierarchy given by the Schmidt Subspace Theorem. As a consequence, we obtain effective control of the parametric families of S-units in $\xplusy$, with direct applications to the proof of $S$-local and global versions of the ABC Theorem over number fields.