= Small archimedean value of a large unit
{title2=$\min_\sigma\log|\sigma(u)|\le-c\|b\|_\infty$}
Write $u=\zeta\prod\varepsilon_i^{b_i}$ in a fixed fundamental-unit basis. The logarithmic embedding is injective on the exponent lattice and lands in the weighted sum-zero hyperplane. Norm equivalence gives a logarithmic coordinate of size at least a constant times $\|b\|_\infty$; the weighted sum-zero relation then forces a negative coordinate of comparable size. There are only finitely many torsion choices. This exponential smallness, together with $\alpha u+\beta v=1$ and a <linear form in logarithms of algebraic numbers> for the nearby value $\beta v$, bounds the <unit> exponents effectively.
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