Content–Height with Absolute Gain in $a+b=c$ and Classical Applications

This technical supplement consolidates the proofs, normalizations, and computational routines that support the main results of the article. The separation into Appendices G–K serves two complementary objectives. First, to make explicit the quantitative dependencies on structural parameters (degrees, heights, prime supports, and universal constants) that, in the body of the text, are employed operationally and sometimes only sketched. Second, to provide a verifiable, non-circular, and self-contained route for the technical passages that link local–global readings (Archimedean and p-adic) with polynomial disaggregations and height normalizations.