A Dynamic Rigidity Proof of the abc Conjecture

We prove the abc conjecture by establishing a dynamic rigidity principle for multiplicatively compressed arithmetic configurations. Any hypothetical violation of the abc inequality is shown to give rise to configurations exhibiting extreme multiplicative alignment, which are unstable under admissible variations preserving prime support. We formalize this obstruction through the Multiplicative Alignment Condition (MAC) and show that the abc conjecture is equivalent to a single rigidity inequality relating height growth to radical compression. The problem is thereby reduced to a finite–dimensional analysis of linear forms in logarithms associated with such variations. Using multiplicative compression, we prove that all relevant linear forms reduce uniformly to bounded dimension. Classical lower bounds for linear forms in logarithms then yield uncondi- tional uniform estimates sufficient to verify MAC. The argument relies only on standard analytic number–theoretic tools and does not invoke p–adic deformation theory, anabelian geometry, or inter–universal constructions. As a result, the rigidity inequality holds unconditionally, and the abc conjecture follows. The proof identifies the abc conjecture as an instance of a general multiplicative rigidity phenomenon, analogous to stability mechanisms familiar from analytic regularity theory.