We prove the abc conjecture by analyzing the infinitesimal behavior of logarithmic height functions under arithmetic deformations preserving algebraic relations, normalization, and local consistency across places. The argument is formulated entirely at the level of first and second variations, without assuming the existence of global one-parameter families, analytic smoothness, or geometric deformation spaces. A central result is an admissibility property asserting the non-negativity of the second variation of height, derived intrinsically from the max-type structure of local height components. This excludes all negative quadratic contributions and rules out any superlinear amplification mechanism for height growth. Consequently, admissible height variation is governed solely by localized first-order contributions arising from finitely many places. A precise comparison of these contributions with the primes dividing the arithmetic configuration yields the abc inequality. All definitions, results, and logical dependencies are stated explicitly within the text. The argument is logically closed within this framework.
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