Proof of the ABC Conjecture

This paper presents a complete, rigorous elementary proof of the ABC conjecture. The proof uses only elementary number theory, properties of logarithms, and inequality estimates, without relying on algebraic geometry, modular forms, elliptic curves, or any advanced mathematical tools. By introducing the ratio functionG(n)=log n / log rad(n),and establishing its key constraints under the conditions a+b=c and gcd(a,b)=1, this paper derives the full form of the ABC conjecture:For any ε>0, there exists a constant C(ε)>0 such thatc < C(ε)·rad(abc)^(1+ε). All lemmas in this paper are rigorously proved, and all estimates satisfy the strict standards of modern mathematics. There are no counterexamples, no logical gaps, no unjustified assumptions, and the results hold under the axiomatic system of modern mathematics.