In this paper, under a unified framework of differential-algebraic closure and finite representation of transcendental functions, we present a constructive approach to prove the ABC conjecture. By applying the theory of universal differential algebraic closure Kn and specialization homomorphism ιf established in previous work, we associate the arithmetic quantities (such as coordinates of points) of the Frey curve Eabc linked to an ABC triple (a,b,c) with differential-algebraic functions of the coefficients a,b,c. By introducing an intrinsically defined differential algebraic height hDA and using the affine estimate of the N´eron–Tate canonical height as its normalization constraint, we precisely translate the ABC inequality into an estimation problem concerning the “formal size” of differential-algebraic expressions. The core proof consists of a combinatorial analysis and algebraic identity derivation for the universal representation element XPabc ∈ Kn of the key point Pabc,strictly obtaining upper and lower bounds for its hDA. By introducing an adjustable parameter λ in the solution formula to rebalance the estimates, we finally derive that for any ε > 0 there exists a constant Cε such that H ≤ Cε ·rad(abc)1+ε. This paper not only provides a proof of the ABC conjecture intrinsic to the differential algebraic representation framework, but also demonstrates the constructibility and computability of the constant Cε within the framework, thereby establishing new explicit connections among differential algebra, transcendental function theory, and Diophantine geometry.
shifa liu (Wed,) studied this question.