The Arithmetic Fiber of a + b = c: A c-Centered Perspective on the ABC Conjecture

This note develops a c-centered perspective on the ABC conjecture by viewing the equation a + b = c as an additive fiber over the parameter c. For fixed c, the complete finite family of positive decompositions a + b = c is considered as a single arithmetic object rather than as isolated triples. The note shows how multiplicative structure, greatest common divisors, divisor structure, prime support, and the radical rad(abc) are organized within this fiber. In particular, an exact decomposition of the fiber into scaled primitive fibers is derived, making explicit how the divisor and prime-factor structure of c determines its internal stratification. This provides a structural interpretation of the form of the ABC conjecture: the conjecture relates the base parameter c to the radical of individual points within the complete additive fiber over c. The perspective is not presented as a new equivalent formulation or proof of the ABC conjecture. Its contribution is a unified c-centered organization of the additive, multiplicative, gcd, divisor, and radical structures involved in the conjecture, together with a finite computational illustration. The work is intended as a conceptual and structural contribution to the study of the arithmetic relationship between addition and multiplication.