Quadratic Root Structures in the Erdős–Obl´ath Equation

This paper establishes a definitive arithmetic bridge between additive and multiplicative polynomials by investigating the three known integer solutions to the Brocard-Ramanujan Diophantine equation, $n! + 1 = m^2$. By restructuring this factorial relation into a generalized quadratic equation, $x^2 - Bx + C = 0$, we derive a rigid polynomial framework that inherently applies to the arithmetic constraints of Erd\H{o}s prime problems. The structural feasibility of this framework is rigorously verified through a characteristic matrix representation, where the trace and determinant govern the root behavior. Ultimately, we demonstrate that additive sequences and multiplicative structures are not fundamentally isolated; rather, they can be strictly converted into one another via these characteristic root solutions, offering a pure algebraic resolution to the longstanding additive-multiplicative barrier in number theory.