NEXO-01-DRD: The Deformation Invariant Theorem – A Structural Invariant in Quadratic Families

This paper presents the Deformation Invariant Theorem (NEXO-DRD) , a fundamental result in the theory of quadratic equations. It proves that for the one-parameter family x^{2} - bx + c = 0, where c \in \mathbb{R} is fixed and b \in \mathbb{R} varies, the product of the two real roots satisfies r_{1} \cdot r_{2} = c for all admissible values of b. Although this property is implicit in Vieta's formulas, it is here examined in depth as a structural feature of quadratic deformations. The proof is elementary, requiring only the quadratic formula and the difference-of-squares identity, and is accompanied by geometric interpretations, historical context, and a comprehensive catalogue of applications across mathematics, physics, engineering, and cryptography. This work serves as the first in a series of ten papers comprising the NEXO Mathematical System.