Supported through a century of investigation, experimental support, and observational evidence, relativity is accepted as being conceptually sound, mathematically correct, and theoretically valid. Because it is believed to be the only theory that quantitatively explains certain experiments and yields E=mc^2, there is widespread support that any improvement will take the form of an enhancement to, rather than a replacement of the theory. Despite this degree of support, the seminal derivation of special relativity theory found in Section 3 of On the Electrodynamics of Moving Bodies contains a mathematical contradiction that must be remediated. Specifically, the τ and ξ equations are expressed in terms of x, y, and t, where Einstein derives ξ as ξ=τc, which is immediately followed by his stating the ξ and τ equations as: τ=β(t-vx/c^2 ) and ξ=β(x-vt), where β=1/SQRT(1-v^2/c^2 ). The contradiction occurs because the mathematical equality of ξ=τc is not maintained when both equations are evaluated using most combinations of x, y, and t. As a concrete example of the contradiction, when x=1, v=0, and t=0, we find that ξ=1 and τ=0, such that ξ≠τc. Here, we introduce Modern Mechanics, a three–system, classical mechanics–based model of moving systems, that does not contain the contradiction. While Modern Mechanics shares a common mathematical kernel with relativity, it uses different equations that can be viewed as an enhancement, or improvement, to the special relativity mathematics. Experimentally, Modern Mechanics yields E=mc^2 and produces a quantitatively better result for the Michelson-Morley experiment. Conceptually, Modern Mechanics differs from relativity because it removes the contradiction, concepts, and restrictions associated with relativity; integrates kinematics and electromagnetism while retaining the translation equation for moving systems; and offers novel insights into Einstein’s two–system relativity theory derivation (including discussing where and how the inequality is introduced).