Classical C1 Smoothness is Impossible at Finite Resolution

Note: This paper has been significantly revised and now forms Part 1 of "The Discrete Reality Tetralogy" (2025). The complete collection includes three additional papers that establish why derivatives cannot exist at finite resolution, present empirical evidence for discrete measurement, and explain why continuous models work through the Universal Averaging Principle. For the full framework, please see [The Discrete Reality Tetralogy].We prove, in three independent ways, that classical C¹ smoothness cannot occur in any discrete model with a nonzero resolution floor. The result formalizes a common intuition: you can refine triangles to approach a sphere but never be one at finite resolution. Our topology proof rules out positive-dimensional smooth structure in discrete metric spaces; our calculus proof shows finite step size prevents derivatives; and our geometry proof shows polyhedra cannot equal S². We conclude that exact smoothness exists only in the limit as ℓ_min → 0, and that discrete analogs are the appropriate tools at fixed scale.