This paper presents a complete elementary reduction of the unsolved {0,1}-digit square case of Guy’s Problem F24. We prove that all non-trivial square solutions composed only of digits 0 and 1 correspond exactly to four exhaustive and mutually exclusive exponential Diophantine families. Two families yield linear exponential Diophantine equations with binary-digit product constraints, while the remaining two yield quadratic parameter conditions. Key elementary arithmetic identities are formally verified in Lean 4. A 10-adic lifting argument shows that purely modular local arguments cannot resolve the problem, requiring global logarithmic form bounds. This reduction fully converts an intractable decimal digit problem into a structured set of explicit Diophantine search problems.