Logical Dependency Factorization: A Formal Framework

This paper introduces a technique for taking classical impossibility theorems—results that say "X cannot exist"—and finding versions of X that do exist in alternative mathematical universes (called topoi). The key insight: Classical impossibility proofs usually work by showing "if X existed, we'd need a constructive proof of something, but no such proof exists, therefore X doesn't exist." This reasoning depends on classical logic principles like the Law of Excluded Middle. But in mathematical worlds where those classical principles fail, the absence of a constructive proof doesn't prove non-existence anymore. That logical gap becomes the exact space where counterexamples can live. The LDF Framework does four things: Extract what logical principles a classical proof actually depends on Find mathematical universes (topoi) where those principles fail Construct counterexamples using the "room" created by the logical failure Systematize this into a repeatable, functorial procedure Instead of seeing classical impossibility as a dead end, LDF treats it as a roadmap: it tells you exactly where to look for counterexamples in non-classical settings.