Three Dimensions from a Recognition Requirement

Why does space have three dimensions? We study a recognizer with finitely many independent binary distinctions, one changed by each elementary act. Its states form a cube graph, and a shortest closed run through every state realizes a circle. We require a second recognition trajectory to represent a nonzero integral first-homology class in the complement of that circle. For a closed connected orientable smooth substrate with vanishing first integral homology, this requirement holds exactly in dimension three. The substrate is then an integral homology sphere, and the complementary class is measured by the linking number. We prove the converse for every substrate in this class and show that closedness and ambient first-homology vanishing are sharp hypotheses. A refinement construction supplies an explicit substrate from weighted binary records. The result concerns homological records; identifying this substrate with observed physical space is a further physical claim. The proof uses classical Alexander–Lefschetz duality.