The Unique Dimension in Which an Embedded Circle Links: A Machine-Checked Proof that Complement H1 Forces D = 3

Say that a natural number D detects nontrivial linking if some topological embedding f: S¹ → S^D has a complement whose first singular homology with integer coefficients is nonzero. We report a machine-checked proof, in Lean 4 over Mathlib, of the two halves of a dimension-forcing statement: D = 3 detects nontrivial linking, and every D that detects nontrivial linking equals 3. The uniqueness half is uniform over all D ∈ ℕ and holds for arbitrary topological embeddings; no smoothness, PL structure, or tameness is assumed anywhere, so wild circles such as those of Antoine and Fox-Artin are covered. The proof does not pass through full Alexander duality. It combines an explicit retraction for existence at D = 3 with a Mayer-Vietoris reduction of circle complements to arc complements and a compact-support bisection proof that embedded arcs in S^D have H1-acyclic complements in every dimension (the arc case of Hatcher's Proposition 2B.1). The target proposition is content-typed on Mathlib's singular homology of the actual complement space, a design forced by two failed earlier formulations that were inhabitable by arithmetic encodings; a standing gate script audits the statement's type-level content against the known cheat shapes. All results carry zero sorry and depend only on the three standard Lean axioms (propext, Classical.choice, Quot.sound). A physical interpretation within the Recognition Science program is stated separately and explicitly quarantined from the mathematical claims.