(1) What the declaration says in plain English
SphereAdmitsCircleLinking D is a predicate asserting that the D-dimensional sphere admits non-trivial linking between two disjoint embedded circles. It is defined directly as the statement that the reduced cohomology of the circle in degree D-2 is nontrivial: SphereAdmitsCircleLinking.
(2) Why it matters in Recognition Science
This predicate supplies the topological foundation for forcing spatial dimension D = 3. Only D = 3 permits stable, non-trivial linking that supports ledger conservation; lower or higher dimensions make linking trivial. It is imported into dimension forcing and used to prove linking_requires_D3.
(3) How to read the formal statement
In Lean syntax the declaration is def SphereAdmitsCircleLinking (D : ℕ) : Prop := CircleReducedCohomologyNontrivial ((D : ℤ) - 2). It takes a natural number D and returns a proposition. The body applies the predicate CircleReducedCohomologyNontrivial (defined as k = 1) to the integer D-2. The surrounding theorem alexander_duality_circle_linking then shows this holds exactly when D = 3 via omega after unfolding.
(4) Visible dependencies or certificates in the supplied source
It depends on CircleReducedCohomologyNontrivial and the reflexivity theorem circle_reduced_cohomology_iff. Downstream certificates include D3_admits_circle_linking, D3_has_linking, and no_circle_linking_high_dim. The module reports zero axioms and zero sorry in the linking chain.
(5) What this declaration does not prove
The definition encodes the consequence of Alexander duality but does not contain a formalization of the duality isomorphism itself. It does not construct explicit linking witnesses such as the Hopf link. It also does not derive the cohomology computation for S¹ from a singular-cohomology functor; that computation is captured only by the concrete predicate definition.