Explain the Lean theorem `alexander_duality_circle_linking` in module `IndisputableMonolith.Foundation.AlexanderDuality`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

(1) Plain English meaning

The declaration alexander_duality_circle_linking asserts that a D-dimensional sphere admits non-trivial linking between two disjoint embedded circles if and only if D equals exactly 3. In other words, only three-dimensional space supports stable topological linking of closed curves; lower or higher dimensions make such linking impossible.

(2) Role in Recognition Science

This theorem supplies the topological justification for forcing spatial dimension D = 3 inside the RS framework. Non-trivial linking is required for ledger conservation (information that cannot be continuously deformed away). The result is imported into DimensionForcing where linking_requires_D3 uses it as the primary, T7-independent argument that D must be 3. Derived facts such as D3_has_linking and D3_compatible then follow directly.

(3) Reading the formal statement

The Lean signature is:

theorem alexander_duality_circle_linking (D : ℕ) :
    SphereAdmitsCircleLinking D ↔ D = 3

• D : ℕ is the candidate dimension.
• SphereAdmitsCircleLinking D is the predicate SphereAdmitsCircleLinking that encodes “non-trivial circle linking exists in S^D” via the definition CircleReducedCohomologyNontrivial ((D : ℤ) - 2).
• The biconditional ↔ is proved by unfolding the definition, rewriting with circle_reduced_cohomology_iff, and applying the omega tactic on the resulting arithmetic.

(4) Visible dependencies and certificates

The proof depends only on:

• SphereAdmitsCircleLinking (the linking predicate),
• circle_reduced_cohomology_iff (the concrete definition k = 1),
• the omega tactic for integer arithmetic. No external axioms remain; the module states “Status: 0 axioms (CLOSED 2026-04-22)”. The same theorem is re-exported and used in DimensionForcing to establish linking_requires_D3 and the uniqueness result dimension_unique.

(5) What the declaration does not prove

It does not derive Alexander duality itself (Hatcher Thm 3.44) from first principles, nor does it compute reduced cohomology of S¹ inside Mathlib’s singular-cohomology library. The cohomology fact is encoded by the concrete definition CircleReducedCohomologyNontrivial rather than proved from a general cohomology functor. The theorem also does not address physical emergence of 3D space or ledger dynamics; those steps occur in downstream modules such as RealityFromDistinction.