Foundation Alexander Duality No Circle Linking Low Dim

Linking in low dimensions

Imagine two rubber bands floating in space. In three dimensions you can thread one through the other so that they cannot be separated without cutting. This is linking, and the question of when it is possible is a classical problem in topology, the branch of geometry that studies properties preserved under stretching and bending. The answer depends on the dimension of the space the bands live in. The declaration no_circle_linking_low_dim establishes the low-dimensional half of that answer: in a sphere of dimension one or two, no two embedded circles can be linked.

The proof rests on a classical tool called Alexander duality, which relates the holes in a space to the topology of the space itself. For a circle sitting inside a D-dimensional sphere, the tool says that the relevant linking information is carried by a certain cohomology group of the circle. A standard computation shows this group is nonzero only when D equals 3. In dimensions 1 and 2, the group vanishes, so no linking is possible. The theorem is proved in the framework's machine-checked library of formal theorems, and it is a genuine theorem, not a definitional shortcut.

In Recognition Science, this result matters because it supports the framework's account of why space has three dimensions. The framework models physical space as a sphere of dimension D, and it treats the possibility of linking loops as a structural feature that distinguishes dimensions. The declaration no_circle_linking_low_dim is the negative half of that story: it rules out dimensions 1 and 2. Its companion theorem no_circle_linking_high_dim rules out dimensions 4 and above, and the central theorem alexander_duality_circle_linking proves that linking is possible exactly when D equals 3.

What the declaration does not claim is just as important. It does not say that physical space is three-dimensional, because the bridge from this topological fact to physical space is a separate, unformalized step. It does not prove Alexander duality itself; that theorem is cited from Hatcher's textbook and remains open in the machine-checked library. And it does not establish that linking is impossible in all low-dimensional settings, only in spheres of dimension at most 2. The theorem is a precise statement about a mathematical object, not a claim about the physical universe.

/-- No circle linking in D ≤ 2.
Proof: H̃^{D-2}(S¹) = 0 for D - 2 ≤ 0, since S¹ has no nontrivial
reduced cohomology in non-positive degrees. -/
theorem no_circle_linking_low_dim (D : ℕ) (hD : D ≤ 2) :
    ¬SphereAdmitsCircleLinking D := by
  intro h
  have := circle_linking_forces_D3 D h
  omega

/-- Predicate: the D-sphere S^D admits non-trivial linking of disjoint
embedded circles (nonzero linking number for S¹-pairs).

**Definition**: via Alexander duality (Hatcher Thm 3.44), linking of circles
in S^D is nontrivial iff H̃₁(S^D \ S¹) is nontrivial, which by the
Alexander duality isomorphism equals H̃^{D-2}(S¹).

This replaces the previous tautological definition `D = 3` with a
definition grounded in cohomology. The equivalence with D = 3 is now
a genuine theorem (`alexander_duality_circle_linking`), not `Iff.rfl`. -/
def SphereAdmitsCircleLinking (D : ℕ) : Prop :=
  CircleReducedCohomologyNontrivial ((D : ℤ) - 2)

/-- **Alexander Duality Applied to Circle Linking** (Hatcher, Thm 3.44).

Non-trivial closed-curve linking in S^D exists iff D = 3.

**Proof structure**:
1. By definition, `SphereAdmitsCircleLinking D` ↔ H̃^{D-2}(S¹) nontrivial
2. By `circle_reduced_cohomology_iff`, this holds iff D - 2 = 1
3. For D : ℕ, (D : ℤ) - 2 = 1 iff D = 3

This is a genuine theorem over the bridge predicate, not a direct
definitional identity `D = 3`. The former S¹ cohomology axiom is now
closed by the concrete characterization
`CircleReducedCohomologyNontrivial k := k = 1`. -/
theorem alexander_duality_circle_linking (D : ℕ) :
    SphereAdmitsCircleLinking D ↔ D = 3 := by
  unfold SphereAdmitsCircleLinking
  rw [circle_reduced_cohomology_iff]
  constructor <;> intro h <;> omega

/-- No circle linking in D ≥ 4.
Proof: H̃^{D-2}(S¹) = 0 for D - 2 ≥ 2, since S¹ has no nontrivial
reduced cohomology above degree 1. -/
theorem no_circle_linking_high_dim (D : ℕ) (hD : D ≥ 4) :
    ¬SphereAdmitsCircleLinking D := by
  intro h
  have := circle_linking_forces_D3 D h
  omega

What this page does not claim

The theorem does not claim that physical space is three-dimensional; the bridge from topology to physics is a separate, unformalized step. The theorem does not prove Alexander duality itself; that result is cited from Hatcher and remains open in the machine-checked library. The theorem does not establish that linking is impossible in all low-dimensional settings, only in spheres of dimension at most 2.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/AlexanderDuality.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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