Foundation Linking Vanishing High Dim Range Arc Plus Inter Arc Minus

Two arcs, two poles

In topology, a circle can be placed inside a higher-dimensional sphere as a curve that does not cross itself. The declaration range_arcPlus_inter_arcMinus concerns two particular such curves on a one-dimensional sphere, the circle. One curve, called arcPlus, is the set of points whose second coordinate is nonnegative; the other, arcMinus, is the set where that coordinate is nonpositive. The statement proves that the intersection of these two sets is exactly the two points where the first coordinate is at its extreme values, conventionally called east and west. In plain language, the two arcs share only their endpoints.

The proof is a lemma in the framework's machine-checked library of formal theorems, part of a file named LinkingVanishingHighDim. It is not an isolated curiosity. The lemma feeds into a larger theorem, forces_D3_of_arcAcyclic, which states that if every such arc in dimensions other than three has a complement with trivial first homology, then any dimension that detects nontrivial linking must be three. The two-arc intersection result is a concrete, checkable piece of that larger claim. It confirms that the two arcs cover the circle when taken together and overlap only at the poles, a fact the larger argument relies on.

What the declaration does not claim is just as important. It does not assert that the two arcs are the only possible curves on a circle, nor that every circle embedded in a higher-dimensional sphere has this property. It does not prove that three-dimensional space is the only possible dimension; that conclusion requires additional hypotheses about arc complements, which the larger theorem states explicitly. The lemma is a geometric fact about two specific maps, not a statement about all possible embeddings.

In the context of the framework, this result is a small but necessary gear. It shows that the two arcs, defined by simple coordinate conditions, behave exactly as expected: they partition the circle and meet at the two poles. This is the kind of foundational fact that a machine-checked proof can verify with certainty, and it is one of many such facts that, taken together, support the framework's larger claims about the structure of space.

/-- The two semicircles meet exactly in the east and west points. -/
lemma range_arcPlus_inter_arcMinus :
    Set.range arcPlus ∩ Set.range arcMinus =
      {(eastP : ↥(Sph 1)), (westP : ↥(Sph 1))} := by
  rw [range_arcPlus, range_arcMinus]
  ext z
  simp only [Set.mem_inter_iff, Set.mem_setOf_eq, Set.mem_insert_iff,
    Set.mem_singleton_iff]
  constructor
  · rintro ⟨h1, h2⟩
    have hz1 : amb z 1 = 0 := le_antisymm h2 h1
    have hz0 : amb z 0 ^ 2 = 1 := by
      have := coord_sq_add_sq z
      nlinarith
    have habs : |amb z 0| = 1 := abs_eq_one_of_sq_eq_one hz0
    rcases (abs_eq zero_le_one).mp habs with h | h
    · left
      apply amb_injective
      apply esp1_ext
      · rw [h, amb_eastP_zero]
      · rw [hz1]
        show (0 : ℝ) = EuclideanSpace.single (0 : Fin 2) (1 : ℝ) 1
        rw [EuclideanSpace.single_apply, if_neg (by decide)]
    · right
      apply amb_injective
      apply esp1_ext
      · rw [h, amb_westP_zero]
      · rw [hz1]
        have hc : amb (westP : ↥(Sph 1)) =
            -(EuclideanSpace.single (0 : Fin 2) (1 : ℝ)) := coe_neg_sphere eastP
        rw [hc, show (-(EuclideanSpace.single (0 : Fin 2) (1 : ℝ))) 1 =
          -(EuclideanSpace.single (0 : Fin 2) (1 : ℝ) 1) from rfl,
          EuclideanSpace.single_apply, if_neg (by decide), neg_zero]
  · rintro (rfl | rfl)
    · constructor
      · rw [show amb (eastP : ↥(Sph 1)) 1 =
          EuclideanSpace.single (0 : Fin 2) (1 : ℝ) 1 from rfl,
          EuclideanSpace.single_apply, if_neg (by decide)]
      · rw [show amb (eastP : ↥(Sph 1)) 1 =
          EuclideanSpace.single (0 : Fin 2) (1 : ℝ) 1 from rfl,
          EuclideanSpace.single_apply, if_neg (by decide)]
    · have hw : amb (westP : ↥(Sph 1)) 1 = 0 := by
        have hc : amb (westP : ↥(Sph 1)) =
            -(EuclideanSpace.single (0 : Fin 2) (1 : ℝ)) := coe_neg_sphere eastP
        rw [hc, show (-(EuclideanSpace.single (0 : Fin 2) (1 : ℝ))) 1 =
          -(EuclideanSpace.single (0 : Fin 2) (1 : ℝ) 1) from rfl,
          EuclideanSpace.single_apply, if_neg (by decide), neg_zero]
      exact ⟨le_of_eq hw.symm, le_of_eq hw⟩

/-- The two semicircles cover the circle. -/
lemma range_arcPlus_union_arcMinus :
    Set.range arcPlus ∪ Set.range arcMinus = Set.univ := by
  rw [range_arcPlus, range_arcMinus]
  ext z
  simp only [Set.mem_union, Set.mem_setOf_eq, Set.mem_univ, iff_true]
  exact le_total 0 (amb z 1)

/-- **The bridge's uniqueness half, conditional on the arc frontier**:
granting arc-complement acyclicity in every dimension `D ≥ 2`, `D ≠ 3`,
nontrivial linking forces `D = 3`.  Dimensions `0` and `1` are the banked
unconditional results (`LinkingVanishingLowDim`). -/
theorem forces_D3_of_arcAcyclic
    (harc : ∀ D, 2 ≤ D → D ≠ 3 → ArcComplementsAcyclic D) :
    ∀ D, DetectsNontrivialLinking D → D = 3 := by
  intro D hdet
  by_contra hne
  match D, hne with
  | 0, _ => exact LinkingVanishingLowDim.not_detects_zero hdet
  | 1, _ => exact LinkingVanishingLowDim.not_detects_one hdet
  | (n + 2), hne =>
      exact not_detects_of_arcAcyclic (n + 2) (by omega) hne
        (harc (n + 2) (by omega) hne) hdet

What this page does not claim

The declaration does not prove that the two arcs are the only possible curves on a circle. It does not show that every circle embedded in a higher-dimensional sphere has this intersection property. It does not by itself force three-dimensional space; that conclusion requires additional hypotheses.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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