Foundation Linking Vanishing High Dim Range Arc Plus Union Arc Minus

Two arcs that cover a circle

A circle is the set of points at a fixed distance from a center. One way to picture it is as the unit circle in the plane, with coordinates (x, y) satisfying x² + y² = 1. The declaration range_arcPlus_union_arcMinus concerns two specific arcs on this circle. The first, arcPlus, is the set of points where the y-coordinate is nonnegative, the upper semicircle from the rightmost point to the leftmost point. The second, arcMinus, is the lower semicircle, where the y-coordinate is nonpositive. The statement is that the union of these two arcs is the entire circle.

This is not a deep geometric fact, but it is a precise one, and the framework's machine-checked library of formal theorems proves it. The proof is not a single step; it relies on the definitions of the arcs and on the fact that for any real number, either it is nonnegative or it is nonpositive. That dichotomy is the entire content of the argument. The result is a lemma, not a theorem about the physical world. It is a statement about a circle in the plane, and it does not by itself say anything about three-dimensional space or about linking.

The arcs themselves are defined with care. Each is the image of a continuous map from the unit interval into the circle, and each is an embedding, meaning the map is injective and its image is a copy of the interval. The two arcs meet only at the two endpoints, the rightmost and leftmost points of the circle. The proof that their union is the whole circle uses the fact that the two ranges, the upper and lower semicircles, have no gap between them.

In Recognition Science, this lemma is a small piece of a larger argument about why nontrivial linking is detected only in dimension three. The framework models a physical recognition event as a discrete record, and it proves that certain topological configurations, such as a circle embedded in a sphere, can be detected only when the ambient dimension is three. The covering lemma is a step in that proof, but it is not the proof itself. It establishes a fact about a circle, not about space.

What the declaration does not claim is equally clear. It does not claim that the two arcs are disjoint; they share two points. It does not claim that the arcs are the only way to cover the circle; many other pairs of arcs would do. It does not claim that the union is connected in any special way; the union is the whole circle, which is connected, but that is a separate fact. The lemma is a narrow, precise statement, and its value lies in being a verified step in a longer chain.

/-- 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 upper semicircle is the range of `arcPlus`. -/
lemma range_arcPlus :
    Set.range arcPlus = {z : ↥(Sph 1) | 0 ≤ amb z 1} := by
  ext z
  constructor
  · rintro ⟨t, rfl⟩
    show (0 : ℝ) ≤ amb (arcFun 1 (one_pow 2) t) 1
    rw [arcFun_coord1, one_mul]
    exact Real.sqrt_nonneg _
  · intro hz
    exact ⟨arcParam z, arcFun_arcParam 1 (one_pow 2) z
      (by rw [one_mul, abs_of_nonneg hz])⟩

/-- The lower semicircle is the range of `arcMinus`. -/
lemma range_arcMinus :
    Set.range arcMinus = {z : ↥(Sph 1) | amb z 1 ≤ 0} := by
  ext z
  constructor
  · rintro ⟨t, rfl⟩
    show amb (arcFun (-1) neg_one_sq t) 1 ≤ 0
    rw [arcFun_coord1]
    have := Real.sqrt_nonneg (1 - (1 - 2 * (t : ℝ)) ^ 2)
    nlinarith
  · intro hz
    exact ⟨arcParam z, arcFun_arcParam (-1) neg_one_sq z
      (by rw [abs_of_nonpos hz]; ring)⟩

/-- 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⟩

What this page does not claim

The two arcs are disjoint; they share two points. The union of the arcs is connected in a way that the arcs themselves are not. The covering lemma by itself establishes anything about three-dimensional space.

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)

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