Foundation Singular Sphere Geometry Sphere Homology Vanish

The vanishing theorem

The theorem sphere_homology_vanish concerns the homology of spheres, a standard tool in topology that counts the holes of each dimension. For the ordinary circle, the theorem says that its one-dimensional hole is real and nonzero, while all other hole dimensions vanish. For the two-dimensional sphere, the two-dimensional hole (the enclosed volume) is nonzero, but the one-dimensional hole vanishes. In general, for an n-dimensional sphere, the theorem states that every homology group in dimension k is zero whenever k is neither 0 nor n.

The proof is fully formal and machine-checked in the framework's library of formal theorems. It builds the sphere as the set of points at distance one from the origin in Euclidean space, then covers it with two open sets: the sphere minus the north pole and the sphere minus the south pole. Each of these sets is contractible, meaning it can be shrunk to a point, and their intersection is homeomorphic to a punctured hyperplane. The theorem then follows from a standard Mayer-Vietoris argument, which computes the homology of a space from the homology of its pieces. The key geometric fact is that the intersection of the two open sets is itself contractible in all dimensions except one, and that remaining dimension is exactly the one that produces the single nonzero hole of the sphere.

In Recognition Science, this result is part of a larger effort to show that ordinary mathematical structures, such as spheres and their topology, are not assumed but derived from the framework's core postulates. The framework models reality as a discrete record of recognition events, and it proves that this record forces a three-dimensional spatial structure. The vanishing theorem is a step in that derivation: it establishes that the topological objects the framework uses behave in the standard way, with no exotic higher-dimensional holes appearing. It is a theorem about the framework's own sphere construction, not a claim about physical space directly.

The theorem does not claim that all homology groups of a sphere vanish. It explicitly excludes the cases k = 0 and k = n, where the homology is nonzero: the zeroth homology counts the connected components (one for a sphere), and the top homology counts the enclosed volume (one for a sphere). It also does not claim anything about the homology of other spaces, nor does it assert that physical space is a sphere. The theorem is a precise statement about the framework's formal sphere, and its proof is complete and machine-checked.

/-- **Stage D vanishing.** `H_k(Sⁿ) = 0` for `1 ≤ k`, `k ≠ n`. -/
theorem sphere_homology_vanish :
    ∀ n k : ℕ, 1 ≤ k → k ≠ n → IsZero (Hgrp (Sph n) k) := by
  intro n
  induction n with
  | zero =>
      intro k hk _
      exact isZero_sph0 (by omega)
  | succ n ih =>
      intro k hk hkn
      match k, hk with
      | 1, _ =>
          have hn : n ≠ 0 := by omega
          obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn
          exact isZero_h1_of_contractible (isOpen_coverU (m + 2))
            (isOpen_coverV (m + 2)) (coverU_union_coverV (m + 2))
      | (k + 2), _ =>
          exact (ih (k + 1) (by omega) (by omega)).of_iso (suspensionIso n k)

/-- **`H₁(S¹) ≠ 0`.** If it vanished, the Mayer-Vietoris connecting map
out of it would be zero, and exactness would kill the point-difference
class, contradicting its nonzero pairing. -/
theorem h1_s1_ne_zero : ¬ IsZero (Hgrp (Sph 1) 1) := by
  intro hZ
  have hδ : mvδ (isOpen_coverU 1) (isOpen_coverV 1)
      (coverU_union_coverV 1) 0 = 0 :=
    hZ.eq_of_src _ _
  have hex := mv_exact₁ (isOpen_coverU 1) (isOpen_coverV 1)
    (coverU_union_coverV 1) 0
  rw [ShortComplex.moduleCat_exact_iff] at hex
  have hker : mvPair (coverU 1) (coverV 1) 0 (diffClass (1 : ℤ)) = 0 := by
    rw [← ModuleCat.comp_apply, diffClass_mvPair, zeroApp]
  obtain ⟨w, hw⟩ := hex (diffClass (1 : ℤ)) hker
  have hw' : mvδ (isOpen_coverU 1) (isOpen_coverV 1)
      (coverU_union_coverV 1) 0 w = diffClass (1 : ℤ) := hw
  rw [hδ, zeroApp] at hw'
  have h1 : augH Wc arcA isClopen_arcA (diffClass (1 : ℤ)) = (1 : ℤ) := by
    rw [← ModuleCat.comp_apply, diffClass_pairing, ModuleCat.id_apply]
  rw [← hw', map_zero] at h1
  exact one_ne_zero h1.symm

What this page does not claim

The theorem does not claim that all homology groups of a sphere vanish. The theorem does not claim that physical space is a sphere. The theorem does not claim anything about the homology of spaces other than spheres.

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

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