Foundation Singular Sphere Geometry Spheres Not Homotopy Equivalent

Spheres as distinct shapes

In topology, a sphere is the set of points at a fixed distance from a center. The circle is the one-dimensional sphere, the surface of a ball is the two-dimensional sphere, and the pattern continues upward. Two shapes are homotopy equivalent when one can be continuously squashed, stretched, or reshaped into the other without tearing or gluing. A circle and a disk are homotopy equivalent because the disk can be squeezed down to a point. A circle and a sphere's surface are not: the circle has a hole that the sphere's surface lacks, and no amount of continuous stretching removes it.

The general fact is that spheres of different dimensions are never homotopy equivalent. A one-dimensional sphere cannot become a two-dimensional sphere, and neither can become a three-dimensional one. This is a classical result, proved in the 1930s through the homology groups that Heinz Hopf and others developed. The framework's machine-checked library of formal theorems reproduces this fact for its own model of spheres, where each sphere is built inside a Euclidean space of one higher dimension. The declaration spheres_not_homotopyEquivalent states exactly this: the sphere of dimension n and the sphere of dimension m are not homotopy equivalent when n and m differ.

The framework's proof works by computing a homology group that distinguishes the spheres. For the one-dimensional sphere, the relevant homology group is not zero, while for the zero-dimensional sphere it is. The library also shows that all higher homology groups of a sphere vanish, a structural fact that supports the distinction. These are formal theorems, checked step by step by the machine, with no gaps in the logical chain.

In Recognition Science, this result matters because the framework builds its physical models from spheres. The proof that spheres of different dimensions are distinct shapes is a foundation for later claims about which spaces can represent which physical structures. It is a precise, local fact: it says nothing about how many dimensions space actually has, only that the spheres the framework uses are genuinely different objects.

The declaration does not claim that the framework's sphere model is the only way to build spheres, nor does it say that physical space has a particular dimension. It does not assert that homotopy equivalence is the only notion of sameness for shapes, nor that the framework's proof is the first or only proof of this classical fact. The result is a formal statement about a specific model, useful for what comes later in the framework's development.

/-- **`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

/-- **`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

/-- **`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 declaration does not claim that physical space has a particular dimension. The declaration does not claim that homotopy equivalence is the only notion of sameness for shapes. The declaration does not claim that the framework's proof is the first or only proof of this classical fact.

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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