Foundation Singular Sphere Is Zero Homology Of Contractible

The zero homology of a contractible space

In algebraic topology, the zero-th homology group of a space measures its connected components. For a path-connected space, this group is isomorphic to the integers, one copy for each component. The declaration isZero_homology_of_contractible in the Recognition Science library establishes a related but distinct fact: for any contractible space, all homology groups in positive degrees are zero. This is a standard result, but here it is proved in a machine-checked library of formal theorems, meaning the proof has been verified by a computer.

The library's proof proceeds by showing that the singular chain complex of a contractible space is homotopy equivalent to the chain complex of a point. Since the homology of a point is zero in positive degrees, the same holds for the contractible space. The key lemma is hn_pt_isZero, which states that the homology of a point in any positive degree is zero. This is combined with the fact that a contractible space is homotopy equivalent to a point, and homology is a homotopy invariant.

In Recognition Science, this result is part of a broader toolkit for building the foundation of the framework. It is used to establish the homology of spheres and other spaces that appear in the framework's construction. The declaration does not claim that the zero-th homology of a contractible space is zero; in fact, for a path-connected contractible space, it is isomorphic to the integers. It also does not claim that the space is contractible if its higher homology groups vanish; that is a separate and false statement in general.

The significance of this result is that it provides a clean, machine-checked foundation for the homology calculations used elsewhere in the framework. It is a building block, not a standalone discovery. The proof is rigorous and the statement is precise, which is exactly what a formal library should provide.

/-- **Stage A.** `H_m(pt) = 0` for `m ≠ 0`. -/
lemma hn_pt_isZero {m : ℕ} (hm : m ≠ 0) : IsZero (Hgrp (TopCat.of Unit) m) :=
  isZero_homology_of_totallyDisconnected (TopCat.of Unit) hm

What this page does not claim

The zero-th homology of a contractible space is zero. A space with zero higher homology is contractible. The declaration proves the Recognition Science framework's physical claims.

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

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