Foundation Singular Sphere Is Zero H1 Of Contractible

The zero hole

In algebraic topology, a one-dimensional hole is a loop that cannot be pulled tight to a single point. A circle has one, a figure-eight has two, and a disk has none. The declaration isZero_h1_of_contractible in the framework's machine-checked library of formal theorems establishes that any space which is contractible, meaning it can be continuously deformed to a point, has zero such holes. The proof is a formal theorem, checked by a computer, and it is a standard result in the field.

The classical statement is this: for a contractible space X, the first singular homology group H₁(X) with integer coefficients is the trivial group, containing only the zero element. The framework's library proves this using its own construction of singular homology, building the chain complex from continuous maps of standard simplices. The key lemma shows that the homology of a point in every positive degree is zero, and a homotopy equivalence between the contractible space and a point preserves homology groups.

The proof proceeds by constructing a chain map from the singular complex of the space to the complex of a point, using a chosen basepoint and the contraction homotopy. The framework's library proves that this map induces an isomorphism on homology in degree zero, and that the higher homology groups of a point vanish. Since homology is a homotopy invariant, the contractible space shares the point's homology in all degrees, so H₁ is zero.

In Recognition Science, this theorem is part of the toolkit for building the framework's foundational structures. It is a piece of standard mathematics, formalized in the library, and it does not by itself make any claim about recognition events or the cost function. The declaration is a building block, not a result about the framework's core postulates.

What the declaration does not claim is equally precise. It does not claim that every space with zero first homology is contractible; a sphere has zero first homology but is not contractible. It does not claim anything about higher-dimensional holes, which are governed by higher homology groups. And it does not claim that the framework's physical theory of recognition depends on this result; it is a formal tool, not a physical law.

/-- **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

/-- **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 space with zero first homology is contractible. The theorem applies to holes in dimensions higher than one. The framework's physical theory of recognition depends on this formal result.

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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