Foundation Circle H1 Computation Circle H1 Ziso Int Of Nonempty Homotopy Equiv O

The circle's first homology

In algebraic topology, the first homology group of a circle, written H₁(S¹; ℤ), counts the one-dimensional holes in a space. For the circle there is exactly one such hole, and the group is the integers ℤ: any loop winds around some integer number of times. This is a classical fact, known since the birth of homology theory in the late 19th century, and it is the foundation for computing homology of higher-dimensional spheres and other spaces built from circles.

The framework's machine-checked library of formal theorems proves this fact for its own finite model of the circle. The model is a chain complex, a sequence of abelian groups connected by boundary maps, with ℤ placed in degree 1 and nothing elsewhere. The proof shows this complex's degree-1 homology is isomorphic to ℤ, and that this model is homotopy equivalent to the standard singular chain complex of the topological circle. A homotopy equivalence preserves homology, so the result transfers: the circle's first homology is ℤ.

In Recognition Science, this computation is a stepping stone. The framework's broader goal is to derive three spatial dimensions from a ledger of recognition events, and the circle's homology is a piece of the topological scaffolding. But the declaration in question, circleH1ZIsoIntOfNonemptyHomotopyEquivOrdinaryCellularAtOne, proves only the algebraic core. It does not connect this finite model to the framework's imported singular homology object for the standard sphere, nor does it feed the strict bridge to three-dimensional space. Those connections remain open targets.

What the result does establish is precise: given a homotopy equivalence between the singular chain complex of the circle and the finite cellular model, the first homology is the integers. The proof is machine-checked, meaning every step is verified by the kernel of the proof assistant. It is not a claim about the physical circle in space, nor about the framework's derivation of dimensions. It is a formal theorem about a specific algebraic model and its homology.

/-- Proposition-facing form of `ordinaryCellularCircleChainModelH1IsoInt`. -/
theorem ordinaryCellularCircleChainModelH1NonemptyIsoInt :
    Nonempty (ordinaryCellularCircleChainModel.homology 1 ≅ ModuleCat.of ℤ ℤ) :=
  ⟨ordinaryCellularCircleChainModelH1IsoInt⟩

/-- The same homotopy-equivalence bridge stated in the exact proposition shape
used by `MathlibCohomologyBridge`. -/
theorem circleH1ZIsoIntOfHomotopyEquivOrdinaryCellular
    (e : HomotopyEquiv sphereOneSingularIntChainComplex ordinaryCellularCircleChainModel)
    [∀ i, sphereOneSingularIntChainComplex.HasHomology i] :
    MathlibCohomologyBridge.circleH1ZIsoInt :=
  circleH1ZIsoIntOfQuasiIsoAtOrdinaryCellular e.hom

/-- The same homotopy-equivalence bridge stated in the exact proposition shape
used by `MathlibCohomologyBridge`. -/
theorem circleH1ZIsoIntOfHomotopyEquivOrdinaryCellular
    (e : HomotopyEquiv sphereOneSingularIntChainComplex ordinaryCellularCircleChainModel)
    [∀ i, sphereOneSingularIntChainComplex.HasHomology i] :
    MathlibCohomologyBridge.circleH1ZIsoInt :=
  circleH1ZIsoIntOfQuasiIsoAtOrdinaryCellular e.hom

What this page does not claim

The declaration does not prove the circle's homology for the standard imported singular homology object, only for the framework's finite model. The declaration does not establish any physical fact about circles in space or the framework's derivation of dimensions. The homotopy equivalence to the standard singular chain complex is assumed, not proved, in this declaration.

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/CircleH1Computation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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