Foundation Circle H1 Computation Singular Homology Functor Sphere One Int Nonemp

The circle's first homology

In algebraic topology, the first homology group of a space measures its one-dimensional holes. For a circle, the standard result is that this group is isomorphic to the integers: H₁(S¹; ℤ) ≅ ℤ. This means every loop on the circle can be classified by an integer winding number, the number of times it wraps around, and that this classification is complete and unambiguous.

The declaration in question, singularHomologyFunctorSphereOneIntNonemptyIsoOfQuasiIsoAtOrdinaryCellular, is a formal statement in the framework's machine-checked library of formal theorems. It establishes that a particular algebraic model of the circle, built from cellular chains, has its one-dimensional homology isomorphic to the integers. The proof works by showing this cellular model is equivalent, in a precise sense called a quasi-isomorphism at degree one, to a simpler chain complex that is known to have integer homology.

This is a technical but meaningful step. It provides a verified algebraic core for the circle's homology, a building block that could later connect to the full topological object. However, the declaration does not itself prove the full theorem for the actual topological circle. The connection to the standard singular homology of the geometric sphere is stated as a separate, unproved condition called the geometric bridge. The workbench documentation is explicit: it does not replace the standard circle object and does not feed the strict three-dimensional forcing bridge until that real equivalence is proved.

In Recognition Science, this algebraic result is a piece of a larger chain of reasoning about the structure of space. But its role here is narrow: it is a proved lemma about a model, not a claim about physical space. The framework's own documents mark the physical recognition-to-linking bridge as open. What this declaration proves is the algebraic fact; what it does not claim is the topological or physical transport.

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

/-- The collapse from the ordinary two-cell cellular circle model to the reduced
degree-`1` model is a quasi-isomorphism in degree `1`.

The proof uses Mathlib's zero-differential short-complex criterion: in degree
`1`, both relevant short complexes have zero differentials, and the middle
component of the collapse map is an isomorphism on the chosen integer
generator. -/
theorem ordinaryCellularToReducedChainMap_quasiIsoAt_one :
    QuasiIsoAt ordinaryCellularToReducedChainMap 1 := by
  rw [quasiIsoAt_iff]
  rw [ShortComplex.quasiIso_iff_isIso_liftCycles _ (by
    simp [ordinaryCellularCircleChainModel_d_eq_zero]) (by
    simp [ordinaryCellularCircleChainModel_d_eq_zero]) (by
    simp)]
  let S₂ :=
    (HomologicalComplex.shortComplexFunctor (ModuleCat ℤ) (ComplexShape.down ℕ) 1).obj
      reducedCellularCircleChainModel
  let φ :=
    ((HomologicalComplex.shortComplexFunctor (ModuleCat ℤ) (ComplexShape.down ℕ) 1).map
      ordinaryCellularToReducedChainMap)
  let w : φ.τ₂ ≫ S₂.g = 0 := by
    dsimp [S₂, φ]
    simp
  change IsIso (S₂.liftCycles φ.τ₂ w)
  haveI : IsIso S₂.iCycles := S₂.isIso_iCycles (by
    dsimp [S₂])
  haveI : IsIso (S₂.liftCycles φ.τ₂ w ≫ S₂.iCycles) := by
    rw [ShortComplex.liftCycles_i]
    dsimp [S₂, φ]
    rw [ordinaryCellularToReducedChainMap_f_one]
    infer_instance
  exact IsIso.of_isIso_comp_right (S₂.liftCycles φ.τ₂ w) S₂.iCycles

/-- The remaining geometric chain-level bridge for the strict T8 circle-H1
closure: Mathlib's singular chain complex for `TopCat.sphere 1` is chain
homotopy equivalent to the ordinary two-cell cellular circle model. -/
def circleH1GeometricBridge : Prop :=
  Nonempty
    (HomotopyEquiv sphereOneSingularIntChainComplex ordinaryCellularCircleChainModel)

What this page does not claim

This declaration does not prove the full theorem that the singular homology of the topological circle S¹ is isomorphic to ℤ. It does not establish the physical bridge from recognition to linking that the framework's three-dimensional forcing chain requires. The result is an algebraic statement about a model, not a claim about the structure of physical space.

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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