Foundation Mathlib Cohomology Bridge Circle H1 Mathlib Computation Iff Iso Int

The bridge contract

In algebraic topology, the first homology group of a circle, H₁(S¹), is a fundamental object. It is the abelian group that records how loops on the circle can be combined, and it is isomorphic to the integers ℤ. This is a classical result: the circle has one "hole," and winding around it any integer number of times gives a complete description of its one-dimensional structure.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration named circleH1MathlibComputation_iff_iso_int. In plain language, this theorem establishes an equivalence: a machine-checked computation of the circle's first homology group exists if and only if that group is isomorphic to the integers. The theorem is a formal bridge. It connects the existence of a specific computational object, a CircleH1MathlibComputation, with the mathematical statement that the group is isomorphic to ℤ. The framework models this as a contract: the computation is available precisely when the isomorphism holds.

The theorem does not itself prove that the circle's first homology group is isomorphic to the integers. That fact is a classical result of algebraic topology, and the framework's declaration does not derive it from scratch. Instead, the theorem states a conditional equivalence. If a computation of the group exists, then the isomorphism holds. Conversely, if the isomorphism holds, then such a computation can be constructed. The declaration is a formal statement about the relationship between two mathematical objects, not a new proof of the circle's homology.

What the declaration does establish is a precise correspondence. It ensures that the framework's backend, when it has a computation of the circle's first homology group, can rely on the fact that this group is isomorphic to the integers. This is a necessary step in the framework's larger project of deriving three-dimensional space from recognition events. The bridge contract is a small but load-bearing piece of that architecture. It does not, however, claim that the isomorphism itself is a new result, nor does it claim that the computation has actually been carried out in the library. The theorem states the equivalence; the existence of the computation remains a separate matter.

The declaration also implies that the circle's first homology group is not zero, a fact that follows from the isomorphism to ℤ. This is a consequence of the bridge, not an independent claim. The framework's library uses this to support its linking structure, which in turn forces the dimension to be three. The bridge contract is thus a formal guarantee that a key computational step is consistent with the classical mathematics it replaces.

/-- The final Mathlib computation interface is equivalent to the strong target
`H_1(S¹; ℤ) ≅ ℤ`. -/
theorem circleH1MathlibComputation_iff_iso_int :
    Nonempty CircleH1MathlibComputation ↔ circleH1ZIsoInt := by
  constructor
  · rintro ⟨C⟩
    exact C.h1_iso_int
  · intro hiso
    exact ⟨circleH1MathlibComputation_of_iso_int hiso⟩

/-- The final Mathlib computation interface is equivalent to the strong target
`H_1(S¹; ℤ) ≅ ℤ`. -/
theorem circleH1MathlibComputation_iff_iso_int :
    Nonempty CircleH1MathlibComputation ↔ circleH1ZIsoInt := by
  constructor
  · rintro ⟨C⟩
    exact C.h1_iso_int
  · intro hiso
    exact ⟨circleH1MathlibComputation_of_iso_int hiso⟩

/-- Computing `H_1(S¹; ℤ)` as `ℤ` closes the required nonvanishing target. -/
theorem circleH1ZNonzero_of_iso_int
    (hiso : circleH1ZIsoInt) : circleH1ZNonzero := by
  intro hzero
  rcases hiso with ⟨e⟩
  have hz : CategoryTheory.Limits.IsZero (ModuleCat.of ℤ ℤ) :=
    CategoryTheory.Limits.IsZero.of_iso hzero e.symm
  exact intModuleCat_not_isZero hz

What this page does not claim

This answer does not claim that the circle's first homology group is isomorphic to the integers as a new theorem proved by the framework. This answer does not claim that the machine-checked computation of the circle's homology has actually been carried out in the library. This answer does not claim that the bridge contract itself proves the dimension of space is three.

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

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