Foundation Circle H1 Computation Ordinary Cellular To Reduced Comp Reduced Cellu

The round trip

The declaration ordinaryCellularToReduced_comp_reducedCellularToOrdinary_f_one is a theorem in the framework's machine-checked library of formal theorems. It states that if you take the ordinary cellular chain model of a circle, map it to the reduced cellular model, and then map back, the result at degree 1 is the identity map. In plainer terms, the two models describe the same one-dimensional piece of the circle, and the round trip leaves it unchanged.

To see why this matters, consider two bookkeeping systems for the same object. The ordinary model records a copy of the integers in degree 0 and another copy in degree 1, with all boundary maps zero. The reduced model keeps only the degree 1 copy. The theorem shows that the map from the ordinary model to the reduced one, followed by the map back, is the identity on that degree 1 piece. This is a coherence check: the two descriptions agree on the part that carries the circle's first homology group, which is the integers.

The theorem is a local result inside a workbench module. It does not claim that either model is the actual topological circle, nor that the first homology group of the true circle is the integers. Those claims live in other declarations, and the bridge to the standard topological object is still an open target. The theorem only certifies the algebraic round trip between the two cellular models.

In the framework's account, this kind of bookkeeping is not idle. The framework models physical structure as a discrete record of recognition events, and the circle's first homology group is a candidate for a fundamental invariant. Before that invariant can be used, the framework must show its internal models are consistent. This theorem is one such consistency check, a small but necessary step toward the larger goal of connecting the algebraic model to the actual sphere.

/-- The other composite need not be the identity on the ordinary two-cell
complex, because the ordinary model has an additional degree-`0` generator.
It is, however, the identity in degree `1`, the degree relevant to the circle
homology computation. -/
theorem ordinaryCellularToReduced_comp_reducedCellularToOrdinary_f_one :
    (ordinaryCellularToReducedChainMap ≫ reducedCellularToOrdinaryChainMap).f 1 =
      𝟙 (ordinaryCellularCircleChainModel.X 1) := by
  simp [ordinaryCellularToReducedChainMap_f_one,
    reducedCellularToOrdinaryChainMap_f_one]

/-- The other composite need not be the identity on the ordinary two-cell
complex, because the ordinary model has an additional degree-`0` generator.
It is, however, the identity in degree `1`, the degree relevant to the circle
homology computation. -/
theorem ordinaryCellularToReduced_comp_reducedCellularToOrdinary_f_one :
    (ordinaryCellularToReducedChainMap ≫ reducedCellularToOrdinaryChainMap).f 1 =
      𝟙 (ordinaryCellularCircleChainModel.X 1) := by
  simp [ordinaryCellularToReducedChainMap_f_one,
    reducedCellularToOrdinaryChainMap_f_one]

What this page does not claim

The theorem does not claim that either cellular model is the actual topological circle. The theorem does not claim that the first homology group of the true circle is the integers. The theorem does not claim the bridge to the standard topological object is complete.

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)

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Derived articles

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