Foundation Circle Param Constant Sphere One Singular One Simplex Face Zero

The constant simplex

In topology, a singular 1-simplex is a continuous map from a line segment into a space. A constant singular 1-simplex is the degenerate case: a map that sends the entire segment to a single point. The declaration constantSphereOneSingularOneSimplex_face_zero is a machine-checked theorem about this object on the unit circle. It states that the left endpoint of this constant path is the basepoint 0-simplex, and its companion theorem states the same for the right endpoint. Together they prove both faces of the constant simplex coincide: the path starts and ends at the same point.

The result is a foundational anchor in the Recognition Science framework's machine-checked library of formal theorems. It is not a statement about the circle's fundamental group or its non-trivial loop. The constant simplex is explicitly not the fundamental generator; it is the first API anchor in a by-hand derivation of the circle's H1 homology. The declaration establishes the exact carrier and a checked basepoint in the singular simplicial set of the circle, so that later once-around simplices live in the same structure and use the same face maps.

What this theorem does not claim is equally precise. It does not prove that the constant simplex is a cycle or that it generates any homology. It does not assert that the circle's fundamental group is trivial, nor that the constant path is homotopic to the once-around loop. The theorem is a definitional identity: it records that the face maps of the constant simplex land on the constant 0-simplex. It is the first step in a larger derivation, not the derivation itself.

/-- The left face of the constant singular 1-simplex is the basepoint
0-simplex. -/
theorem constantSphereOneSingularOneSimplex_face_zero :
    (TopCat.toSSet.obj (TopCat.sphere 1)).δ (0 : Fin 2)
      constantSphereOneSingularOneSimplex =
        constantSphereOneSingularZeroSimplex := by
  rfl

/-- The right face of the constant singular 1-simplex is the basepoint
0-simplex. -/
theorem constantSphereOneSingularOneSimplex_face_one :
    (TopCat.toSSet.obj (TopCat.sphere 1)).δ (1 : Fin 2)
      constantSphereOneSingularOneSimplex =
        constantSphereOneSingularZeroSimplex := by
  rfl

/-- The constant singular 1-simplex at `sphereOneBasepoint`, in the actual
singular simplicial set of `TopCat.sphere 1`. -/
def constantSphereOneSingularOneSimplex :
    (TopCat.toSSet.obj (TopCat.sphere 1)).obj (op (SimplexCategory.mk 1)) :=
  (TopCat.toSSetObjEquiv (TopCat.sphere 1) (op (SimplexCategory.mk 1))).symm
    (ContinuousMap.const _ sphereOneBasepoint)

What this page does not claim

The constant simplex is a cycle or generates any homology group. The circle's fundamental group is trivial. The constant path is homotopic to the once-around loop.

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/CircleParam.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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