Foundation Circle Param Constant Sphere One Singular One Simplex Faces Eq

The constant simplex faces

A singular 1-simplex is a continuous map from a line segment into a space. In the singular simplicial set of the circle, the constant 1-simplex is the map that sends the whole segment to the chosen basepoint. Its two faces, the images of the segment's two endpoints, are both the constant 0-simplex at that same basepoint. The theorem constantSphereOneSingularOneSimplex_faces_eq states that these two faces are equal: the left face and the right face of the constant 1-simplex coincide.

This is a foundational identity in the framework's machine-checked library of formal theorems. It is not a deep geometric fact but a necessary consistency check: the constant simplex must behave identically at both ends, or the whole singular simplicial set construction would be incoherent. The proof is immediate, by reducing both faces to the same constant 0-simplex. The declaration serves as the first API anchor for later once-around simplices, which will live in the same singular simplicial set and use the same face maps.

In Recognition Science, the ledger (a discrete record of events) is built from such simplicial primitives. The constant simplex is not the fundamental generator of the circle's homology; it is the trivial case that must be settled before any nontrivial loop can be defined. The theorem does not claim that the constant simplex generates the circle's fundamental group, nor that it represents a nontrivial cycle. It only pins down the face structure of the simplest possible simplex, so that later constructions have a solid basepoint to start from.

/-- The two faces of the constant singular 1-simplex coincide. -/
theorem constantSphereOneSingularOneSimplex_faces_eq :
    (TopCat.toSSet.obj (TopCat.sphere 1)).δ (0 : Fin 2)
      constantSphereOneSingularOneSimplex =
        (TopCat.toSSet.obj (TopCat.sphere 1)).δ (1 : Fin 2)
          constantSphereOneSingularOneSimplex := by
  rw [constantSphereOneSingularOneSimplex_face_zero,
    constantSphereOneSingularOneSimplex_face_one]

What this page does not claim

The constant simplex is not the fundamental generator of the circle's homology. The theorem does not establish any property of nontrivial loops or the fundamental group. The declaration does not connect the circle's simplicial structure to the framework's cost function or forcing chain.

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)

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