Foundation Circle Param Constant Sphere One Singular One Simplex Face One

A constant path's two ends

A singular 1-simplex is a continuous path, a line segment mapped into a space. The declaration in question concerns the constant path on a circle: the path that sits motionless at a single chosen point, the basepoint. The theorem proves that the right end of this constant path, its face at index one, is exactly the constant 0-simplex at that same basepoint. A 0-simplex is a point, so the statement says, in plain terms, that the path's endpoint is the point where it started.

This fact is not a discovery about circles; it is a definitional check. The proof is by reflexivity, meaning the two sides of the equality are the same by construction. The declaration's importance is architectural. It lives in a machine-checked library of formal theorems, a collection where every statement is verified by a computer. The library is building a careful, by-hand derivation of the circle's fundamental group, the mathematical object that counts how many times a loop winds around. Before that derivation can proceed, the library must establish the exact carrier: the precise metric sphere, a chosen basepoint, and the basic simplices that will serve as building blocks.

The constant simplex is not the fundamental generator of the circle's topology. A constant path winds zero times; it is the trivial loop. The declaration establishes that this trivial loop behaves correctly under the face maps, the operations that extract a path's endpoints. It is the first API anchor, a fixed point from which all later once-around simplices, the paths that actually wind around the circle, can be defined and checked against the same face maps.

In Recognition Science, the circle is not just a geometric object; it is a candidate for a recognition cycle, a discrete record of events. The framework's broader project is to show that such cycles, and the constants that govern them, are forced by a single cost function. This declaration is a small but necessary step in that project: it ensures that the formal language used to describe the circle is precise enough to support the larger claims. The theorem itself proves nothing about recognition or cost; it proves that a constant path has both ends at its starting point.

What the declaration does not claim is equally clear. It does not claim that the constant simplex generates the circle's fundamental group, nor that it represents a nontrivial loop. It does not claim anything about the Recognition Science framework's broader theorems, such as the forcing of the golden ratio or the eight-tick cycle. It is a foundational lemma, a piece of scaffolding, not a result about the physical world. Its value is that it makes the formal environment reliable enough for the larger derivation to proceed.

/-- 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 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 generates the circle's fundamental group. The theorem establishes any property of the Recognition Science framework's cost function or forcing chain. The declaration proves that the circle has a nontrivial topology.

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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