Foundation Circle Param Sphere One Base Vector Mem Sphere

The checked basepoint

The unit circle in the plane is the set of points at distance exactly 1 from the origin. Its most natural starting point is the vector (1, 0), the point one unit to the right. The declaration sphereOneBaseVector_mem_sphere is a machine-checked proof that this vector indeed belongs to the unit circle, in the precise sense that its distance from the origin equals 1. The proof is a direct computation: it expands the definition of distance and simplifies the result to 1 = 1.

This fact matters because it anchors the whole circle in a formal setting. The declaration establishes the exact carrier: the circle is the metric sphere of radius 1 around the origin in the two-dimensional Euclidean plane. It also fixes a basepoint, the point (1, 0), as a distinguished element of that sphere. The proof is recorded in a machine-checked library of formal theorems, meaning every step is verified by a computer kernel rather than accepted by hand.

In Recognition Science, the circle is not yet a recognition ledger. The framework models reality as a discrete record of events, and the circle here is a geometric object, not a ledger. The declaration does not claim that the circle is a recognition cycle, nor that the basepoint has any special recognition meaning. It only establishes a geometric fact: the chosen vector lies on the unit circle. The trigonometric parametrization (cos t, sin t) also lies on the same circle, and the basepoint is the value of that parametrization at t = 0.

The practical consequence is a solid foundation for further work. With the basepoint checked, later constructions can refer to it without re-proving its membership. The constant singular simplex at the basepoint, a degenerate loop that stays at one point, is built on this foundation. The declaration is a small but necessary first step: before one can talk about going around the circle, one must know where the circle is and where the starting point sits.

/-- The first coordinate unit vector lies on Mathlib's unit circle. -/
theorem sphereOneBaseVector_mem_sphere :
    sphereOneBaseVector ∈ Metric.sphere (0 : SphereOneAmbient) 1 := by
  change dist (EuclideanSpace.single (0 : Fin 2) (1 : ℝ) : SphereOneAmbient) 0 = 1
  simp

/-- The first coordinate unit vector lies on Mathlib's unit circle. -/
theorem sphereOneBaseVector_mem_sphere :
    sphereOneBaseVector ∈ Metric.sphere (0 : SphereOneAmbient) 1 := by
  change dist (EuclideanSpace.single (0 : Fin 2) (1 : ℝ) : SphereOneAmbient) 0 = 1
  simp

/-- The exact metric-sphere carrier under the `ULift` in `TopCat.sphere 1`. -/
abbrev SphereOneCarrier : Type :=
  Metric.sphere (0 : SphereOneAmbient) 1

What this page does not claim

The declaration does not claim that the circle is a recognition ledger or that the basepoint has any recognition meaning. The declaration does not prove that the trigonometric parametrization is continuous or that it returns to the basepoint after one full turn. The declaration does not establish any property of the fundamental group or higher homotopy groups of the circle.

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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