Foundation Circle Covering Carrier Covering Val

The trigonometric parametrization

The unit circle in the plane is the set of points at distance 1 from the origin. A standard way to trace it out is the trigonometric parametrization, which sends a real number t to the point (cos t, sin t). This map wraps the real line around the circle infinitely many times, and it is the starting point for defining the winding number: the integer that counts how many times a closed curve goes around the origin.

The declaration carrierCovering_val in the Recognition Science framework's machine-checked library of formal theorems establishes that this familiar map is an honest covering map of the metric unit circle. A covering map is a continuous surjection with a special local property: every point of the circle has a neighborhood that is evenly covered by disjoint open pieces of the real line. The declaration proves, pointwise, that the map t ↦ (cos t, sin t) agrees with the covering map that the library constructs by transporting the exponential map from the complex numbers to the circle.

This agreement is not a new geometric discovery. It is a formal verification that the concrete trigonometric parametrization already used in the framework's circle parameterization satisfies the exact definition of a covering map. The proof proceeds by transporting a known covering map of the complex unit circle along an isometry to the real plane, and then along a homeomorphism to the precise carrier set of the topological sphere. No axioms beyond the standard ones of the ambient type theory are used, and no project-local replacements for the circle are introduced.

The consequence is that the winding number, defined by lifting singular simplices through a covering map, can be built directly on the ordinary trigonometric parametrization. This is the covering-space foundation for the homology of the circle, and it is a necessary step in the framework's derivation of the integer homology group H₁(S¹; ℤ) ≅ ℤ.

The declaration does not claim that the trigonometric parametrization is the only covering map of the circle, nor does it assert any property of the winding number itself. It only establishes the covering-map status of one concrete map and its pointwise agreement with the trigonometric vector. The construction of the degree or winding invariant, and the proof that it is well-defined and integer-valued, are separate steps that build on this foundation.

/-- The carrier covering agrees with the ambient trigonometric vector
`(cos t, sin t)`. -/
theorem carrierCovering_val (t : ℝ) :
    (carrierCovering t : SphereOneAmbient) = trigCircleVector t := by
  show isoE (Circle.exp t : ℂ) = trigCircleVector t
  rw [show (Circle.exp t : ℂ) = Complex.exp (t * Complex.I) from Circle.coe_exp t,
      show isoE (Complex.exp (t * Complex.I))
        = Complex.orthonormalBasisOneI.repr (Complex.exp (t * Complex.I)) from rfl]
  ext i
  rw [Complex.orthonormalBasisOneI_repr_apply]
  fin_cases i
  · simp [trigCircleVector, Complex.exp_ofReal_mul_I_re]
  · simp [trigCircleVector, Complex.exp_ofReal_mul_I_im]

/-- **The trigonometric parametrization `t ↦ (cos t, sin t)` of the imported
`TopCat.sphere 1` object is a covering map.**  This is the covering-space
foundation for the winding / degree invariant on singular `1`-chains. -/
theorem isCoveringMap_trigCirclePoint :
    IsCoveringMap CircleParam.trigCirclePoint := by
  rw [← ulift_carrierCovering_eq_trig]
  exact isCoveringMap_carrierCovering.homeomorph_comp
    (Homeomorph.ulift (X := SphereOneCarrier)).symm

What this page does not claim

The declaration does not prove that the trigonometric parametrization is the unique covering map of the circle. The declaration does not define or prove properties of the winding number itself. The declaration does not establish any result about the homology of higher-dimensional spheres.

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/CircleCovering.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.

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