Foundation Circle Winding Path Lift Shifted Exists Norm Bound

The bounded lift

In mathematics, a path on a circle can be represented as a path on the real line, a process called lifting. The circle is the set of points at a fixed distance from a center, and the real line is the familiar number line. The lifting is not unique: starting at a different point on the line changes the whole lift. The declaration pathLift_shifted_exists_norm_bound concerns the difference between the lift at any time and the lift at the start time.

The theorem states that for any continuous path on the circle, there exists a real number C such that for every time t, the absolute value of the difference between the lift at time t and the lift at time 0 is at most C. In plainer terms, the unwound path never wanders infinitely far from its starting point; it stays within a finite band. This is a consequence of the path being continuous and the time interval being compact, meaning closed and bounded.

The proof in the machine-checked library of formal theorems proceeds by noting that the function t ↦ lift(t) - lift(0) is continuous on the compact unit interval. A continuous function on a compact set is bounded, which yields the desired constant C. The theorem does not provide a specific value for C; it only establishes that such a bound exists.

This result is a technical lemma that supports later work. It is used to show that the winding number, which counts how many times a loop goes around the circle, is well-defined and integer-valued. The boundedness ensures that the displacement of the lift, which is the difference between its endpoints, is a finite number that can be divided by 2π to give the winding number.

The theorem does not claim that the lift itself is bounded, only that the difference from its starting value is. A lift can drift by a large amount over a long path, but the drift is always finite. It also does not claim that the bound C is uniform across all paths; different paths may require different bounds. The result is purely existential, not constructive.

In Recognition Science, this lemma is part of the framework's foundational toolkit. It helps establish the properties of the circle as a fundamental object, which in turn supports the framework's derivation of the golden ratio and the eight-tick cycle. The boundedness of lifts is a prerequisite for defining the winding number rigorously, which is used to distinguish different classes of loops on the circle.

/-- The canonical lift, shifted by its initial value, is uniformly bounded.  This
is the exact bound consumed by the cone formula
`L₀ + (1 - x₂) * (L(coneBaseParam x) - L₀)`. -/
theorem pathLift_shifted_exists_norm_bound (γ : C(I, SphereOne)) :
    ∃ C : ℝ, ∀ t : I, ‖pathLift γ t - pathLift γ 0‖ ≤ C := by
  obtain ⟨C, hC⟩ := isCompact_univ.exists_bound_of_continuousOn
    (s := (Set.univ : Set I)) (f := fun t : I => pathLift γ t - pathLift γ 0)
    (by exact ((pathLift γ).continuous.sub continuous_const).continuousOn)
  refine ⟨C, ?_⟩
  intro t
  exact hC t (by simp)

/-- The canonical lift, shifted by its initial value, is uniformly bounded.  This
is the exact bound consumed by the cone formula
`L₀ + (1 - x₂) * (L(coneBaseParam x) - L₀)`. -/
theorem pathLift_shifted_exists_norm_bound (γ : C(I, SphereOne)) :
    ∃ C : ℝ, ∀ t : I, ‖pathLift γ t - pathLift γ 0‖ ≤ C := by
  obtain ⟨C, hC⟩ := isCompact_univ.exists_bound_of_continuousOn
    (s := (Set.univ : Set I)) (f := fun t : I => pathLift γ t - pathLift γ 0)
    (by exact ((pathLift γ).continuous.sub continuous_const).continuousOn)
  refine ⟨C, ?_⟩
  intro t
  exact hC t (by simp)

/-- The canonical lift, shifted by its initial value, is uniformly bounded.  This
is the exact bound consumed by the cone formula
`L₀ + (1 - x₂) * (L(coneBaseParam x) - L₀)`. -/
theorem pathLift_shifted_exists_norm_bound (γ : C(I, SphereOne)) :
    ∃ C : ℝ, ∀ t : I, ‖pathLift γ t - pathLift γ 0‖ ≤ C := by
  obtain ⟨C, hC⟩ := isCompact_univ.exists_bound_of_continuousOn
    (s := (Set.univ : Set I)) (f := fun t : I => pathLift γ t - pathLift γ 0)
    (by exact ((pathLift γ).continuous.sub continuous_const).continuousOn)
  refine ⟨C, ?_⟩
  intro t
  exact hC t (by simp)

What this page does not claim

The lift itself is bounded, only the difference from its starting value. The bound C is uniform across all paths. The theorem provides a constructive method for finding the bound.

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/CircleWinding.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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