Convex model contact variation narrow winding interval
ProvedBirkhoffGlobalSection.convex_model_contact_variation_narrow_winding_intervaldynamical-systemstopology
For supplied identity-initialized convex model contact variational data, the directional winding interval has width strictly below one half and satisfies the displayed one-turn threshold equivalence. This does not identify a Conley–Zehnder index.
All displayed regularity, tangency, normalization and supplied-data hypotheses of the exact formal statement are retained. This is a conditional result for the defined convex regularization model, not a global-section or index-existence theorem.
Preamble
import Theorems.Thm_BirkhoffGlobalSection_contact_winding_width_bundle open BirkhoffGlobalSection Set Filter Topology open scoped ContDiff Topology set_option linter.unusedVariables false set_option autoImplicit false set_option maxHeartbeats 800000
Formal statement
theorem BirkhoffGlobalSection.convex_model_contact_variation_narrow_winding_interval :
∀ {μ c : ℝ}
(M : ConvexRegularizationModel μ c) (q : ℝ → Phase)
(Z : ℝ → (Phase →L[ℝ] Phase)) (hs : ∀ t, q t ∈ unitThreeSphere)
(hq : ∀ t, HasDerivAt q (convexModelSphereReebField M (q t)) t)
(hZ : ∀ t, HasDerivAt Z ((fderiv ℝ (convexModelSphereReebField M) (q t)).comp (Z t)) t)
(hZ0 : Z 0 = ContinuousLinearMap.id ℝ Phase) (T : ℝ),
∃ (ρ θ : ℝ × ℝ → ℝ) (a b : ℝ), Continuous ρ ∧ Continuous θ ∧
(∀ x, 0 < ρ x) ∧ (∀ s, θ (0, s) = s) ∧
(∀ x, convexModelContactVariationPath M q Z x.1 ![Real.cos x.2, Real.sin x.2] =
![ρ x * Real.cos (θ x), ρ x * Real.sin (θ x)]) ∧
a ≤ b ∧
(fun s => (θ (T, s) - s) / (2 * Real.pi)) '' Icc 0 (2 * Real.pi) = Icc a b ∧
b - a < 1 / 2 ∧
((3 : ℤ) ≤ Int.ceil a + Int.ceil b - 1 ↔
∀ s ∈ Icc 0 (2 * Real.pi), 2 * Real.pi < θ (T, s) - s) := by sorrySource
Independent extraction of component 7 of the Proved theorem BirkhoffGlobalSection.contact_winding_width_bundle. Original verified source: Solutions/Sol_BirkhoffGlobalSection_contact_winding_width_bundle.lean. Exact original component retained.