Convex model contact variation path area preserved
ProvedBirkhoffGlobalSection.convex_model_contact_variation_path_area_preservedconvex-geometryreal-analysis
The supplied identity-initialized convex model variational action preserves standard signed plane area under the stated sphere Reeb differential hypotheses.
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_variational_symplecticity_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_path_area_preserved :
∀ {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) (p r : Plane),
convexModelContactVariationPath M q Z t p 0 * convexModelContactVariationPath M q Z t r 1 -
convexModelContactVariationPath M q Z t p 1 * convexModelContactVariationPath M q Z t r 0 =
p 0 * r 1 - p 1 * r 0 := by sorrySource
Independent extraction of component 9 of the Proved theorem BirkhoffGlobalSection.variational_symplecticity_bundle. Original verified source: Solutions/Sol_BirkhoffGlobalSection_variational_symplecticity_bundle.lean. Exact original component retained.