Convex model sphere weight regular
ProvedBirkhoffGlobalSection.convex_model_sphere_weight_regularconvex-geometryreal-analysis
The squared inverse-gauge sphere weight of a convex model is smooth and positive at every sphere point.
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_smooth_sphere_reeb_bundle open BirkhoffGlobalSection Set Filter SignType open scoped ContDiff Topology set_option linter.unusedVariables false set_option autoImplicit false set_option maxHeartbeats 800000
Formal statement
theorem BirkhoffGlobalSection.convex_model_sphere_weight_regular :
∀ {μ c : ℝ}
(M : ConvexRegularizationModel μ c) (u : Phase) (hu : u ∈ unitThreeSphere),
let f : Phase → ℝ := fun x => ((gauge M.body x)⁻¹)^2
ContDiffAt ℝ ∞ f u ∧ 0 < f u := by sorrySource
Independent extraction of component 5 of the Proved theorem BirkhoffGlobalSection.smooth_sphere_reeb_bundle. Original verified source: Solutions/Sol_BirkhoffGlobalSection_smooth_sphere_reeb_bundle.lean. Exact original component retained.