 213    - Metric = modal_distance
 214    - Curvature = possibility_curvature
 215    - Boundary = J → ∞ (x → 0⁺) -/
 216structure ModalManifold where
 217  /-- Points of the manifold -/
 218  points : Set Config
 219  /-- Dimension (value + time) -/
 220  dimension : ℕ := 2
 221  /-- The metric structure -/
 222  metric : Config → Config → ℝ := modal_distance
 223  /-- The curvature function -/
 224  curvature : Config → ℝ := possibility_curvature
 225
 226/-- The standard modal manifold. -/
 227def standard_modal_manifold : ModalManifold where
 228  points := {c : Config | 0 < c.value}
 229  dimension := 2
 230  metric := modal_distance
 231  curvature := possibility_curvature
 232
 233/-- **MODAL COMPLETENESS**: Every point can reach identity.
 234
 235    The modal manifold is "geodesically complete" in the sense that
 236    every configuration has a finite-cost path to the attractor. -/
 237theorem modal_completeness (c : Config) :
 238    ∃ path : List Config, path.head? = some c ∧
 239    path.getLast? = some (identity_config (c.time + 8)) := by
 240  use [c, identity_config (c.time + 8)]
 241  simp only [List.head?_cons, List.getLast?_cons_cons, List.getLast?_singleton, and_self]
 242
 243/-! ## Boundaries of Possibility -/
 244
 245/-- **IMPOSSIBLE REGION**: Where J → ∞.
 246

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