namespace Semantics.ENE

-- Universality
--
-- Captures substrate-independent dynamical behavior and scaling laws.
-- A structure is admissible only if it preserves its universality-class
-- identity under projection, collapse, and evolution.

/-- A universality class captures substrate-independent dynamical behavior. -/
inductive UniversalityClass
| kpz                  -- KPZ universal scaling (interface growth / roughness)
| directedPercolation  -- Directed percolation universality
| ising                -- Ising critical behavior
| mott                 -- Mott transition
| genericDiffusion     -- Simple diffusive behavior
| custom (name : String)
deriving Repr, BEq

/-- A scaling invariant is a quantity that remains unchanged under renormalization. -/
structure ScalingInvariant where
  name : String
  exponent : Float
  description : String
deriving Repr, BEq

/-- A universal law governs dynamics across substrates. -/
structure UniversalLaw where
  name : String
  invariant : ScalingInvariant
  univClass : UniversalityClass
  statement : String
deriving Repr, BEq

/-- Classified dynamics bind a concrete process to a universality class. -/
structure ClassifiedDynamics where
  processName : String
  universalityClass : UniversalityClass
  law : UniversalLaw
  preservedUnderProjection : Bool
  preservedUnderCollapse : Bool
  preservedUnderEvolution : Bool
deriving Repr, BEq

/-- A projection preserves universality only if the dynamics classification is maintained. -/
def projectionPreservesUniversality (cd : ClassifiedDynamics) : Prop :=
  cd.preservedUnderProjection = true

/-- A scalar collapse preserves universality only if the class survives scalarization. -/
def collapsePreservesUniversality (cd : ClassifiedDynamics) : Prop :=
  cd.preservedUnderCollapse = true

/-- Evolution preserves universality only if the class remains unchanged. -/
def evolutionPreservesUniversality (cd : ClassifiedDynamics) : Prop :=
  cd.preservedUnderEvolution = true

/-- No admissible structure may lose its universality-class identity
under projection, collapse, or evolution. -/
theorem no_universality_loss
  (cd : ClassifiedDynamics)
  (h1 : projectionPreservesUniversality cd)
  (h2 : collapsePreservesUniversality cd)
  (h3 : evolutionPreservesUniversality cd) :
  cd.preservedUnderProjection = true ∧
  cd.preservedUnderCollapse = true ∧
  cd.preservedUnderEvolution = true := by
  exact ⟨h1, h2, h3⟩

end Semantics.ENE
