import Mathlib.Data.Real.Basic

-- The Information Manifold Φ structure
structure Phi where
  feature     : ℕ -- F ∈ {0..7}
  tau         : ℝ -- Torsion/Complexity
  consistency : ℕ -- c ∈ {0..5}
  delta       : ℝ -- Local variance

-- Similarity flow based on Fisher-Rao gradient and torsion correction
-- SimFlowPhi = (Fisher-Rao Distance) + (τ * Consistency)
def simFlowPhi (p1 p2 : Phi) : ℝ :=
  let fisherRao := (p1.tau - p2.tau).sqr -- Simplified Fisher-Rao for this context
  (fisherRao + (p1.tau * toReal p1.consistency)) - (p2.tau * toReal p2.consistency)

-- Convergence predicate: Is the state stable?
def isConvergent (p : Phi) : Prop :=
  p.consistency > 0 && p.delta < 1.0

-- Example instances of S1-S4 specializations
def s1_phi : Phi := {feature := 0, tau := 0, consistency := 1, delta := 0}
def s2_phi : Phi := {feature := 1, tau := 8, consistency := 1, delta := 0.5}
def s3_phi : Phi := {feature := 2, tau := 4, consistency := 1, delta := 0.2}
def s4_phi : Phi := {feature := 3, tau := 8, consistency := 1, delta := 0.8}

-- Example usage: check if S2 is convergent
example (p : Phi) : Prop := isConvergent p2_phi