/-
Semantics/ManifoldFlow.lean - Anisotropically Frustrated Torsional Gradient Flow

This module formalizes the "n-space foldback-lock" equation as the authoritative 
governing physics for the Sovereign Informatic Manifold. 

Governing Equation:
∂_t ϕ = ∇_i(M^ij ∇_j δF/δϕ) - σ ∂ϕ/∂I_lock
∂_t X^A = -Γ^A_BC ∂_i X^B ∂_i X^C - Λ^AB(X^B - X_0^B) - δF/δX^A + τ T^A

Lean is the source of truth.
-/

import Semantics.DynamicCanal
import Semantics.BraidBracket

namespace Semantics.ManifoldFlow

open DynamicCanal
open Semantics.BraidBracket

-- =============================================================================
-- 1. TENSOR FIELDS (Q16.16)
-- =============================================================================

/-- Anisotropic Tensor A^ij -/
structure AnisotropyTensor where
  xx : Fix16
  xy : Fix16
  yy : Fix16
  deriving Repr, DecidableEq, BEq

/-- Metric Tensor g_ij -/
structure MetricTensor where
  xx : Fix16
  xy : Fix16
  yy : Fix16
  deriving Repr, DecidableEq, BEq

/-- Torsion Tensor T^k_ij (for 2D manifold, k=1,2) -/
structure TorsionTensor where
  t1_12 : Fix16 -- T^1_{12}
  t2_12 : Fix16 -- T^2_{12}
  deriving Repr, DecidableEq, BEq

-- =============================================================================
-- 2. MANIFOLD STATE (Fabric + Hyperfluid)
-- =============================================================================

/-- Manifold State at coordinate x -/
structure ManifoldPoint where
  phi    : Fix16       -- Hyperfluid Phase Field
  x_pos  : PhaseVec    -- Embedding X^A in ambient 2-space
  x0_pos : PhaseVec    -- Preferred "fold-back" location X_0^A
  g      : MetricTensor
  t      : TorsionTensor
  a      : AnisotropyTensor
  deriving Repr, DecidableEq, BEq

-- =============================================================================
-- 3. ENERGY FUNCTIONAL (F)
-- =============================================================================

/-- Locking potential W(z; A) = w * (1 - cos(k * z)) approximation -/
def lockingPotential (z : Fix16) (weight : Fix16) : Fix16 :=
  -- Periodic frustration: Using a simplified multiwell 
  -- Fix16 approximation of (1 - cos(z))
  let z_mod := Fix16.mk (z.raw % 0x00010000) -- mod 1.0
  Fix16.mul weight (Fix16.mul z_mod (Fix16.sub Fix16.one z_mod))

/-- Interlocking energy I_lock for recursive deposition -/
def interlockingEnergy (x x_prev : PhaseVec) (a : AnisotropyTensor) : Fix16 :=
  let dx := Fix16.sub x.x x_prev.x
  let dy := Fix16.sub x.y x_prev.y
  -- Frustration modulated by anisotropy
  let frustration := Fix16.add (Fix16.mul a.xx dx) (Fix16.mul a.yy dy)
  lockingPotential frustration (Fix16.mk 0x00008000) -- weight 0.5

/-- Torsional Stress Σ^ij(T) contribution -/
def torsionalStress (t : TorsionTensor) : Fix16 :=
  -- χ * T^i_ab T^jab ... simplified to magnitude squared
  Fix16.add (Fix16.mul t.t1_12 t.t1_12) (Fix16.mul t.t2_12 t.t2_12)

-- =============================================================================
-- 4. EVOLUTION DYNAMICS (OISC Target)
-- =============================================================================

/-- Compute the next Phase Field state (ϕ_{t+1}) via gradient descent -/
def stableDt (dt : Fix16) : Fix16 :=
  if dt.isNeg then Fix16.zero
  else if dt.raw > Fix16.one.raw then Fix16.one
  else dt

/-- CFL-style stability guard for the evolution step size. -/
def cflSatisfied (dt : Fix16) : Bool :=
  (stableDt dt).raw = dt.raw

/-- Compute the next Phase Field state (ϕ_{t+1}) via gradient descent -/
def flowPhi (p : ManifoldPoint) (dt : Fix16) : Fix16 :=
  let dt' := stableDt dt
  let gradient := Fix16.sub p.phi (Fix16.mk 0x00008000) -- simplified δF/δϕ
  -- ϕ' = ϕ - dt * (Mobility * gradient)
  Fix16.sub p.phi (Fix16.mul dt' gradient)

/-- Compute the next Embedding state (X_{t+1}) via fold-back dynamics -/
def flowEmbedding (p : ManifoldPoint) (dt : Fix16) (prevX : PhaseVec) : PhaseVec :=
  let dt' := stableDt dt
  -- Tendency to return to X0: Pull = -Λ(X - X0)
  let pullX := Fix16.mul (Fix16.mk 0x00004000) (Fix16.sub p.x_pos.x p.x0_pos.x)
  let pullY := Fix16.mul (Fix16.mk 0x00004000) (Fix16.sub p.x_pos.y p.x0_pos.y)
  
  -- Frustration from locking: snagging on previous pattern
  let snag := interlockingEnergy p.x_pos prevX p.a
  
  -- Torsional forcing: τ * T
  let forceX := Fix16.mul (Fix16.mk 0x00002000) p.t.t1_12
  let forceY := Fix16.mul (Fix16.mk 0x00002000) p.t.t2_12
  
  { x := Fix16.sub p.x_pos.x (Fix16.mul dt' (Fix16.add (Fix16.add pullX snag) forceX))
  , y := Fix16.sub p.x_pos.y (Fix16.mul dt' (Fix16.add (Fix16.add pullY snag) forceY)) : PhaseVec }

end Semantics.ManifoldFlow
