import Mathlib.LinearAlgebra.Matrix
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

-- Your actual GTCT operator modules
import GCTC.Operators.Compress
import GCTC.Operators.Threshold
import GCTC.Operators.Fold
import GCTC.Operators.Unfold

namespace GCTC.Conformal

/-!
# Conformal Model for Molecular Geometry (GCTC Integration)

Credit: Jesus Camargo, Carlile Lavor, Michael Souza.
"Conformal Coordinates for Molecular Geometry: from 3D to 5D"
arXiv:2408.16188v2 [physics.chem-ph], 11 Nov 2025.

This module provides the 5D conformal engine for GTCT operators,
n-reconfigurations, dm³ helical attractor lift, polylaminin networks,
and protein folding pathways.
-/

/-! ## Conformal Metric -/

def MetricMatrix : Matrix (Fin 5) (Fin 5) ℝ :=
  !![1, 0, 0, 0, 0;
     0, 1, 0, 0, 0;
     0, 0, 1, 0, 0;
     0, 0, 0, 0, -1;
     0, 0, 0, -1, 0]

def conformalInner (u v : Fin 5 → ℝ) : ℝ := 
  Matrix.dotProduct u (MetricMatrix.mulVec v)

def e0 : Fin 5 → ℝ := ![0, 0, 0, 1, 0]
def eInf : Fin 5 → ℝ := ![0, 0, 0, 0, 1]

/-! ## C-Matrix (exact from paper §3.2) -/

def CMatrix (d θ ω : ℝ) : Matrix (Fin 5) (Fin 5) ℝ :=
  let cθ := Real.cos θ; let sθ := Real.sin θ
  let cω := Real.cos ω; let sω := Real.sin ω
  !![ -cθ,          -sθ,         0,         -d * cθ,                  0;
      sθ * cω,     -cθ * cω,    -sω,       d * sθ * cω,               0;
      sθ * sω,     -cθ * sω,     cω,       d * sθ * sω,               0;
      0,            0,           0,        1,                         0;
      d,            0,           0,        d * d / 2,                  1 ]

/-! ## Orthogonality (honest sorry) -/

theorem c_matrix_orthogonal {d θ ω : ℝ} :
  (CMatrix d θ ω).transpose ⬝ MetricMatrix ⬝ (CMatrix d θ ω) = MetricMatrix := by
  simp only [CMatrix, MetricMatrix]
  norm_num [Matrix.mul_fin_three, Real.sin_sq, Real.cos_sq]
  ring_nf
  simp [Real.sin_sq_add_cos_sq, Real.cos_sq_add_sin_sq]
  sorry  -- Entrywise trig expansion needed (25 cases). Completable later.

/-! ## Distance Formula -/

def chainProduct (matrices : List (Matrix (Fin 5) (Fin 5) ℝ)) : Matrix (Fin 5) (Fin 5) ℝ :=
  matrices.foldr (· ⬝ ·) (Matrix.one)

def squaredDistance (B : Matrix (Fin 5) (Fin 5) ℝ) : ℝ :=
  2 * Matrix.dotProduct eInf (B.mulVec e0)

theorem squaredDistance_eq_euclidean (d θ ω : ℝ) :
    squaredDistance (CMatrix d θ ω) = d ^ 2 := by
  simp [squaredDistance, CMatrix, e0, eInf, Matrix.dotProduct, Matrix.mulVec,
        Fin.sum_univ_five]
  ring

/-! ## dm³ Helical Attractor Lift -/

-- Local stub (real definition lives in ThreeM/DM3)
structure DM3Parameters where
  d : ℝ
  θ : ℝ
  ω : ℝ

variable (dm3_params : DM3Parameters)

def dm3_gtct_step : Matrix (Fin 5) (Fin 5) ℝ :=
  CMatrix dm3_params.d dm3_params.θ dm3_params.ω

theorem dm3_step_orthogonal :
  dm3_gtct_step.transpose ⬝ MetricMatrix ⬝ dm3_gtct_step = MetricMatrix :=
  c_matrix_orthogonal

-- TODO: bridge lemma
-- "Applying GChain.apply to a helical state corresponds to left-multiplying
--  the conformal lift by the appropriate CMatrix."

-- STUB: replace with actual displacement formula from 3M ODE analysis
noncomputable def dm3_expected_displacement (n : ℕ) : ℝ := 
  sorry  -- links to ThreeM.DM3.Attractor.spiralDisplacement

-- OPEN CONJECTURE: Spiral return after G⁶⁴ preserves refined Gronwall bound
theorem dm3_spiral_return_conformal (n : ℕ) :
  let chain := List.range n |>.map (fun _ => dm3_gtct_step)
  let B := chainProduct chain
  squaredDistance B = dm3_expected_displacement n := by
  sorry

/-! ## Polylaminin Triangular Unit Cell -/

structure PolyLamininTriangle where
  d : ℝ := 30.0
  θ : ℝ := 2 * Real.pi / 3
  ω : ℝ := 0.0

def polyLaminin_unit_cell : List PolyLamininTriangle :=
  List.replicate 3 { d := 30.0, θ := 2 * Real.pi / 3, ω := 0.0 }

-- Cross-contact assumes both chains share