import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Analysis.Complex.Exponential
import Mathlib.RingTheory.RootsOfUnity.Complex
import QFD.Lepton.Generations

/-!
# The Geometric Koide Relation

**Status**: ✅ Mathematical Proof Complete (0 sorries)
**Purpose**: Rigorously prove that Q = 2/3 follows from the symmetric mass parametrization

## What This Proof Establishes

### QFD Hypothesis (Physical Assumption - NOT proven in Lean)
Lepton masses follow the parametrization:
```
m_k = μ(1 + √2·cos(δ + 2πk/3))²  for k = 0, 1, 2
```
where μ is a mass scale and δ is a phase angle.

**Status**: Physical hypothesis fitted to experimental lepton masses (δ ≈ 0.222)

### Lean-Verified Mathematical Consequence (Proven)

**Main Result**: `theorem koide_relation_is_universal`
```
IF masses satisfy the hypothesis above
THEN KoideQ = (Σm_k)/(Σ√m_k)² = 2/3  (exactly)
```

**What this proves**: The Koide quotient Q = 2/3 is a *mathematical necessity* given
the parametrization, not a numerical coincidence or empirical fit.

**What this does NOT prove**:
- That leptons must follow this parametrization (physical claim, not mathematical)
- That the parametrization arises from Cl(3,3) geometry (QFD interpretation)
- That δ is determined by fundamental principles (it's fitted to data)

## Proven Theorems (0 sorries)

1. **`omega_is_primitive_root`**: ω = exp(2πi/3) is primitive 3rd root of unity
2. **`sum_third_roots_eq_zero`**: 1 + ω + ω² = 0 (from Mathlib)
3. **`sum_cos_symm`**: cos(δ) + cos(δ+2π/3) + cos(δ+4π/3) = 0
4. **`sum_cos_sq_symm`**: cos²(δ) + cos²(δ+2π/3) + cos²(δ+4π/3) = 3/2
5. **`koide_relation_is_universal`**: Parametrization → Q = 2/3

## Proof Strategy

- **Numerator**: Σm_k = 6μ (using trigonometric sum identities)
- **Denominator**: (Σ√m_k)² = 9μ (using square root algebra)
- **Result**: 6μ/9μ = 2/3 (exact, not approximate)

All steps verified in Lean 4 with Mathlib, zero sorries.
-/

namespace QFD.Lepton.KoideRelation

open QFD.Lepton.Generations
open Real
open ComplexConjugate

/-- The empirical Koide Ratio -/
noncomputable def KoideQ (m1 m2 m3 : ℝ) : ℝ :=
  (m1 + m2 + m3) / (sqrt m1 + sqrt m2 + sqrt m3)^2

def generationIndex (g : GenerationAxis) : ℕ :=
  match g with | .x => 0 | .xy => 1 | .xyz => 2

/-- Geometric Mass Function --/
noncomputable def geometricMass (g : GenerationAxis) (mu delta : ℝ) : ℝ :=
  let k := (generationIndex g : ℝ)
  let term := 1 + sqrt 2 * cos (delta + k * (2 * Real.pi / 3))
  mu * term^2

/-! ## 1. Verified Trig Identities -/

/-- The primitive 3rd root of unity -/
noncomputable def omega : ℂ := Complex.exp (2 * Real.pi * Complex.I / 3)

/-- omega is a primitive 3rd root of unity (Mathlib theorem) -/
lemma omega_is_primitive_root : IsPrimitiveRoot omega 3 := by
  rw [omega]
  exact Complex.isPrimitiveRoot_exp 3 (by norm_num)

/-- Sum of 3rd roots of unity equals zero (Mathlib IsPrimitiveRoot.geom_sum_eq_zero) -/
lemma sum_third_roots_eq_zero : 1 + omega + omega^2 = 0 := by
  have h := omega_is_primitive_root
  -- Use Mathlib's theorem about sum of primitive roots
  have h_sum : (Finset.range 3).sum (fun i => omega ^ i) = 0 := by
    apply IsPrimitiveRoot.geom_sum_eq_zero h
    norm_num
  -- Expand the sum: range 3 = {0, 1, 2}
  have : (Finset.range 3).sum (fun i => omega ^ i) = omega^0 + omega^1 + omega^2 := by
    simp only [Finset.sum_range_succ, Finset.sum_range_zero, pow_zero, pow_one]
    ring
  rw [this] at h_sum
  simpa using h_sum

/-- Sum of cosines separated by 2π/3 equals zero (from roots of unity) -/
lemma sum_cos_symm (delta : ℝ) :
  cos delta + cos (delta + 2*Real.pi/3) + cos (delta + 4*Real.pi/3) = 0 := by
  -- Strategy: Use Euler's formula cos(x) = Re(exp(ix))
  -- Then: sum of cosines = Re(sum of exponentials) = Re(exp(iδ)(1 + ω + ω²)) = Re(0) = 0

  -- Step 1: Euler's formula
  have h_cos_formula : ∀ x : ℝ, Real.cos x = (Complex.exp (Complex.I * x)).re := by
    intro x
    -- Real.cos x = (Complex.cos ↑x).re by definition
    conv_lhs => rw [show Real.cos x = (Complex.cos x).re from rfl]
    -- Complex.cos z = (exp(I*z) + exp(-I*z))/2
    rw [Complex.cos, mul_comm]
    -- Now: ((exp(Complex.I * ↑x) + exp(-↑x * Complex.I)) / 2).re = (exp(Complex.I * ↑x)).re
    -- Key: for real x, exp(-x*I) = conj(exp(x*I)), so (exp(x*I) + conj(exp(x*I)))/2 has Re = Re(exp(x*I))
    have h_conj : Complex.exp (-↑x * Complex.I) = conj (Complex.exp (↑x * Complex.I)) := by
      rw [← Complex.exp_conj]
      congr 1
      simp [Complex.conj_ofReal, Complex.conj_I]
    rw [h_conj, mul_comm Complex.I]
    -- Re((z + conj(z))/2) = Re(z)
    simp [Complex.add_re, Complex.conj_re, Complex.div_re]

  -- Step 2: The complex sum equals zero (from roots of unity) ✅ PROVEN
  have h_complex_zero :
    Complex.exp (Complex.I * delta) +
    Complex.exp (Complex.I * (delta + 2*Real.pi/3)) +
    Complex.exp (Complex.I * (delta + 4*Real.pi/3)) = 0 := by
      -- This equals exp(iδ) * (1 + ω + ω²) = exp(iδ) * 0 = 0
      have h_factor :
        Complex.exp (Complex.I * delta) * (1 + omega + omega^2) =
        Complex.exp (Complex.I * delta) +
        Complex.exp (Complex.I * (delta + 2*Real.pi/3)) +
        Complex.exp (Complex.I * (delta + 4*Real.pi/3)) := by
          rw [omega, mul_add, mul_add, mul_one]
          congr 1
          · -- exp(iδ) * exp(2πi/3) = exp(i(δ + 2π/3)) ✅ PROVEN
            rw [← Complex.exp_add]
            congr 1
            push_cast
            ring_nf
          · -- exp(iδ) * exp(2πi/3)² = exp(i(δ + 4π/3)) ✅ PROVEN
            rw [sq, mul_comm, mul_assoc, ← Complex.exp_add, ← Complex.exp_add]
            congr 1
            push_cast
            ring_nf
      rw [← h_factor, sum_third_roots_eq_zero, mul_zero]  -- Uses Mathlib roots of unity ✅

  -- Step 3: Connect via real parts
  rw [h_cos_formula delta, h_cos_formula (delta + 2*Real.pi/3), h_cos_formula (delta + 4*Real.pi/3)]
  rw [← Complex.add_re, ← Complex.add_re]
  -- Need to show: (exp(I*↑delta) + exp(I*↑(delta+2π/3)) + exp(I*↑(delta+4π/3))).re = 0
  -- This follows from h_complex_zero by ofReal_add: ↑(a+b) = ↑a + ↑b
  rw [show Complex.I * ↑(delta + 2*Real.pi/3) = Complex.I * (delta + 2*Real.pi/3) by simp [Complex.ofReal_add]]
  rw [show Complex.I * ↑(delta + 4*Real.pi/3) = Complex.I * (delta + 4*Real.pi/3) by simp [Complex.ofReal_add]]
  rw [h_complex_zero]
  simp

/--
Sum of squared cosines at symmetric angles equals 3/2.
This is the key algebraic step for proving the Koide relation.

Uses double-angle formula cos²x = (1 + cos(2x))/2 and sum_cos_symm.
-/
lemma sum_cos_sq_symm (δ : ℝ) :
  cos δ ^ 2 + cos (δ + 2*π/3) ^ 2 + cos (δ + 4*π/3) ^ 2 = 3/2 := by
  -- Apply double-angle formula to each term
  rw [cos_sq δ, cos_sq (δ + 2*π/3), cos_sq (δ + 4*π/3)]
  -- Normalize angles: 2*(δ + 2π/3) = 2*δ + 4*π/3 and 2*(δ + 4*π/3) = 2*δ + 8*π/3
  have h1 : (2:ℝ) * (δ + 2*π/3) = 2*δ + 4*π/3 := by ring
  have h2 : (2:ℝ) * (δ + 4*π/3) = 2*δ + 8*π/3 := by ring
  rw [h1, h2]
  -- Use periodicity: cos(2δ + 8π/3) = cos(2δ + 2π/3) since 8π/3 - 2π = 2π/3
  have sum_cos_zero : cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 8*π/3) = 0 := by
    have h_period : cos (2*δ + 8*π/3) = cos (2*δ + 2*π/3) := by
      have : (2:ℝ)*δ + 8*π/3 = 2*δ + 2*π/3 + 2*π := by ring
      rw [this, cos_add_two_pi]
    calc cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 8*π/3)
        = cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 2*π/3) := by rw [h_period]
      _ = cos (2*δ) + cos (2*δ + 2*π/3) + cos (2*δ + 4*π/3) := by ring
      _ = 0 := sum_cos_symm (2*δ)
  -- Final algebraic simplification: combine fractions and apply sum_cos_zero
  calc 1 / 2 + cos (2 * δ) / 2 + (1 / 2 + cos (2 * δ + 4 * π / 3) / 2)
          + (1 / 2 + cos (2 * δ + 8 * π / 3) / 2)
      = (1 + 1 + 1 + cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 8*π/3)) / 2 := by ring
    _ = (3 + (cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 8*π/3))) / 2 := by ring
    _ = (3 + 0) / 2 := by rw [sum_cos_zero]
    _ = 3 / 2 := by norm_num

/--
**Theorem: Koide Quotient from Symmetric Mass Parametrization**

**QFD Hypothesis (assumed, not proven)**:
Lepton masses follow the parametrization:
  m_k = μ(1 + √2·cos(δ + 2πk/3))²  for k = 0, 1, 2

**Mathematical Consequence (proven in this theorem)**:
Given the parametrization above, the Koide quotient Q = 2/3 exactly.

**What this proves**: Q = 2/3 is mathematically necessary given the parametrization,
not a numerical coincidence or empirical parameter.

**What this does NOT prove**:
- That leptons must follow this parametrization (physical claim)
- That the parametrization arises from Cl(3,3) geometry (QFD interpretation)
- That δ is determined by fundamental principles (δ ≈ 0.222 is fitted to data)

**Proof strategy**:
- Numerator: Σm_k = μ·[3 + 2√2·Σcos θ_k + 2·Σcos²θ_k] = μ·[3 + 0 + 3] = 6μ
- Denominator: (Σ√m_k)² = (√μ·[3 + √2·Σcos θ_k])² = (3√μ)² = 9μ
- Result: Q = 6μ/9μ = 2/3

**Parameters**:
- `mu` : mass scale (must be positive)
- `delta` : phase angle (empirically δ ≈ 0.222 for leptons)
- `h_pos0, h_pos1, h_pos2` : positivity requirements for square root extraction
-/
theorem koide_relation_is_universal (mu delta : ℝ) (h_mu : mu > 0)
  (h_pos0 : 1 + sqrt 2 * cos delta > 0)
  (h_pos1 : 1 + sqrt 2 * cos (delta + 2 * π / 3) > 0)
  (h_pos2 : 1 + sqrt 2 * cos (delta + 4 * π / 3) > 0) :
  let m_e   := geometricMass .x   mu delta
  let m_mu  := geometricMass .xy  mu delta
  let m_tau := geometricMass .xyz mu delta
  KoideQ m_e m_mu m_tau = 2/3 := by
  unfold KoideQ geometricMass generationIndex
  simp only [Nat.cast_zero, Nat.cast_one, Nat.cast_ofNat, zero_mul, add_zero]

  -- Trig identities
  have h_sum_cos := sum_cos_symm delta
  have h_sum_cos_sq := sum_cos_sq_symm delta

  -- Abbreviations
  let c0 := cos delta
  let c1 := cos (delta + 2 * π / 3)
  let c2 := cos (delta + 4 * π / 3)

  -- Expand (1 + √2*c)²
  have h_sq : ∀ c, (1 + sqrt 2 * c)^2 = 1 + 2 * sqrt 2 * c + 2 * c^2 := by
    intro c
    have : sqrt 2 ^ 2 = 2 := Real.sq_sqrt (by norm_num)
    ring_nf; rw [this]; ring

  -- NUMERATOR = 6μ (geometric cancellation proof)
  have h_num : mu * (1 + sqrt 2 * c0)^2 + mu * (1 + sqrt 2 * c1)^2 +
      mu * (1 + sqrt 2 * c2)^2 = 6 * mu := by
    simp only [h_sq]
    calc mu * (1 + 2 * sqrt 2 * c0 + 2 * c0^2) + mu * (1 + 2 * sqrt 2 * c1 + 2 * c1^2)
            + mu * (1 + 2 * sqrt 2 * c2 + 2 * c2^2)
        = mu * (3 + 2 * sqrt 2 * (c0 + c1 + c2) + 2 * (c0^2 + c1^2 + c2^2)) := by ring
      -- Geometric cancellation: the linear sum (c0 + c1 + c2) = 0 (vectors sum to zero)
      _ = mu * (3 + 2 * sqrt 2 * 0 + 2 * (3/2)) := by
          rw [h_sum_cos]    -- Apply sum_cos_symm: cross-terms cancel
          rw [h_sum_cos_sq] -- Apply sum_cos_sq_symm: quadratic terms sum to 3/2
      -- Simplify: 0 from linear terms, 3 from quadratic terms
      _ = mu * (3 + 0 + 2 * (3/2)) := by ring
      _ = mu * (3 + 3) := by ring
      _ = mu * 6 := by ring
      _ = 6 * mu := by ring

  -- DENOMINATOR = 9μ
  have h_den : (sqrt (mu * (1 + sqrt 2 * c0)^2) + sqrt (mu * (1 + sqrt 2 * c1)^2)
      + sqrt (mu * (1 + sqrt 2 * c2)^2))^2 = 9 * mu := by
    have h_sqrt : ∀ c, 1 + sqrt 2 * c > 0 →
        sqrt (mu * (1 + sqrt 2 * c)^2) = sqrt mu * (1 + sqrt 2 * c) := by
      intro c hc
      rw [Real.sqrt_mul (le_of_lt h_mu), Real.sqrt_sq (le_of_lt hc)]
    rw [h_sqrt c0 h_pos0, h_sqrt c1 h_pos1, h_sqrt c2 h_pos2]
    have h_sum : sqrt mu * (1 + sqrt 2 * c0) + sqrt mu * (1 + sqrt 2 * c1)
        + sqrt mu * (1 + sqrt 2 * c2) = 3 * sqrt mu := by
      calc sqrt mu * (1 + sqrt 2 * c0) + sqrt mu * (1 + sqrt 2 * c1)
              + sqrt mu * (1 + sqrt 2 * c2)
          = sqrt mu * (3 + sqrt 2 * (c0 + c1 + c2)) := by ring
        _ = sqrt mu * (3 + sqrt 2 * 0) := by rw [h_sum_cos]
        _ = sqrt mu * 3 := by ring
        _ = 3 * sqrt mu := by ring
    rw [h_sum]
    have : (sqrt mu)^2 = mu := Real.sq_sqrt (le_of_lt h_mu)
    calc (3 * sqrt mu)^2 = 9 * (sqrt mu)^2 := by ring
      _ = 9 * mu := by rw [this]

  -- FINAL: Q = 6μ / 9μ = 2/3
  -- The goal uses explicit cosines; simplify angle coefficients first
  have ha1 : (1:ℝ) * (2 * π / 3) = 2 * π / 3 := by ring
  have ha2 : (2:ℝ) * (2 * π / 3) = 4 * π / 3 := by ring
  simp only [ha1, ha2]
  -- Now the goal matches our c0, c1, c2 abbreviations
  rw [h_num, h_den]
  field_simp
  norm_num

end QFD.Lepton.KoideRelation
