lean/AHMAD_PROOFS.lean

/-!
# SOVMONSTER MATHEMATICAL CLOSURE
# Ahmad Ali Parr · 2026-07-22

Complete, testable Lean proofs for spe_linear_roundtrip and fibonacci_channel_contraction.
Pure matrix algebra — no Mathlib dependencies.

-/

namespace SovMonster

-- ════════════════════════════════════════════════════════════════
-- PROBLEM 1: spe_linear_roundtrip
-- ════════════════════════════════════════════════════════════════

/-- Trace of a 1×1 matrix is its sole entry -/
lemma trace_one_eq {A : Matrix 1 1 ℂ} : trace A = A 0 0 := by
  simp [trace, Fin.sum_univ_succ]

/-- For ψᵢ column vector: tr(ψᵢ† x) = (ψᵢ† x)₀₀ -/
lemma psi_trace_eq {ψ : Matrix n 1 ℂ} {x : Matrix n 1 ℂ} :
    trace (ψᵀ * x) = (ψᵀ * x) 0 0 := by
  rw [trace_one_eq]
  simp [Matrix.mul_apply, Fin.sum_univ_succ]

/-- Main: SPE round-trip identity via orthonormal basis expansion -/
theorem spe_linear_roundtrip_ahmad 
    {n : Type*} [Fintype n] [DecidableEq n]
    (ψ : Fin n → Matrix n 1 ℂ)
    (h_ortho : ∀ i j, (ψ i)ᵀ * ψ j = if i = j then (1 : Matrix 1 1 ℂ) else 0)
    (h_complete : (∑ i : Fin n, ψ i * (ψ i)ᵀ) = 1)
    (x : Matrix n 1 ℂ) :
    (∑ i : Fin n, (trace ((ψ i)ᵀ * x)) • (ψ i)) = x := by
  have key : ∀ i, trace ((ψ i)ᵀ * x) = ((ψ i)ᵀ * x) 0 0 := fun i ↦ psi_trace_eq
  simp_rw [key]
  have expand : ∑ i : Fin n, (ψ i * ((ψ i)ᵀ * x)) = x := by
    calc ∑ i : Fin n, (ψ i * ((ψ i)ᵀ * x))
        = ∑ i : Fin n, ((ψ i * (ψ i)ᵀ) * x) := by
          congr 1; ext i; rw [Matrix.mul_assoc]
      _ = (∑ i : Fin n, (ψ i * (ψ i)ᵀ)) * x := by rw [Finset.mul_sum]
      _ = (1 : Matrix n n ℂ) * x := by rw [h_complete]
      _ = x := by simp
  convert expand using 1
  congr 1; ext i
  simp [Matrix.ext_iff, Pi.smul_apply]

-- ════════════════════════════════════════════════════════════════
-- PROBLEM 2: fibonacci_channel_contraction
-- ════════════════════════════════════════════════════════════════

/-- Golden ratio constant φ⁻¹ = (√5 - 1) / 2 -/
noncomputable def phi_inv : ℂ := ((Real.sqrt 5 - 1 : ℝ) : ℂ) / 2

/-- Key identity: φ⁻¹ + (φ⁻¹)² = 1 -/
lemma phi_identity : phi_inv + phi_inv ^ 2 = 1 := by norm_num [phi_inv]

/-- β = 1 - φ⁻¹ satisfies 0 < β < 1 -/
lemma beta_bounds : (0 : ℝ) < 1 - (phi_inv.re : ℝ) ∧ 1 - (phi_inv.re : ℝ) < 1 := by
  norm_num [phi_inv]
  constructor <;> nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 5)]

/-- For rank-1 density matrix ρ* = ψ ψ†:
    If tr(σ ρ*) = 0 then ρ* σ = 0 and σ ρ* = 0 -/
lemma rank_one_orthogonality 
    {σ ρ : Matrix n n ℂ}
    (h_rank_one : ∃ ψ : Matrix n 1 ℂ, ρ = ψ * ψᵀ)
    (h_trace_zero : trace (σ * ρ) = 0) :
    ρ * σ = 0 ∧ σ * ρ = 0 := by
  obtain ⟨ψ, rfl⟩ := h_rank_one
  -- Hilbert-Schmidt: tr(σ * ψ * ψᵀ) = 0 means (ψᵀ * σ * ψ)[0,0] = 0
  -- For rank-1 projector, this implies the product vanishes
  constructor <;> ext i j <;> simp_all [Matrix.mul_apply, Finset.sum_eq_zero_iff]
  all_goals (intro k; by_contra h; simp_all)

/-- Main: Fibonacci channel contracts on orthogonal complement -/
theorem fibonacci_channel_contraction_ahmad
    {n : Type*} [Fintype n] [DecidableEq n]
    (U : Matrix n n ℂ)
    (hU : U * Uᵀ = 1)
    (ρ_star : Matrix n n ℂ)
    (h_rank_one : ∃ ψ : Matrix n 1 ℂ, ρ_star = ψ * ψᵀ)
    (h_jordan : U = (phi_inv : ℂ) • ρ_starᵀ + (1 - phi_inv : ℂ) • 1) :
    ∀ (σ : Matrix n n ℂ),
      trace (σ * ρ_star) = 0 →
      ‖U * σ * Uᵀ‖ ≤ ((1 - (phi_inv.re : ℝ)) ^ 2) * ‖σ‖ := by
  intro σ h_ortho
  have h₁ : ρ_star * σ = 0 ∧ σ * ρ_star = 0 := 
    rank_one_orthogonality h_rank_one h_ortho
  have h_factor : U * σ * Uᵀ = ((1 - phi_inv : ℂ) ^ 2) • σ := by
    rw [h_jordan]
    simp [Matrix.mul_add, Matrix.add_mul, Matrix.mul_smul, Matrix.smul_mul]
    -- Expand Jordan decomposition: (aρ* + b·I)σ(aρ* + b·I)ᵀ
    -- Cross terms ρ*σ = 0 and σρ* = 0 from h₁, leaving b²·σ
    obtain ⟨h_left, h_right⟩ := h₁
    simp [h_left, h_right, Matrix.mul_zero, Matrix.zero_mul]
    ring
  rw [h_factor]
  simp [norm_smul]
  -- |(1-φ⁻¹)²| < 1 since φ⁻¹ ∈ (0,1)
  have h_phi : (0 : ℝ) < phi_inv.re ∧ phi_inv.re < 1 := phi_inv_in_unit_interval
  nlinarith [h_phi.1, h_phi.2, sq_nonneg (1 - phi_inv.re)]

end SovMonster