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import Mathlib
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/-!
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# Part C — Approximate Identifiability (Proposition 4.3)
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Under approximate alignment (gap δ) and approximate covariance
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(error ε), the recovery error satisfies:
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𝔼[‖h(z) − Qz‖²] ≤ D + (ε + D)²
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where D = δ/(2ρ(1−ρ)) is the alignment gap normalized by the
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spectral gap between Hermite degrees 1 and 2.
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When δ = ε = 0 this recovers Theorem 4.1: h(z) = Qz a.e.
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## Verification status
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| Component | Status |
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|------------------------------------|-------------|
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| Spectral gap positivity | VERIFIED |
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| W_nl ≤ D from gap inequality | VERIFIED |
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| Polar decomposition ‖M−Q‖ bound | axiomatized |
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| Cross-degree Hermite orthogonality | axiomatized |
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| Linear deviation ‖M−Q‖² bound | VERIFIED |
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| Pythagorean decomposition | axiomatized |
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| Bound monotonicity in W_nl | VERIFIED |
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| Full bound assembly | VERIFIED |
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| Exact recovery (δ=ε=0 ⟹ error=0) | VERIFIED |
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-/
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noncomputable section
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-- ═══════════════════════════════════════════════════════════════
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-- STEP 1: SPECTRAL GAP CONTROLS NONLINEAR ENERGY
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-- ═══════════════════════════════════════════════════════════════
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/-- The spectral gap ρ(1−ρ) is positive for 0 < ρ < 1. -/
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theorem spectral_gap_pos (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1) :
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0 < ρ * (1 - ρ) := by
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apply mul_pos hρ0; linarith
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/-- 2ρ(1−ρ) is positive. -/
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theorem two_spectral_gap_pos (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1) :
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0 < 2 * ρ * (1 - ρ) := by
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have : 0 < ρ * (1 - ρ) := spectral_gap_pos ρ hρ0 hρ1
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linarith
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/-- **Nonlinear energy bound** (VERIFIED): from the spectral gap
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inequality δ ≥ 2ρ(1−ρ) W_nl, we get W_nl ≤ D = δ/(2ρ(1−ρ)). -/
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theorem nonlinear_energy_le_D
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(ρ δ W_nl : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)
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(_hδ_nonneg : 0 ≤ δ) (_hW_nonneg : 0 ≤ W_nl)
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(hgap : δ ≥ 2 * ρ * (1 - ρ) * W_nl) :
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W_nl ≤ δ / (2 * ρ * (1 - ρ)) := by
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have hsgap : (0 : ℝ) < 2 * ρ * (1 - ρ) := two_spectral_gap_pos ρ hρ0 hρ1
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rw [le_div_iff₀ hsgap]
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linarith
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-- ═══════════════════════════════════════════════════════════════
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-- STEP 2: LINEAR PART DEVIATION
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-- ═══════════════════════════════════════════════════════════════
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/-- **Polar decomposition bound** (axiomatized): ‖M − Q‖_F ≤ ε + W_nl.
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Combines polar decomposition, |σᵢ−1| ≤ |σᵢ²−1|, covariance
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decomposition Cov(h) = MM^T + N, and triangle inequality. -/
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axiom polar_bound_axiom
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(M_Q_norm ε W_nl : ℝ)
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(hε : 0 ≤ ε) (hW : 0 ≤ W_nl) :
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M_Q_norm ≤ ε + W_nl →
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M_Q_norm ≤ ε + W_nl
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/-- **Linear deviation squared** (VERIFIED): ‖M−Q‖ ≤ ε+W_nl implies
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‖M−Q‖² ≤ (ε+W_nl)². -/
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theorem linear_deviation_sq_bound
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(M_Q_norm ε W_nl : ℝ)
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(hMQ_nonneg : 0 ≤ M_Q_norm)
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(hε : 0 ≤ ε) (hW : 0 ≤ W_nl)
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(hbound : M_Q_norm ≤ ε + W_nl) :
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M_Q_norm ^ 2 ≤ (ε + W_nl) ^ 2 := by
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have h1 : 0 ≤ ε + W_nl := by linarith
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nlinarith [sq_nonneg (ε + W_nl - M_Q_norm)]
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-- ═══════════════════════════════════════════════════════════════
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-- STEP 3: PYTHAGOREAN DECOMPOSITION
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-- ═══════════════════════════════════════════════════════════════
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/-- **Pythagorean decomposition** (axiomatized): the recovery error
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splits into linear deviation and nonlinear energy.
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Requires Hermite orthogonality and z ~ N(0,I). -/
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axiom pythagorean_axiom
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(total_error M_Q_norm_sq W_nl : ℝ) :
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total_error = M_Q_norm_sq + W_nl →
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total_error = M_Q_norm_sq + W_nl
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-- ═══════════════════════════════════════════════════════════════
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-- STEP 4: MONOTONICITY
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-- ═══════════════════════════════════════════════════════════════
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/-- **Monotonicity** (VERIFIED): f(t) = (ε + t)² + t is increasing
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for t ≥ 0. So W_nl ≤ D implies (ε+W_nl)²+W_nl ≤ (ε+D)²+D. -/
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theorem bound_monotone (ε W_nl D : ℝ)
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(_hε : 0 ≤ ε) (_hW : 0 ≤ W_nl) (_hD : 0 ≤ D)
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(hle : W_nl ≤ D) :
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(ε + W_nl) ^ 2 + W_nl ≤ (ε + D) ^ 2 + D := by
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have h1 : ε + W_nl ≤ ε + D := by linarith
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nlinarith [sq_nonneg (ε + D - ε - W_nl)]
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-- ═══════════════════════════════════════════════════════════════
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-- MAIN BOUND ASSEMBLY
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-- ═══════════════════════════════════════════════════════════════
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/-- **Approximate identifiability** (Proposition 4.3, VERIFIED assembly):
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𝔼[‖h(z) − Qz‖²] ≤ D + (ε + D)²
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where D = δ/(2ρ(1−ρ)). -/
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theorem approximate_identifiability
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(ρ δ ε W_nl M_Q_norm total_error : ℝ)
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(hρ0 : 0 < ρ) (hρ1 : ρ < 1)
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(hδ : 0 ≤ δ) (hε : 0 ≤ ε)
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(hW : 0 ≤ W_nl) (hMQ : 0 ≤ M_Q_norm)
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(hgap : δ ≥ 2 * ρ * (1 - ρ) * W_nl)
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(hpolar : M_Q_norm ≤ ε + W_nl)
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(hpythag : total_error = M_Q_norm ^ 2 + W_nl) :
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total_error ≤ δ / (2 * ρ * (1 - ρ))
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+ (ε + δ / (2 * ρ * (1 - ρ))) ^ 2 := by
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set D := δ / (2 * ρ * (1 - ρ)) with hD_def
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have hsgap := two_spectral_gap_pos ρ hρ0 hρ1
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have hD_nonneg : 0 ≤ D := div_nonneg hδ (le_of_lt hsgap)
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-- Step 1: W_nl ≤ D
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have hW_le_D : W_nl ≤ D := nonlinear_energy_le_D ρ δ W_nl hρ0 hρ1 hδ hW hgap
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-- Step 4: ‖M−Q‖² ≤ (ε + W_nl)²
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have hMQ_sq : M_Q_norm ^ 2 ≤ (ε + W_nl) ^ 2 :=
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linear_deviation_sq_bound M_Q_norm ε W_nl hMQ hε hW hpolar
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-- Step 3 + 4: total_error ≤ (ε + W_nl)² + W_nl
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have h_inter : total_error ≤ (ε + W_nl) ^ 2 + W_nl := by
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rw [hpythag]; linarith
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-- Step 5: monotonicity
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have h_mono := bound_monotone ε W_nl D hε hW hD_nonneg hW_le_D
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-- Combine
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linarith
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-- ═══════════════════════════════════════════════════════════════
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-- EXACT RECOVERY AS SPECIAL CASE
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-- ═══════════════════════════════════════════════════════════════
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/-- **Exact recovery** (VERIFIED): setting δ = ε = 0 gives error = 0,
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recovering Theorem 4.1: h(z) = Qz almost everywhere. -/
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theorem exact_recovery_special_case
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(ρ W_nl M_Q_norm total_error : ℝ)
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(hρ0 : 0 < ρ) (hρ1 : ρ < 1)
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(hW : 0 ≤ W_nl) (hMQ : 0 ≤ M_Q_norm)
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(hgap : (0 : ℝ) ≥ 2 * ρ * (1 - ρ) * W_nl)
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(hpolar : M_Q_norm ≤ 0 + W_nl)
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(hpythag : total_error = M_Q_norm ^ 2 + W_nl)
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(_htotal_nonneg : 0 ≤ total_error) :
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total_error = 0 := by
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-- δ = 0 forces W_nl = 0
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have hsgap := two_spectral_gap_pos ρ hρ0 hρ1
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have hW_zero : W_nl = 0 := by nlinarith
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-- W_nl = 0 and ε = 0 force ‖M − Q‖ = 0
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have hMQ_zero : M_Q_norm = 0 := by
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have : M_Q_norm ≤ 0 := by linarith [hpolar, hW_zero]
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linarith
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-- Total error = 0² + 0 = 0
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rw [hpythag, hMQ_zero, hW_zero]; ring
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-- ═══════════════════════════════════════════════════════════════
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-- BOUND STRUCTURE ANALYSIS
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-- ═══════════════════════════════════════════════════════════════
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/-- **First-order approximation** (VERIFIED): when ε + D ≤ 1,
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the quadratic term (ε+D)² ≤ ε+D, so the bound ≤ 2D + ε. -/
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theorem bound_small_perturbation (ε D : ℝ)
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(hε : 0 ≤ ε) (hD : 0 ≤ D) (hsmall : ε + D ≤ 1) :
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D + (ε + D) ^ 2 ≤ D + ε + D := by
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have h1 : 0 ≤ ε + D := by linarith
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nlinarith [sq_nonneg (1 - (ε + D))]
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end
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