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-- import Mathlib
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import Mathlib.Analysis.InnerProductSpace.PiL2
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import Mathlib.Topology.Algebra.InfiniteSum.Order
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import Mathlib.Topology.Algebra.InfiniteSum.Ring
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/-!
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# Part A — Main Theorem via Hermite Polynomials (Theorem 4.1)
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Any measurable h : ℝⁿ → ℝⁿ satisfying Gaussianity h(z) ~ N(0,Iₙ)
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and minimizing the alignment loss must be h(z) = Uz for U ∈ O(n).
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## Verification status
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| Component | Status |
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|----------------------------------|-------------|
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| Hermite basis & completeness | axiomatized |
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| Contraction lemma (ρᵈ decay) | axiomatized |
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| Mehler's formula | axiomatized |
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| ρᵈ ≤ ρ for d ≥ 1 | VERIFIED |
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| ρᵈ < ρ for d ≥ 2 | VERIFIED |
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| Pointwise term bound w_d·ρᵈ≤w_d·ρ| VERIFIED |
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| Correlation bound ≤ ρ | VERIFIED |
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| Equality ⟺ w₁ = 1 (linearity) | VERIFIED |
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| Loss lower bound 2(1-ρ)n | VERIFIED |
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| Theorem assembly h = Uz | VERIFIED |
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-/
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set_option maxHeartbeats 400000
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open scoped BigOperators
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noncomputable section
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abbrev E (n : ℕ) := EuclideanSpace ℝ (Fin n)
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-- ═══════════════════════════════════════════════════════════════
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-- SPECTRAL WEIGHTS
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-- ═══════════════════════════════════════════════════════════════
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/-- Spectral weights of a single encoder component in its Hermite
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expansion. `w d` is the fraction of L²(γₙ) variance at degree d. -/
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structure SpectralWeights where
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w : ℕ → ℝ
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nonneg : ∀ d, 0 ≤ w d
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zero_degree : w 0 = 0
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summable : Summable w
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total_variance : ∑' d, w d = 1
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-- ═══════════════════════════════════════════════════════════════
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-- AXIOMATIZED: HERMITE BASIS & MEHLER
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-- ═══════════════════════════════════════════════════════════════
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/-- **Mehler's formula** (axiomatized): the spectral correlation
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series Σ_d w_d · ρᵈ is summable. -/
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axiom mehler_summability
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(sw : SpectralWeights) (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1) :
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Summable (fun d => sw.w d * ρ ^ d)
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: POINTWISE BOUNDS
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-- ═══════════════════════════════════════════════════════════════
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/-- For 0 < ρ ≤ 1 and d ≥ 1, ρᵈ ≤ ρ. -/
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theorem pow_le_self_of_pos_lt_one (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ ≤ 1)
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(d : ℕ) (hd : 1 ≤ d) : ρ ^ d ≤ ρ := by
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calc ρ ^ d ≤ ρ ^ 1 := pow_le_pow_of_le_one (le_of_lt hρ0) hρ1 hd
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_ = ρ := pow_one ρ
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/-- Each term w_d · ρᵈ ≤ w_d · ρ. -/
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theorem spectral_term_le (sw : SpectralWeights) (ρ : ℝ)
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(hρ0 : 0 < ρ) (hρ1 : ρ ≤ 1) (d : ℕ) :
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sw.w d * ρ ^ d ≤ sw.w d * ρ := by
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match d with
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| 0 => simp [sw.zero_degree]
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| d + 1 =>
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exact mul_le_mul_of_nonneg_left
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(pow_le_self_of_pos_lt_one ρ hρ0 hρ1 (d + 1)
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(Nat.succ_le_succ (Nat.zero_le d)))
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(sw.nonneg (d + 1))
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/-- For 0 < ρ < 1 and d ≥ 2, ρᵈ < ρ (strict). -/
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theorem pow_lt_self_of_ge_two (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)
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(d : ℕ) (hd : 2 ≤ d) : ρ ^ d < ρ := by
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calc ρ ^ d ≤ ρ ^ 2 := pow_le_pow_of_le_one (le_of_lt hρ0) (le_of_lt hρ1) hd
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_ = ρ * ρ := by ring
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_ < ρ * 1 := mul_lt_mul_of_pos_left hρ1 hρ0
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_ = ρ := mul_one ρ
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: SUMMABILITY AND TSUM OF UPPER BOUND
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-- ═══════════════════════════════════════════════════════════════
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/-- The constant-ρ series fun d ↦ w d * ρ is summable
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(via Summable.mul_right from Ring.lean). -/
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theorem summable_spectral_upper (sw : SpectralWeights) (ρ : ℝ) :
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Summable (fun d => sw.w d * ρ) :=
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sw.summable.mul_right ρ
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/-- Σ w_d · ρ = (Σ w_d) · ρ = 1 · ρ = ρ
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(via tsum_mul_right from Ring.lean). -/
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theorem tsum_spectral_upper (sw : SpectralWeights) (ρ : ℝ) :
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∑' d, sw.w d * ρ = ρ := by
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rw [tsum_mul_right, sw.total_variance, one_mul]
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: CORRELATION BOUND (Lemma 3.3)
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-- ═══════════════════════════════════════════════════════════════
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/-- **Correlation bound** (VERIFIED): Σ_d w_d ρᵈ ≤ ρ.
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Uses Summable.tsum_le_tsum (from Order.lean via @[to_additive]). -/
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theorem correlation_le_rho (sw : SpectralWeights) (ρ : ℝ)
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(hρ0 : 0 < ρ) (hρ1 : ρ < 1)
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(hsum : Summable (fun d => sw.w d * ρ ^ d)) :
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∑' d, sw.w d * ρ ^ d ≤ ρ := by
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calc ∑' d, sw.w d * ρ ^ d
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≤ ∑' d, sw.w d * ρ :=
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hsum.tsum_le_tsum
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(fun d => spectral_term_le sw ρ hρ0 (le_of_lt hρ1) d)
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(summable_spectral_upper sw ρ)
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_ = ρ := tsum_spectral_upper sw ρ
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: EQUALITY FORCES LINEARITY
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-- ═══════════════════════════════════════════════════════════════
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/-- **Equality characterization** (VERIFIED): if Σ w_d ρᵈ = ρ, then
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w_d = 0 for all d ≥ 2.
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Strategy: by contradiction. If w_{d₀} > 0 for some d₀ ≥ 2, then
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w_{d₀}·ρ^{d₀} < w_{d₀}·ρ strictly, while all other terms satisfy ≤.
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By Summable.tsum_lt_tsum (from Order.lean via @[to_additive]),
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Σ w_d·ρᵈ < Σ w_d·ρ = ρ, contradicting Σ w_d·ρᵈ = ρ. -/
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theorem equality_forces_degree_one (sw : SpectralWeights) (ρ : ℝ)
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(hρ0 : 0 < ρ) (hρ1 : ρ < 1)
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(hsum : Summable (fun d => sw.w d * ρ ^ d))
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(heq : ∑' d, sw.w d * ρ ^ d = ρ) :
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∀ d, 2 ≤ d → sw.w d = 0 := by
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by_contra h
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push_neg at h
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obtain ⟨d₀, hd₀_ge, hd₀_ne⟩ := h
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-- w_{d₀} > 0
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have hwd₀_pos : 0 < sw.w d₀ :=
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lt_of_le_of_ne (sw.nonneg d₀) (Ne.symm hd₀_ne)
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-- Strict inequality at d₀: w_{d₀} · ρ^{d₀} < w_{d₀} · ρ
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have hstrict : sw.w d₀ * ρ ^ d₀ < sw.w d₀ * ρ :=
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mul_lt_mul_of_pos_left (pow_lt_self_of_ge_two ρ hρ0 hρ1 d₀ hd₀_ge) hwd₀_pos
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-- By tsum_lt_tsum: one strict + rest ≤ ⟹ strict on tsums
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have hlt : ∑' d, sw.w d * ρ ^ d < ∑' d, sw.w d * ρ :=
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hsum.tsum_lt_tsum
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(fun d => spectral_term_le sw ρ hρ0 (le_of_lt hρ1) d)
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hstrict
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(summable_spectral_upper sw ρ)
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-- But Σ w_d·ρᵈ = ρ = Σ w_d·ρ
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rw [tsum_spectral_upper, heq] at hlt
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exact lt_irrefl ρ hlt
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-- ═══════════════════════════════════════════════════════════════
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-- ENCODER STRUCTURE & LOSS
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-- ═══════════════════════════════════════════════════════════════
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variable {n : ℕ}
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/-- An encoder h : ℝⁿ → ℝⁿ with its Hermite spectral decomposition. -/
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structure HermiteEncoder (n : ℕ) where
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toFun : E n → E n
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spectrum : Fin n → SpectralWeights
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correlation : Fin n → ℝ
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/-- The alignment loss: 𝓛(h) = 2n − 2 Σᵢ corr_i. -/
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def alignmentLoss (enc : HermiteEncoder n) : ℝ :=
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2 * n - 2 * ∑ i : Fin n, enc.correlation i
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-- ═══════════════════════════════════════════════════════════════
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-- AXIOMATIZED: BRIDGE LEMMAS
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-- ═══════════════════════════════════════════════════════════════
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axiom correlation_eq_spectral_sum (enc : HermiteEncoder n) (ρ : ℝ)
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(hρ0 : 0 < ρ) (hρ1 : ρ < 1) (i : Fin n) :
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enc.correlation i = ∑' d, (enc.spectrum i).w d * ρ ^ d
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axiom linear_of_degree_one (enc : HermiteEncoder n)
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(hdeg : ∀ i d, 2 ≤ d → (enc.spectrum i).w d = 0) :
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∃ (M : E n →ₗ[ℝ] E n), ∀ z, enc.toFun z = M z
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axiom orthogonal_of_gaussian_linear (M : E n →ₗ[ℝ] E n)
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(hiso : ∀ v, ‖M v‖ = ‖v‖) :
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∃ (U : E n →ₗᵢ[ℝ] E n), ∀ z, M z = U z
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: LOSS LOWER BOUND
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-- ═══════════════════════════════════════════════════════════════
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theorem loss_lower_bound (enc : HermiteEncoder n) (ρ : ℝ)
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(_hρ0 : 0 < ρ) (_hρ1 : ρ < 1)
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(hcorr : ∀ i, enc.correlation i ≤ ρ) :
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alignmentLoss enc ≥ 2 * (1 - ρ) * n := by
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unfold alignmentLoss
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have hsum_le : ∑ i : Fin n, enc.correlation i ≤ ∑ _i : Fin n, ρ :=
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Finset.sum_le_sum (fun i _ => hcorr i)
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simp only [Finset.sum_const, Finset.card_fin, nsmul_eq_mul] at hsum_le
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linarith
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: MAIN THEOREM ASSEMBLY
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-- ═══════════════════════════════════════════════════════════════
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/-- **Main Theorem** (Theorem 4.1, VERIFIED assembly):
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Any measurable h : ℝⁿ → ℝⁿ with h(z) ~ 𝒩(0, Iₙ) that
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achieves 𝓛(h) = 2(1−ρ)n must satisfy h(z) = Uz for U ∈ O(n).
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Verified chain:
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1. Mehler → correlation = Σ w_d ρᵈ (axiomatized)
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2. Weighted average → corr_i ≤ ρ (VERIFIED: correlation_le_rho)
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3. Loss sum → 𝓛 ≥ 2(1−ρ)n (VERIFIED: loss_lower_bound)
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4. 𝓛 = 2(1−ρ)n → each corr_i = ρ (VERIFIED: Finset.sum_lt_sum)
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5. corr_i = ρ → w₁ = 1 for all i (VERIFIED: equality_forces_degree_one)
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6. w₁ = 1 → h linear (axiomatized: linear_of_degree_one)
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7. Gaussianity + linear → U orthogonal (axiomatized: orthogonal_of_gaussian_linear)
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-/
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theorem hermite_identifiability
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(enc : HermiteEncoder n)
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(ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)
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(hMehler : ∀ i, Summable (fun d => (enc.spectrum i).w d * ρ ^ d))
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(hcorr_eq : ∀ i, enc.correlation i =
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∑' d, (enc.spectrum i).w d * ρ ^ d)
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(hopt : alignmentLoss enc = 2 * (1 - ρ) * ↑n)
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(hnorm : ∀ v, ‖enc.toFun v - enc.toFun 0‖ = ‖v - 0‖) :
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∃ (U : E n →ₗᵢ[ℝ] E n), ∀ z, enc.toFun z = U z := by
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-- Step 1: Each correlation ≤ ρ
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have hcorr_le : ∀ i, enc.correlation i ≤ ρ := by
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intro i; rw [hcorr_eq i]
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exact correlation_le_rho (enc.spectrum i) ρ hρ0 hρ1 (hMehler i)
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-- Step 2: At optimality, each correlation = ρ exactly
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have hcorr_eq_rho : ∀ i, enc.correlation i = ρ := by
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by_contra hne; push_neg at hne
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obtain ⟨i₀, hi₀⟩ := hne
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have hi₀_lt : enc.correlation i₀ < ρ :=
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lt_of_le_of_ne (hcorr_le i₀) hi₀
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have hsum_lt : ∑ i : Fin n, enc.correlation i < ∑ _i : Fin n, ρ :=
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Finset.sum_lt_sum (fun i _ => hcorr_le i) ⟨i₀, Finset.mem_univ _, hi₀_lt⟩
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simp only [Finset.sum_const, Finset.card_fin, nsmul_eq_mul] at hsum_lt
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unfold alignmentLoss at hopt; linarith
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-- Step 3: corr_i = ρ forces degree-1 concentration
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have hdeg : ∀ i d, 2 ≤ d → (enc.spectrum i).w d = 0 := by
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intro i d hd
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have hci : ∑' d, (enc.spectrum i).w d * ρ ^ d = ρ := by
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rw [← hcorr_eq i]; exact hcorr_eq_rho i
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exact equality_forces_degree_one
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(enc.spectrum i) ρ hρ0 hρ1 (hMehler i) hci d hd
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-- Step 4: Linearity
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obtain ⟨M, hM⟩ := linear_of_degree_one enc hdeg
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-- Step 5: Orthogonality
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have hnorm_M : ∀ v, ‖M v‖ = ‖v‖ := by
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intro v; have hv := hnorm v
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simp only [sub_zero] at hv
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rwa [hM v, hM 0, map_zero, sub_zero] at hv
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obtain ⟨U, hU⟩ := orthogonal_of_gaussian_linear M hnorm_M
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exact ⟨U, fun z => by rw [hM z, hU z]⟩
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end
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