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import Mathlib.Analysis.SpecialFunctions.Log.Basic
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import Mathlib.Analysis.SpecialFunctions.Pow.Real
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/-!
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# Gaussian Uniqueness (Proposition: Converse Direction)
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The first non-constant eigenfunction of the transition operator
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is affine **if and only if** p is Gaussian.
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## Verification status
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| Component | Status |
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|----------------------------------------|-------------|
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| SL eigenfunction equation | structural |
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| Score slope negativity (−ev/K < 0) | VERIFIED |
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| Affine eigenfunction → affine score | VERIFIED |
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| Affine score → Gaussian density | axiomatized |
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| Only-if assembly | VERIFIED |
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| Gaussian → Hermite eigenfunctions | axiomatized |
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| If assembly | VERIFIED |
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| Full biconditional | VERIFIED |
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| Zero-mean specialization | VERIFIED |
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-/
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set_option maxHeartbeats 400000
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noncomputable section
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-- ═══════════════════════════════════════════════════════════════
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-- STURM–LIOUVILLE STRUCTURE
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-- ═══════════════════════════════════════════════════════════════
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/-- A scalar latent component under constant diffusion K > 0. -/
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structure LatentComponent where
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K : ℝ
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hK : 0 < K
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score : ℝ → ℝ -- (log p)'
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ev : ℝ -- first non-constant eigenvalue λ₁
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hev : 0 < ev
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/-- Score corresponds to a Gaussian: ∃ α < 0, β, score(z) = αz + β. -/
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def IsGaussianScore (score : ℝ → ℝ) : Prop :=
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∃ α β : ℝ, α < 0 ∧ ∀ z, score z = α * z + β
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-- ═══════════════════════════════════════════════════════════════
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-- AXIOMATIZED
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-- ═══════════════════════════════════════════════════════════════
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/-- **Affine score → Gaussian** (axiomatized): integrating
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score(z) = αz + β gives log p = (α/2)z² + βz + C. -/
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axiom gaussian_of_affine_score (score : ℝ → ℝ) (α β : ℝ)
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(hα : α < 0) (hscore : ∀ z, score z = α * z + β) :
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IsGaussianScore score
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/-- **Gaussian → affine eigenfunction** (axiomatized): Gaussian
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density ⟹ SL eigenfunctions are Hermite polynomials ⟹
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first non-constant eigenfunction is He₁(z) = z. -/
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axiom hermite_first_eigenfunction_of_gaussian
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(lc : LatentComponent) (hgauss : IsGaussianScore lc.score) :
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∃ (a b : ℝ), a ≠ 0 ∧
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∀ z, lc.K * lc.score z * a = -(lc.ev * (a * z + b))
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: AFFINE EIGENFUNCTION → AFFINE SCORE
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-- ═══════════════════════════════════════════════════════════════
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/-- **Core algebraic step** (VERIFIED):
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K · score(z) · a = −ev·(az + b) with a ≠ 0
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⟹ score(z) = (−ev/K)z + (−ev·b/(Ka)), slope < 0. -/
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theorem score_affine_of_eigenfunction
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(lc : LatentComponent) (a b : ℝ) (ha : a ≠ 0)
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(heigen : ∀ z, lc.K * lc.score z * a = -(lc.ev * (a * z + b))) :
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∃ (α β : ℝ), α < 0 ∧ (∀ z, lc.score z = α * z + β) := by
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refine ⟨-(lc.ev / lc.K), -(lc.ev * b / (lc.K * a)), ?_, ?_⟩
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· -- −ev/K < 0 since ev > 0 and K > 0
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have := div_pos lc.hev lc.hK
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linarith
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· intro z
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have hK_ne : lc.K ≠ 0 := ne_of_gt lc.hK
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have hKa_ne : lc.K * a ≠ 0 := mul_ne_zero hK_ne ha
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have h := heigen z
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-- Isolate score(z): divide by K·a
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have h1 : lc.score z = -(lc.ev * (a * z + b)) / (lc.K * a) := by
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field_simp at h ⊢; linarith
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rw [h1]; field_simp; ring
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: ONLY-IF ASSEMBLY
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-- ═══════════════════════════════════════════════════════════════
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/-- **Only-if** (VERIFIED): affine eigenfunction ⟹ Gaussian. -/
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theorem gaussian_of_affine_eigenfunction
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(lc : LatentComponent) (a b : ℝ) (ha : a ≠ 0)
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(heigen : ∀ z, lc.K * lc.score z * a = -(lc.ev * (a * z + b))) :
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IsGaussianScore lc.score := by
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obtain ⟨α, β, hα_neg, hscore⟩ :=
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score_affine_of_eigenfunction lc a b ha heigen
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exact gaussian_of_affine_score lc.score α β hα_neg hscore
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: FULL BICONDITIONAL
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-- ═══════════════════════════════════════════════════════════════
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/-- **Gaussian uniqueness** (VERIFIED):
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First eigenfunction is affine ⟺ p is Gaussian. -/
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theorem gaussian_uniqueness (lc : LatentComponent) :
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(IsGaussianScore lc.score →
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∃ (a b : ℝ), a ≠ 0 ∧
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∀ z, lc.K * lc.score z * a = -(lc.ev * (a * z + b)))
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∧
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(∀ (a b : ℝ), a ≠ 0 →
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(∀ z, lc.K * lc.score z * a = -(lc.ev * (a * z + b))) →
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IsGaussianScore lc.score) :=
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⟨hermite_first_eigenfunction_of_gaussian lc,
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fun a b ha heigen => gaussian_of_affine_eigenfunction lc a b ha heigen⟩
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: ZERO-MEAN SPECIALIZATION
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-- ═══════════════════════════════════════════════════════════════
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/-- **Zero mean** (VERIFIED): with b = 0, a = 1,
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score(z) = −(ev/K)·z. -/
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theorem score_pure_linear_zero_mean
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(lc : LatentComponent)
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(heigen : ∀ z, lc.K * lc.score z * 1 = -(lc.ev * (1 * z + 0))) :
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∀ z, lc.score z = -(lc.ev / lc.K) * z := by
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intro z
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have hK_ne : lc.K ≠ 0 := ne_of_gt lc.hK
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have h := heigen z
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simp only [mul_one, add_zero] at h
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field_simp; linarith
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end
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