| import OAI.Probability.InvariantIsing.Cavity.CavityBasePerturbedLaw |
| import OAI.Probability.InvariantIsing.Arrays.TensorPerturbedWardLimit |
|
|
| namespace OAI |
|
|
| /-! Both Ward identities for the physical base array. The SO law is |
| derived from its actual Haar eigenbasis, and its Gaussian realization is |
| identified by the full array-law equality. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter |
| open scoped Topology |
|
|
| namespace InvariantIsing |
|
|
| theorem cavity_rotation_ward_limits |
| (N : ℕ → ℕ) (hN : ∀ k, 0 < N k) (hNlim : Tendsto N atTop atTop) (m depth : ℕ) |
| (μ : (k : ℕ) → Measure (Orthogonal (N k))) [∀ k, IsProbabilityMeasure (μ k)] |
| [∀ k, (μ k).IsMulRightInvariant] |
| (eig : (k : ℕ) → Fin (N k) → ℝ) |
| (I : (k : ℕ) → Fin m → Finset (Fin (N k))) |
| (hdis : ∀ k, Set.PairwiseDisjoint (Set.univ : Set (Fin m)) (I k)) |
| (hcover : ∀ k, Finset.univ.biUnion (I k) = Finset.univ) |
| (lam : Fin m → ℝ) (hlam : ∀ k a i, i ∈ I k a → eig k i = lam a) |
| (u : (k : ℕ) → Fin (N k) → ℝ) (hu : ∀ k j, |u k j| ≤ 2) |
| (v : ℕ → Fin m → ℝ) (hv : ∀ k a, |v k a| ≤ 2) |
| (t : ℕ → ℝ) {t₀ : ℝ} (ht : Tendsto t atTop (𝓝 t₀)) (b : ℕ → ℝ) |
| (Q : ProbabilityMeasure (SpectralArray (m+1))) |
| (hL : Tendsto (fun k => cavityRotationArrayLaw (μ k) |
| (labeledCascadeLaw depth b : Measure (LabeledTree depth)) |
| (diagonalPerturbedEigenvalues (eig k) (I k) (v k) (t k)) (I k) (cavityBaseAmplitude (u k))) |
| atTop (𝓝 Q)) |
| (ρ : Fin m → ℝ) |
| (hρ : Tendsto (fun k a => ((I k a).card : ℝ)/N k) atTop (𝓝 ρ)) : |
| (∀ a b₀, ∀ Φ : ℝ → ℝ, Continuous Φ → ∀ B : ℝ, 0 ≤ B → (∀ r, |Φ r| ≤ B) → |
| spectralOffWardResidual Q ρ (fun a => t₀*lam a) a b₀ Φ = 0) ∧ |
| (∀ a b₀, spectralDiagonalWardResidual Q ρ (fun a => t₀*lam a) a b₀ = 0) := by |
| have hEq k := cavity_base_perturbed_array_law (depth := depth) |
| (hN k) (μ k) (eig k) (I k) (u k) (v k) (t k) b |
| simp_rw [hEq] at hL |
| exact tensorPerturbedArrayLaw_ward_limits N hN hNlim m depth |
| (fun k => cavityOrientedBaseLaw (hN k) (μ k)) |
| (fun k => cavityOrientedBaseLaw_leftInvariant (hN k) (μ k)) |
| eig (fun _ _ => 0) I hdis hcover lam hlam u hu v hv t ht b |
| (fun _ _ => 0) (fun _ => monotone_const) (fun _ => le_rfl) Q hL ρ hρ |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
