openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Cavity /CavityCanonicalSynchronization.lean
| import OAI.Probability.InvariantIsing.Cavity.CavityCanonicalLabels | |
| import OAI.Probability.InvariantIsing.Spectral.SpectralCanonicalIdentification | |
| namespace OAI | |
| /-! The limiting Ward equations give the canonical monotone spectral | |
| labels on every off-diagonal entry, not only along quantile parameters. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter | |
| open scoped BigOperators Topology | |
| namespace InvariantIsing | |
| theorem cavity_canonical_group_synchronization {m : ℕ} | |
| {Q : ProbabilityMeasure (SpectralArray (m + 1))} | |
| (hgg : HasEntryGhirlandaGuerra (fun x i j => x (i,j)) (Q : Measure (SpectralArray (m + 1)))) | |
| (hG : ∀ᵐ x ∂(Q : Measure (SpectralArray (m + 1))), SpectralGram x) | |
| (q : Fin (m + 1) → ℝ) (hq : ∀ a, 0 ≤ q a) | |
| (hd : ∀ᵐ x ∂(Q : Measure (SpectralArray (m + 1))), ∀ i a, (x (i,i) a : ℝ) = q a) | |
| (hE : ∀ e : ℕ → ℕ, Function.Injective e → | |
| (Q : Measure (SpectralArray (m + 1))).map (permuteSpectralArray e) = Q) | |
| (hP : ∀ᵐ x ∂(Q : Measure (SpectralArray (m + 1))), SpectralPartitionGeometry m x) | |
| (hn : ∀ᵐ x ∂(Q : Measure (SpectralArray (m + 1))), ∀ a, 0 ≤ (x (0,1) a : ℝ)) | |
| (ρ eig : Fin m → ℝ) (hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) | |
| (hoff : ∀ a b, ∀ Φ : ℝ → ℝ, Continuous Φ → ∀ C : ℝ, 0 ≤ C → (∀ r, |Φ r| ≤ C) → | |
| spectralOffWardResidual Q ρ eig a b Φ = 0) | |
| (hdiag : ∀ a b, spectralDiagonalWardResidual Q ρ eig a b = 0) : | |
| let p := spectralSpinQuantilePath Q hP hn | |
| (∀ a, q a.castSucc = spectralGroupDiagonal ρ eig hρ hρsum p a) ∧ | |
| ∀ᵐ x ∂(Q : Measure (SpectralArray (m + 1))), ∀ i j, i ≠ j → ∀ a : Fin m, | |
| (x (i,j) a.castSucc : ℝ) = | |
| cavityCanonicalCoordinate ρ eig hρ hρsum p a (spectralSpinArray x i j) := by | |
| intro p | |
| obtain ⟨hdq, hc⟩ := spectralCanonical_identification hgg hG q hq hd hE hP hn | |
| ρ eig hρ hρsum hoff hdiag | |
| refine ⟨hdq, ?_⟩ | |
| have hall (a : Fin m) : ∀ᵐ x ∂(Q : Measure (SpectralArray (m + 1))), ∀ i j, i ≠ j → | |
| (x (i,j) a.castSucc : ℝ) = | |
| cavityCanonicalCoordinate ρ eig hρ hρsum p a (spectralSpinArray x i j) := by | |
| apply spectral_factor_all_off_diagonal hE a _ | |
| (continuous_cavityCanonicalCoordinate ρ eig hρ hρsum p a) | |
| have hsingle : ∀ b, 0 ≤ (Pi.single a.castSucc 1 : Fin (m + 1) → ℝ) b := by | |
| intro b | |
| simp only [Pi.single_apply] | |
| split_ifs <;> norm_num | |
| have hp : ∀ᵐ x ∂(Q : Measure (SpectralArray (m + 1))), | |
| spectralLinearArray (spectralSpinWeight m) x 0 1 ∈ Icc (0 : ℝ) 1 := by | |
| simpa only [spectralLinearArray_spinWeight] using spectralPartition_spin_unit hP hn | |
| have he := factor_of_pair_law pathMeasure (Q : Measure (SpectralArray (m + 1))) | |
| p (spectralGroupQuantilePath Q hn a) | |
| (fun x => spectralLinearArray (spectralSpinWeight m) x 0 1) | |
| (fun x => spectralLinearArray (Pi.single a.castSucc 1) x 0 1) | |
| p.measurable (spectralGroupQuantilePath Q hn a).measurable | |
| (by unfold spectralLinearArray spectralLinearEntry; fun_prop) | |
| (by unfold spectralLinearArray spectralLinearEntry; fun_prop) | |
| (spectralQuantilePath_pair_law hgg hG q hq hd (fun e => hE e e.injective) | |
| (spectralSpinWeight m) (Pi.single a.castSucc 1) (spectralSpinWeight_nonneg m) | |
| hsingle hp (spectralCoordinate_unit hn a.castSucc)) | |
| (cavityCanonicalCoordinate ρ eig hρ hρsum p a) | |
| (continuous_cavityCanonicalCoordinate ρ eig hρ hρsum p a) | |
| (by | |
| filter_upwards [hc] with s hs | |
| exact (hs a).trans (cavityCanonicalCoordinate_at_path ρ eig hρ hρsum p a s).symm) | |
| simpa only [spectralLinearArray_spinWeight, spectralLinearArray_single, | |
| spectralCoordinateArray] using he | |
| filter_upwards [ae_all_iff.mpr hall] with x hx | |
| intro i j hij a | |
| exact hx a i j hij | |
| end InvariantIsing | |
| end | |
| end OAI | |