openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticBlockDiagonalLower.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticDiagonalSelection | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticBlockTrialLower | |
| namespace OAI | |
| /-! Weak tests on growing physical blocks yield the magnetic variational trial lower bound. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter | |
| open scoped Topology BigOperators BoundedContinuousFunction | |
| namespace InvariantIsing | |
| theorem magnetic_block_diagonal_trial_lower {m : ℕ} | |
| (ρ lam : Fin m → ℝ) (hρ : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a=1) | |
| {K : ℝ} (hK : 0 ≤ K) (hlam : ∀ a, |lam a| ≤ K) | |
| {A : Type*} [Fintype A] [DecidableEq A] | |
| (N : ℕ → ℕ) (hN : ∀ r, 0 < N r) (hNlim : Tendsto N atTop atTop) | |
| (group : ∀ r, Fin (N r) → A) (k : ℕ → A → ℕ) | |
| (hk : ∀ r a, k r a ≤ spinGroupSize (group r) a) | |
| (γ mag : A → ℝ) (hγ : ∀ a, 0 ≤ γ a) (hγsum : ∑ a, γ a=1) | |
| (hcount : ∀ r a, (spinGroupSize (group r) a : ℝ)=N r*γ a) | |
| {s : ℝ} (hs : s < 1) (hmag : ∀ a, |mag a| ≤ s) | |
| (hc : ∀ r a, (k r a : ℝ)=spinGroupSize (group r) a*((1+mag a)/2)) | |
| (base : ℕ → OverlapPath) | |
| (htest : ∀ r (Φ : ℝ →ᵇ ℝ), Tendsto (fun j => ∫ t, Φ (cavityStrictUniformPath (base r) j t)* | |
| (restrictedBlockOverlapPath (hN r) (spinGroupSlice (group r) (k r)) | |
| (spinGroupSlice_nonempty (group r) (k r) (hk r)) | |
| (cavityStrictUniformField ρ lam hρ hsum (base r) j) t- | |
| cavityStrictUniformPath (base r) j t) ∂pathMeasure) atTop (𝓝 0)) : | |
| ∃ d φ : ℕ → ℕ, StrictMono d ∧ StrictMono φ ∧ | |
| ∀ ε > 0, ∀ᶠ r in atTop, | |
| (magneticVariationalFunctional (finiteR ρ lam hρ hsum) γ mag).toReal-ε ≤ constrainedBlockValue (spinGroupSlice (group (φ r)) (k (φ r))) | |
| (cavityStrictUniformField ρ lam hρ hsum (base (φ r)) (d (φ r)))+ | |
| fieldPairing (cavityStrictUniformPath (base (φ r)) (d (φ r))) | |
| (cavityStrictUniformField ρ lam hρ hsum (base (φ r)) (d (φ r)))/2+ | |
| spectralFunctional (finiteR ρ lam hρ hsum) | |
| (cavityStrictUniformPath (base (φ r)) (d (φ r))) := by | |
| obtain ⟨d,φ,p,hd,hφ,hq,hself⟩ := exists_magnetic_consistent_diagonal | |
| ρ lam hρ hsum hK hlam N hN hNlim group k hk (fun _ => mag) hs | |
| (fun _ => hmag) hc base htest | |
| refine ⟨d,φ,hd,hφ,?_⟩ | |
| exact magnetic_block_trial_eventually_lower ρ lam hρ hsum | |
| (fun r => N (φ r)) (fun r => hN (φ r)) (hNlim.comp hφ.tendsto_atTop) | |
| (fun r => group (φ r)) (fun r => k (φ r)) (fun r => hk (φ r)) γ mag hγ hγsum | |
| (fun r => hcount (φ r)) hs hmag (fun r => hc (φ r)) | |
| (fun r => cavityStrictUniformField ρ lam hρ hsum (base (φ r)) (d (φ r))) | |
| (fun r => cavityStrictUniformField_height_le ρ lam hρ hsum hK hlam | |
| (base (φ r)) (d (φ r)) (Fin.last (d (φ r)))) p | |
| (fun r => cavityStrictUniformPath (base (φ r)) (d (φ r))) hself hq | |
| end InvariantIsing | |
| end | |
| end OAI | |