| import OAI.Probability.DilutedSpin.JointGridSelection |
| import OAI.Probability.DilutedSpin.SpinRootData |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace DilutedSpinGlass.DepthAverage |
| open Filter |
| open scoped Topology BigOperators |
| variable {α : Type} [Fintype α] [DecidableEq α] {L : ℕ} [NeZero L] |
|
|
| /-- At each fixed grid, there are only finitely many depth assignments. Thus |
| the signed physical identity error can be averaged BEFORE the size limit. -/ |
| theorem tendsto_average_zero (D : (α → Fin L) → Prop) |
| (F : ℕ → (α → Fin L) → ℝ) |
| (hF : ∀ q, Tendsto (fun n => F n q) atTop (𝓝 0)) : |
| Tendsto (fun n => average D (F n)) atTop (𝓝 0) := by |
| classical |
| simp only [average_eq] |
| have hs := tendsto_finsetSum (Finset.univ.filter D) (fun q _ => hF q) |
| have hh := hs.div_const ((L:ℝ)^Fintype.card α) |
| simpa only [Finset.sum_const_zero,zero_div] using hh |
|
|
| end DilutedSpinGlass.DepthAverage |
| namespace DilutedSpinGlass.UniversalDictionary |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory HeterogeneousMarks PhysicalRoot PrescribedTree ConcreteReservoir Filter Set |
| open scoped NNReal BigOperators Topology |
| variable {α : Type} [Fintype α] [DecidableEq α] {L p : ℕ} |
|
|
| /-- The low-increment grid sequence simultaneously has its regular singleton |
| base case and a vanishing normalized regular-cube matrix error. The old tree, |
| target shape, labeling and charging denominator may all vary with the complete |
| depth assignment. This is the actual shared-root signed covariance, not an |
| absolute coefficient majorant or a diagonal restriction of two averages. -/ |
| theorem joint_grid_averaged_selection (M : Model p) {C H : ℝ} (hC : 0 ≤ C) (hH : 0 ≤ H) |
| (hθ : ∀ᵐ z ∂M.disorder.toMeasure, ∀ σ, |z.1 σ| ≤ C) |
| (hh : ∀ᵐ h ∂M.field.toMeasure, |h| ≤ H) |
| (hθi : Integrable (fun z : InteractionSample p => ‖z.1‖) M.disorder.toMeasure) |
| (hhi : Integrable (fun h : ℝ => |h|) M.field.toMeasure) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ (Ns : ℕ → ℕ) (us : ℕ → Spec L×ℕ → ℝ), StrictMono Ns ∧ |
| (∀ n i, us n i ∈ Icc (probeLow i.2) (probeHigh i.2)) ∧ |
| (∀ n, ConcreteReservoir.increment (weights L) prior (gridExponents L) direction anchor M (Ns n) (us n) ≤ |
| liminf (pressure M) atTop+ε) ∧ |
| (∀ η : ℝ, 0 < η → ∀ t : ℝ, 0 < t → ∀ᶠ n in atTop, |
| ∀ d ∈ regularSingletonDepths L η, |
| physicalSingletonEnergy (gridExponents L) M C H (Ns n+1) (us n) d ≤ |
| ((L+1:ℕ):ℝ)⁻¹/η+t) ∧ |
| (∀ (D : (α → Fin (L+1)) → Prop) |
| (S T : (α → Fin (L+1)) → PrescribedTree (L+1)) |
| (a : (Q : α → Fin (L+1)) → (S Q).Leaf) |
| (k : (α → Fin (L+1)) → ℕ), (∀ Q, 0<k Q) → |
| ∀ (q : (Q : α → Fin (L+1)) → Option (Fin (k Q)) → (T Q).Leaf) |
| (denom : (α → Fin (L+1)) → ℝ), |
| Tendsto (fun n => DepthAverage.average D (fun Q => |
| |physicalTreeMatrixCovariance (gridExponents L) (S Q) (a Q) M C H |
| (Ns n+1) (us n) (T Q) (q Q)|/|denom Q|)) atTop (𝓝 0)) := by |
| obtain ⟨Ns,us,hNs,hus,hinc,henergy,hmatrix⟩ := joint_grid_selection M hC hH hθ hh hθi hhi hε |
| refine ⟨Ns,us,hNs,hus,hinc,henergy,?_⟩ |
| intro D S T a k hk q denom |
| apply DepthAverage.tendsto_average_zero |
| intro Q |
| have hh := (hmatrix (S Q) (a Q) (k Q) (hk Q) (T Q) (q Q)).abs.div_const |denom Q| |
| simpa only [abs_zero,zero_div] using hh |
|
|
| end DilutedSpinGlass.UniversalDictionary |
| end |
|
|
| end |
|
|
| end OAI |
|
|
