| import OAI.Probability.DilutedSpin.PhysicalMultileafMoments |
| import OAI.Probability.DilutedSpin.ScheduledRegularFrame |
| import OAI.Probability.DilutedSpin.TwoPointLabeling |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace DilutedSpinGlass.ReducedTopology |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory PrescribedTree HeterogeneousMarks |
| open scoped BigOperators |
| variable {α Ω J X Y : Type} [Fintype α] [DecidableEq α] |
| [Fintype Ω] {Mark : J → Type} [∀ j, Fintype (Mark j)] [Countable J] |
| [MeasurableSpace J] [MeasurableSingletonClass J] [MeasurableSpace X] [MeasurableSpace Y] |
| {M N t : ℕ} |
| attribute [local irreducible] projectionShiftError matrixProjectionError oldProjectionError |
| matrixObservableHistory KernelTower.halfTripleDifferenceAt splitProjector |
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| |
| theorem canonical_regular_root_multileaf (μ : Measure (FullRootState Y X J M)) |
| [IsProbabilityMeasure μ] |
| (H r d : ℕ) (k : ℕ+) (hk : 2≤(k:ℕ)) (a : α) (C : Fin k → ReducedTopology) |
| (e : (j : Fin k) → (C j).Vertex → {j : α // j≠a}) |
| (Q : α → Fin (H+1+1+r+1+d)) (ha : (Q a).val=r+1+d) |
| (had : ∀ j, Admissible (C j) (fun v => (Q (e j v)).val) (r+1+d+1) (r+1+d+1+H)) |
| {η : ℝ} (hlarge : 1<η*(H+1+1+r+1+d:ℕ)) (hr : DepthAverage.Regular η Q) |
| (T : KernelTower Ω (H+1+1+r+1+d)) (P : (j : J) → Fin (H+1+1+r+1+d) → FiniteLaw (Mark j)) |
| (m : Fin (H+1+1+r+1+d) → ℝ) |
| (base : RootPath Y M → (k : ℕ) → RootPath X k → FinitePath Ω (H+1+1+r+1+d) → ℝ) |
| (old : (j : J) → FinitePath Ω (H+1+1+r+1+d) → FinitePath (Mark j) (H+1+1+r+1+d) → ℝ) |
| (V : FinitePath Ω (H+1+1+r+1+d) → Fin N → ℝ) |
| (hb : ∀ k y, Measurable (fun z : RootPath Y M × RootPath X k => base z.1 k z.2 y)) |
| (hV : ∀ y i, |V y i|≤1) : |
| let K := rootTower T P m base old |
| let f := rootVector V (I := J) (A := Mark) (X := X) (Y := Y) (M := M) |
| let L := H+1+1+r+1+d |
| let ht := shiftedFrameHeight H r d |
| let OldChild := fun j => realize (H+1) (r+1+d+1) (C j) (fun v => (Q (e j v)).val) |
| let NewChild := fun j => realize H (r+1+1+d+1) (C j) (fun v => (Q (e j v)).val+1) |
| let OldNode := PrescribedTree.node k OldChild |
| let NewNode := PrescribedTree.node k NewChild |
| let BaseNode := PrescribedTree.node k (fun j => realize H (r+1+d+1) (C j) (fun v => (Q (e j v)).val)) |
| let B := shiftedPrefixDepths (stem BaseNode r) d |
| let S := splitFrame OldNode r d |
| let RawTarget := splitFrame NewNode (r+1) d |
| let Target := heightCast ht RawTarget |
| let K' := fun z => kernelHeightCast ht.symm (K z) |
| let f' := fun z => vectorHeightCast ht.symm (f z) |
| let W := (k:ℝ)*∑ j, Real.sqrt (∫ z, descendantEnergyAt (OldChild j) r d (K z) (f z) ∂μ) |
| let W' := (k:ℝ)*∑ j, Real.sqrt (∫ z, descendantEnergyAt (NewChild j) (r+1) d (K' z) (f' z) ∂μ) |
| let h := fun z => projectionShiftError a C e (K z) (f z) Q |
| let _ := fun z x i => splitProjector OldNode r d (K z) (fun y => f z y i) x |
| let A := fun z => spatialProduct (fun _ : Option (Fin (t+1)) => f z) |
| ∀ (b : OldNode.Leaf) (b' : NewNode.Leaf) (q0 : Option (Fin (t+1)) → RawTarget.Leaf), |
| Function.Bijective q0 → q0 none=splitFrameLeaf NewNode (r+1) d 0 b' → |
| q0 (some (Fin.last t))=splitFrameLeaf NewNode (r+1) d 1 b' → |
| let v := some (Fin.last t) |
| let cs := canonicalMatrixTail t |
| let q := fun j => leafHeightCast ht RawTarget (q0 j) |
| let x := splitFrameLeaf OldNode r d 0 b |
| let _ := splitFrameLeaf OldNode r d 1 b |
| let m := grid L 0 L |
| let J := partialKappa Target m (Finset.univ.image q)/ |
| partialKappa Target m ((insert v ({none}:Finset (Option (Fin (t+1))))).image q) |
| (((d:ℝ)+2)/(L:ℝ))*(∫ z, shapeEnergyAt (stem OldNode r) d (K z) (f z) ∂μ) ≤ |
| 2*(2*(L:ℝ)⁻¹ + |matrixRootCovariance μ (rootAlphabet (Ω := Ω) (A := Mark)) Target S q K m (v::cs) x A |
| (fun z => treeOverlap S (f z))|/|J|+ |
| (2*Real.sqrt W'+(∫ z, h z ∂μ))*shiftedCharge B Target S (v::cs).length/|J|+ |
| pairHistoryMass S m x*(2*Real.sqrt W))+ |
| (((d:ℝ)+2)/(L:ℝ))*(16*W) := by |
| dsimp only |
| intro b b' q0 hq hu hv |
| have hQ : ∀ j v, (Q a).val+1≤(Q (e j v)).val := by |
| intro j v |
| rw [ha] |
| exact admissible_lower (C j) _ (had j) v |
| let BaseNode := PrescribedTree.node k (fun j => realize H (r+1+d+1) (C j) (fun v => (Q (e j v)).val)) |
| let B := shiftedPrefixDepths (stem BaseNode r) d |
| have hS := scheduled_regular_old_count H r d k hk a C e Q ha had hlarge hr |
| exact physical_scheduled_root_multileaf_moments μ H r d k hk a C e Q ha hQ T P m base old V hb hV |
| b b' q0 hq none (some (Fin.last t)) (by simp) hu hv B hS |
| (canonicalMatrixTail t) (canonicalMatrixTail_nodup t) (canonicalMatrixTail_not_mem t) |
| (canonicalMatrixTail_full t) |
|
|
| end DilutedSpinGlass.ReducedTopology |
| end |
|
|
| end |
|
|
| end OAI |
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|
