| import OAI.Analysis.DiamondDistortion.HeadProjections |
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| namespace OAI |
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| universe u v |
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| noncomputable section |
| open scoped BigOperators Classical Topology |
| open Set |
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| namespace DiamondDistortion |
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| open scoped ENNReal |
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| namespace PathModel |
| open RecursiveModel DirectLayers SignCube |
| /-- Ancestral coordinate projections for the literal linear path cost are |
| contractive, jointly cofinal, and nested in both orders. -/ |
| theorem linear_regular : Regular linearHeads where |
| nonempty := head_nonempty |
| upper := head_upper |
| nested_left H K hHK x := linearData.projection_nested H.val K.val hHK x |
| nested_right H K hHK x := linearData.projection_nested_right H.val K.val hHK x |
| contractive := linear_projection_contractive |
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| theorem adaptive_graph (C : ℝ) (hin : SignInput linearHeads.projection C) |
| {D s : ℝ} (k : ℕ) (f : Diamond.Vertex k → LinearSpace) |
| (hf : Diamond.HasDistortion k f D s) : |
| (k:ℝ)/16 ≤ C*(D^2+(k:ℝ)*((2:ℝ)^k)⁻¹^2) ∧ |
| C*(D^2+(k:ℝ)*((2:ℝ)^k)⁻¹^2) ≤ C*(D^2+1) := |
| DirectLayers.adaptive_graph linearHeads linear_regular hin k f hf |
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| theorem predictable_495_graph (hin : PredictableInput linearHeads.projection 495) |
| {D s : ℝ} (k : ℕ) (f : Diamond.Vertex k → LinearSpace) |
| (hf : Diamond.HasDistortion k f D s) : (k:ℝ)/4 ≤ 495*(D+1)^2 := |
| DirectLayers.predictable_graph linearHeads linear_regular hin k f hf |
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| theorem variation_252_graph (hin : PredictableInput linearHeads.projection 252) |
| {D s : ℝ} (k : ℕ) (f : Diamond.Vertex k → LinearSpace) |
| (hf : Diamond.HasDistortion k f D s) : (k:ℝ)/(16*D^2) ≤ 1008 := |
| DirectLayers.variation_graph linearHeads linear_regular hin k f hf |
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| theorem leakage_1024_graph (hin : LeakageInput linearHeads.projection) |
| {D s : ℝ} (k : ℕ) (hk : 1 ≤ k) (f : Diamond.Vertex k → LinearSpace) |
| (hf : Diamond.HasDistortion k f D s) : |
| (k:ℝ)/8 ≤ 1024*(D^2+(k:ℝ)*(leakageTolerance k)^2) ∧ |
| 1024*(D^2+(k:ℝ)*(leakageTolerance k)^2) ≤ 1024*(D^2+1) := |
| DirectLayers.leakage_graph linearHeads linear_regular hin k hk f hf |
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| theorem squared_gap_400_graph (hin : GapInput linearHeads.projection) |
| {D s γ : ℝ} (k : ℕ) (f : Diamond.Vertex k → LinearSpace) |
| (hf : Diamond.HasDistortion k f D s) (hγ : 0 < γ) |
| (hγa : γ < (2/D)/8) (hγk : γ < 1/((k:ℝ)+1)) : |
| ∃ T : Protocol LinearSpace ForestHead k, |
| T.Martingale ∧ T.Nested ∧ T.Anchored ∧ T.CommonHeads ∧ T.Supported linearHeads.projection ∧ |
| T.terminalEnergy ≤ 4 ∧ T.initialEnergy+(k:ℝ)*((2/D)/8)^2/400 ≤ T.terminalEnergy ∧ |
| (k:ℝ)*((2/D)/8)^2/400 ≤ T.initialEnergy+(k:ℝ)*((2/D)/8)^2/400 := |
| DirectLayers.gap_graph linearHeads linear_regular hin k f hf hγ hγa hγk |
| end PathModel |
|
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| namespace DirectSegmentApplications |
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| theorem deterministic_descent {D s : ℝ} (k : ℕ) (f : Diamond.Vertex k → ForestDual) |
| (hf : Diamond.HasDistortion k f D s) : (k:ℝ) ≤ 256*D^2 := by |
| have h := headline.1 k f D s hf |
| nlinarith [sq_nonneg D] |
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| theorem bin_descent {D : ℝ} (k : ℕ) (hk : 1 ≤ k) (f : Diamond.Vertex k → ForestDual) |
| (hl : ∀ u v,Diamond.distance k u v ≤ ‖f u-f v‖) |
| (hu : ∀ u v,‖f u-f v‖ ≤ D*Diamond.distance k u v) : Real.sqrt (k:ℝ)/144 < D := by |
| have hsl := Diamond.slope_lower f (s := 1) (by simpa only [one_mul] using hl) (Diamond.baseEdge k) |
| have hsu := Diamond.slope_upper f hu (Diamond.baseEdge k) |
| have hD : 1 ≤ D := le_trans hsl hsu |
| have hdist : Diamond.HasDistortion k f D 1 := ⟨hD,by norm_num,by simpa only [one_mul,mul_one] using fun u v => And.intro (hl u v) (hu u v)⟩ |
| have hh := headline.1 k f D 1 hdist |
| have hs := Real.sq_sqrt (Nat.cast_nonneg (α := ℝ) k) |
| have hsp := Real.sqrt_nonneg (k:ℝ) |
| have hk0 : (1:ℝ) ≤ k := by exact_mod_cast hk |
| apply (div_lt_iff₀ (by norm_num : (0:ℝ)<144)).mpr |
| nlinarith |
| end DirectSegmentApplications |
|
|
| namespace Diamond |
| variable {X : Type*} [NormedAddCommGroup X] |
| /-- The redundant conventional condition D≥1 follows from the |
| all-pairs inequalities alone; thus HasDistortion adds no standing hypothesis. -/ |
| lemma hasDistortion_iff_all_pairs {k : ℕ} (f : Vertex k → X) (D s : ℝ) : |
| HasDistortion k f D s ↔ 0 < s ∧ ∀ u v, |
| s*distance k u v ≤ ‖f u-f v‖ ∧ ‖f u-f v‖ ≤ D*s*distance k u v := by |
| constructor |
| · exact fun h => h.2 |
| · rintro ⟨hs,h⟩ |
| have hh := (h (endpoint k (baseEdge k) false) (endpoint k (baseEdge k) true)).1.trans |
| (h (endpoint k (baseEdge k) false) (endpoint k (baseEdge k) true)).2 |
| rw [distance_edge] at hh |
| have hp : 0 < s*(1/(2:ℝ)^k) := by positivity |
| have hD : 1 ≤ D := (mul_le_mul_iff_left₀ hp).mp (by calc |
| (1:ℝ)*(s*(1/(2:ℝ)^k)) = s*(1/(2:ℝ)^k) := one_mul _ |
| _ ≤ D*s*(1/(2:ℝ)^k) := hh |
| _ = D*(s*(1/(2:ℝ)^k)) := by ring) |
| exact ⟨hD,hs,h⟩ |
| end Diamond |
|
|
| theorem headline_all_pairs : |
| (∀ (k : ℕ) (f : Diamond.Vertex k → ForestDual) (D s : ℝ), |
| 0 < s → (∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧ |
| ‖f u-f v‖ ≤ D*s*Diamond.distance k u v) → 1+(k:ℝ)/4 ≤ D^2) ∧ |
| (∀ (k : ℕ) (f : Diamond.Vertex k → InfinitePredual) (D s : ℝ), |
| 0 < s → (∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧ |
| ‖f u-f v‖ ≤ D*s*Diamond.distance k u v) → 1+(k:ℝ)/4 ≤ D^2) := by |
| constructor |
| · intro k f D s hs h |
| exact headline.1 k f D s ((Diamond.hasDistortion_iff_all_pairs f D s).mpr ⟨hs,h⟩) |
| · intro k f D s hs h |
| exact headline.2 k f D s ((Diamond.hasDistortion_iff_all_pairs f D s).mpr ⟨hs,h⟩) |
|
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| theorem direct_atomic_obstruction_positive : |
| ¬ ∃ D : ℝ,∀ k : ℕ,1 ≤ k → ∃ (f : Diamond.Vertex k → ForestDual) (s : ℝ), |
| 0 < s ∧ ∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧ |
| ‖f u-f v‖ ≤ D*s*Diamond.distance k u v := by |
| rintro ⟨D,h⟩ |
| obtain ⟨n,hn⟩ := exists_nat_gt (4*D^2) |
| obtain ⟨f,s,hs,hf⟩ := h (n+1) (by omega) |
| have hh := headline_all_pairs.1 (n+1) f D s hs hf |
| push_cast at hh |
| nlinarith |
|
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| end DiamondDistortion |
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| end |
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| end OAI |
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