| import OAI.NumberTheory.Jacobsthal.Estimates.EndpointSoftConclusion |
| import OAI.NumberTheory.Jacobsthal.Partitions.EligibleBoxLists |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
|
|
| namespace NumberTheoryLean.FilteredBoxWordMass |
| open ErdosCofactorChoices ErdosSubsetWord FiniteBoxWordMass |
| open ErdosPrimeInputs.PrimePrefixMass |
| open GeometricBoxImages ExpandedCompactPartition GeometricRegularWords ActualWordSelection |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
| attribute [local instance] Classical.propDecidable |
|
|
| noncomputable def selectedWordMass {n : ℕ} (P : Fin n → Finset ℕ) (m : Fin n → ℕ) |
| (F : Finset (List ℕ)) : ℝ := |
| ∑ f∈(selections P m).filter (fun f => descendingWord f∈F),(selectionProduct f:ℝ)⁻¹ |
|
|
| theorem filtered_word_box_mass {n : ℕ} (P : Fin n → Finset ℕ) (m : Fin n → ℕ) |
| (hd : Pairwise (fun i j => Disjoint (P i) (P j))) (F : Finset (List ℕ)) : |
| (∑ ps∈(wordBox P m).filter (fun ps => ps∈F),prefixWeight ps)=selectedWordMass P m F := by |
| have he : (wordBox P m).filter (fun ps => ps∈F)= |
| ((selections P m).filter (fun f => descendingWord f∈F)).image descendingWord := by |
| ext ps |
| simp only [wordBox,Finset.mem_filter,Finset.mem_image] |
| constructor |
| · rintro ⟨⟨f,hf,rfl⟩,hF⟩ |
| exact ⟨f,⟨hf,hF⟩,rfl⟩ |
| · rintro ⟨f,⟨hf,hF⟩,rfl⟩ |
| exact ⟨⟨f,hf,rfl⟩,hF⟩ |
| rw [he,Finset.sum_image] |
| · apply Finset.sum_congr rfl |
| intro f hf |
| rw [StoppedVertexHistory.prefixWeight_product, |
| descendingWord_product P hd f (fun k => ((mem_selections _ _ _).mp (Finset.mem_filter.mp hf).1 k).1)] |
| · intro f hf g hg heq |
| exact word_injective P hd f g |
| (fun k => ((mem_selections _ _ _).mp (Finset.mem_filter.mp hf).1 k).1) |
| (fun k => ((mem_selections _ _ _).mp (Finset.mem_filter.mp hg).1 k).1) heq |
|
|
| theorem filtered_box_mass {n : ℕ} (P : Fin n → Finset ℕ) |
| (hd : Pairwise (fun i j => Disjoint (P i) (P j))) |
| (M : Finset (Fin n → ℕ)) (F : Finset (List ℕ)) (hF : F ⊆ boxWords P M) : |
| (∑ ps∈F,prefixWeight ps)=∑ m∈M,selectedWordMass P m F := by |
| have he : F=M.biUnion (fun m => (wordBox P m).filter (fun ps => ps∈F)) := by |
| ext ps |
| constructor |
| · intro hp |
| obtain ⟨m,hm,hps⟩ := Finset.mem_biUnion.mp (hF hp) |
| exact Finset.mem_biUnion.mpr ⟨m,hm,Finset.mem_filter.mpr ⟨hps,hp⟩⟩ |
| · intro hp |
| obtain ⟨_m,_hm,hps⟩ := Finset.mem_biUnion.mp hp |
| exact (Finset.mem_filter.mp hps).2 |
| calc |
| _ = ∑ ps∈M.biUnion (fun m => (wordBox P m).filter (fun ps => ps∈F)),prefixWeight ps := by rw [← he] |
| _ = ∑ m∈M,∑ ps∈(wordBox P m).filter (fun ps => ps∈F),prefixWeight ps := by |
| apply Finset.sum_biUnion |
| intro m _hm m' _hm' hne |
| exact (wordBoxes_disjoint P hd m m' hne).mono (Finset.filter_subset _ _) (Finset.filter_subset _ _) |
| _ = _ := Finset.sum_congr rfl (fun m _ => filtered_word_box_mass P m hd F) |
| |
| theorem unlisted_geometric_box_mass {w top xi Cs C B K L alpha beta : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (a : ℕ → ℕ) (z : PrimeHistories.Node) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (F : Finset (List ℕ)) : |
| let U := unlistedGeometric hw htop hxi Cs C B K L alpha beta a z Q F |
| (∑ ps∈U,prefixWeight ps)=∑ m∈geometricBoxes hw htop hxi C B K L alpha beta z, |
| selectedWordMass (globalBins w top xi) m U := by |
| dsimp only |
| apply filtered_box_mass _ (fun k l hkl => bins_pairwise_disjoint (zero_lt_one.trans hw) htop hxi k l hkl) |
| intro ps hp |
| have hg := (Finset.mem_inter.mp (Finset.mem_filter.mp hp).1).2 |
| have hr := (Finset.mem_filter.mp hg).1 |
| have hd := mem_decreasingPrefixes.mp (Finset.mem_filter.mp hr).1 |
| apply Finset.mem_biUnion.mpr |
| refine ⟨wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) ps, |
| Finset.mem_image.mpr ⟨ps,hg,rfl⟩,?_⟩ |
| exact Finset.mem_image.mpr ⟨wordSelection (label (zero_lt_one.trans hw) htop hxi) ps, |
| wordSelection_mem hw htop hxi ps hd.2,wordSelection_recovers _ ps hd.1⟩ |
| end NumberTheoryLean.FilteredBoxWordMass |
| |
| |
| |
| namespace NumberTheoryLean.SoftWitnessCapture |
| open ErdosCofactorChoices ErdosSubsetWord ErdosInverseAlignment |
| open ActualWordSelection FiniteBoxWordMass FullIsolatedCofactor EligibleBoxLists |
| open ExpandedCompactPartition ActualBinOwners SingletonBinSelection TwoPrimeObservableSum |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosPrimeInputs.HarmonicPrimeMeasure |
| attribute [local instance] Classical.propDecidable |
| |
| theorem word_selection_descending {w top xi : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f∈selections (globalBins w top xi) m) : |
| wordSelection (label (zero_lt_one.trans hw) htop hxi) (descendingWord f)=f := by |
| have hrec := wordSelection_mem hw htop hxi (descendingWord f) (word_source_membership hw htop hxi m f hf) |
| apply word_injective (globalBins w top xi) |
| (fun k l hkl => bins_pairwise_disjoint (zero_lt_one.trans hw) htop hxi k l hkl) |
| _ f (fun k => ((mem_selections _ _ _).mp hrec k).1) (fun k => ((mem_selections _ _ _).mp hf k).1) |
| exact wordSelection_recovers _ _ (descendingWord_strict f) |
| |
| theorem word_multiplicity_descending {w top xi : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f∈selections (globalBins w top xi) m) : |
| wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) (descendingWord f)=m := by |
| funext k |
| unfold wordMultiplicity |
| rw [word_selection_descending hw htop hxi m f hf] |
| exact ((mem_selections _ _ _).mp hf k).2 |
| |
| theorem soft_full_word_listed {w top xi Cs : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Y : ℝ) |
| (M : Finset (Fin (binCount w top xi) → ℕ)) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) |
| (m : Fin (binCount w top xi) → ℕ) (hm : m∈M) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f∈selections (globalBins w top xi) m) (i : Fin (binCount w top xi)) (hi : m i=1) |
| (a : ℕ → ℕ) (q : ℚ) (hq : q∈Q m) (helig : eligible Y w Cs q) |
| (ha : ∀ p∈(pickedPrime f i*selectionProduct (eraseSelection f i)).primeFactors, |
| Cs < primeExponent w p → aligns (SourceStopPredicate.sourceClass a) q p) : |
| listedWord hw htop hxi Cs a (eligibleLists Y w Cs M Q) (descendingWord f) := by |
| unfold listedWord |
| rw [word_multiplicity_descending hw htop hxi m f hf] |
| exact ⟨q,soft_witness_in_eligibleLists Y w Cs M Q m hm q hq helig, |
| full_cofactor_alignment_word hw htop hxi m f hf i hi _ q ha⟩ |
| |
| theorem unlisted_has_no_soft_witness {w top xi Cs C B K L alpha beta : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Y : ℝ) |
| (M : Finset (Fin (binCount w top xi) → ℕ)) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) |
| (m : Fin (binCount w top xi) → ℕ) (hm : m∈M) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f∈selections (globalBins w top xi) m) (i : Fin (binCount w top xi)) (hi : m i=1) |
| (a : ℕ → ℕ) (z : PrimeHistories.Node) (F : Finset (List ℕ)) |
| (hu : descendingWord f∈unlistedGeometric hw htop hxi Cs C B K L alpha beta a z (eligibleLists Y w Cs M Q) F) |
| (q : ℚ) (hq : q∈Q m) : |
| ¬(eligible Y w Cs q ∧ |
| ∀ p∈(pickedPrime f i*selectionProduct (eraseSelection f i)).primeFactors, |
| Cs < primeExponent w p → aligns (SourceStopPredicate.sourceClass a) q p) := by |
| rintro ⟨helig,ha⟩ |
| exact (Finset.mem_filter.mp hu).2 (soft_full_word_listed hw htop hxi Y M Q m hm f hf i hi a q hq helig ha) |
| end NumberTheoryLean.SoftWitnessCapture |
| |
| |
| |
| namespace ErdosVarianceEligible |
| open NumberTheoryLean ErdosCofactorChoices ErdosSubsetWord SingletonBinSelection TwoPrimeObservableSum |
| FullIsolatedCofactor ActualBinOwners SourceStopPredicate ActualCountErrorEdges |
| ErdosVarianceSmallPrime ErdosVarianceCommon ErdosInverseAlignment |
| LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinPartition |
| open ErdosPrimeInputs.HarmonicPrimeMeasure |
| attribute [local instance] Classical.propDecidable |
| |
| def eligibleWordWitness (Y : ℕ) (w Cs : ℝ) (residue : ℕ → ℕ) (Q : Finset ℚ) (ps : List ℕ) : Prop := |
| ∃ r ∈ Q,eligible (Y : ℝ) w Cs r ∧ |
| ∀ p ∈ ps,Cs < primeExponent w p → aligns (sourceClass residue) r p |
| |
| theorem eligible_word_witness_of_endpoint {w top xi Cs : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Y : ℕ) |
| (m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f ∈ selections (globalBins w top xi) m) (i : Fin (binCount w top xi)) (hi : m i = 1) |
| (residue : ℕ → ℕ) (Q : Finset ℚ) (r : ℚ) (hr : r ∈ Q) |
| (hsoft : (r.den : ℝ) ≤ w^Cs ∧ (r.num.natAbs : ℝ) ≤ (Y : ℝ)*w^(Cs+2) ∧ |
| ∀ t ∈ (pickedPrime f i*selectionProduct (eraseSelection f i)).primeFactors, |
| Cs < Real.logb w (t : ℝ) → aligns (fun u => (residue u : ℤ)) r t) : |
| eligibleWordWitness Y w Cs residue Q (descendingWord f) := by |
| refine ⟨r,hr,⟨hsoft.1,hsoft.2.1⟩,?_⟩ |
| apply full_cofactor_alignment_word hw htop hxi m f hf i hi (sourceClass residue) r |
| intro p hp hhigh |
| exact hsoft.2.2 p hp hhigh |
| |
| theorem word_error_bad_aligning_prime {w top xi : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Y : ℕ) |
| (m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f ∈ selections (globalBins w top xi) m) (i : Fin (binCount w top xi)) (hi : m i = 1) |
| (residue : ℕ → ℕ) (r : ℚ) (eps : ℝ) (halign : aligns (sourceClass residue) r (pickedPrime f i)) |
| (hbad : eps < |wordCountError Y (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) residue |
| (SmallSieveFinite.smallEuler ⌊w⌋₊) (descendingWord f)|) : |
| pickedPrime f i ∈ badAligningPrimes (globalBins w top xi i) Y w residue r |
| (selectionProduct (eraseSelection f i)) eps := by |
| obtain ⟨p,hp,hfi,_hfill⟩ := singleton_selection_recovery _ m i hi f hf |
| have hpmem : pickedPrime f i ∈ globalBins w top xi i := by simpa [pickedPrime,hfi] using hp |
| have hprod := full_word_isolated_product hw htop hxi m f hf i hi |
| apply (mem_badAligningPrimes _ _ _ _ _ _ _ _).mpr |
| refine ⟨hpmem,halign,?_⟩ |
| simpa only [wordCountError,relativeCountError,hprod,Nat.cast_mul,referenceCount] using hbad |
| |
| end ErdosVarianceEligible |
| |
| |
| |
| namespace NumberTheoryLean.EligibleListWord |
| open ErdosCofactorChoices ErdosSubsetWord ErdosInverseAlignment |
| open ActualWordSelection SoftWitnessCapture EligibleBoxLists ExpandedCompactPartition ActualBinOwners |
| open LogarithmicBinScale LogarithmicBinLabels |
| open ErdosPrimeInputs.HarmonicPrimeMeasure |
| attribute [local instance] Classical.propDecidable |
| |
| theorem listed_word_eligible_iff {w top xi Cs : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Y : ℝ) |
| (M : Finset (Fin (binCount w top xi) → ℕ)) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (a : ℕ → ℕ) (ps : List ℕ) : |
| listedWord hw htop hxi Cs a (eligibleLists Y w Cs M Q) ps ↔ |
| wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) ps∈M ∧ |
| ∃ q∈Q (wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) ps),eligible Y w Cs q ∧ |
| ∀ p∈ps,Cs < primeExponent w p → aligns (SourceStopPredicate.sourceClass a) q p := by |
| unfold listedWord |
| constructor |
| · rintro ⟨q,hq,ha⟩ |
| obtain ⟨hm,hq,he⟩ := (mem_eligibleLists _ _ _ _ _ _ _).mp hq |
| exact ⟨hm,q,hq,he,ha⟩ |
| · rintro ⟨hm,q,hq,he,ha⟩ |
| exact ⟨q,(mem_eligibleLists _ _ _ _ _ _ _).mpr ⟨hm,hq,he⟩,ha⟩ |
| |
| theorem listed_selection_eligible_iff {w top xi Cs : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Y : ℝ) |
| (M : Finset (Fin (binCount w top xi) → ℕ)) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) |
| (m : Fin (binCount w top xi) → ℕ) (hm : m∈M) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f∈selections (globalBins w top xi) m) (a : ℕ → ℕ) : |
| listedWord hw htop hxi Cs a (eligibleLists Y w Cs M Q) (descendingWord f) ↔ |
| ∃ q∈Q m,eligible Y w Cs q ∧ |
| ∀ p∈descendingWord f,Cs < primeExponent w p → aligns (SourceStopPredicate.sourceClass a) q p := by |
| rw [listed_word_eligible_iff hw htop hxi Y M Q a,word_multiplicity_descending hw htop hxi m f hf] |
| exact and_iff_right hm |
| |
| theorem unlisted_compact_selected_data {w top xi Cs C B K L alpha beta eps : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Y : ℕ) |
| (M : Finset (Fin (binCount w top xi) → ℕ)) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) |
| (m : Fin (binCount w top xi) → ℕ) (hm : m∈M) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f∈selections (globalBins w top xi) m) (a : ℕ → ℕ) (z : PrimeHistories.Node) |
| (F : Finset (List ℕ)) |
| (hu : descendingWord f∈unlistedGeometric hw htop hxi Cs C B K L alpha beta a z (eligibleLists Y w Cs M Q) |
| (CountErrorClassification.badCompactWords w K eps Y (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) a |
| (SmallSieveFinite.smallEuler ⌊w⌋₊) z F)) : |
| descendingWord f∈GeometricRegularWords.geometricWords hw htop hxi C B K L alpha beta z ∧ |
| eps < |ActualCountErrorEdges.wordCountError Y (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) a |
| (SmallSieveFinite.smallEuler ⌊w⌋₊) (descendingWord f)| ∧ |
| ¬∃ q∈Q m,eligible Y w Cs q ∧ |
| ∀ p∈descendingWord f,Cs < primeExponent w p → aligns (SourceStopPredicate.sourceClass a) q p := by |
| obtain ⟨hmem,hnot⟩ := Finset.mem_filter.mp hu |
| have hh := Finset.mem_inter.mp hmem |
| refine ⟨hh.2,(Finset.mem_filter.mp hh.1).2.2,?_⟩ |
| exact fun hwit => hnot ((listed_selection_eligible_iff hw htop hxi Y M Q m hm f hf a).mpr hwit) |
| end NumberTheoryLean.EligibleListWord |
| |
| |
| |
| namespace ErdosVarianceEligible |
| open NumberTheoryLean FinitePathGeometry PrimeHistories PrimeBinMembership ActualCountErrorEdges |
| GeometricRecentWitness GeometricRegularWords BoundedEdgeBins |
| ErdosCofactorChoices ErdosSubsetWord ErdosInverseCounts TwoPrimeObservableSum ParentTailPartition |
| LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
| attribute [local instance] Classical.propDecidable |
| |
| theorem geometric_selection_recent_witness (K eps : ℝ) (hK : 3 ≤ K) (heps : 0 < eps) : |
| ∃ Kstar B0 w0 : ℝ,K+10 < Kstar ∧ 3 ≤ B0 ∧ 1 < w0 ∧ |
| ∀ (B w top : ℝ),B0 ≤ B → w0 ≤ w → ∀ (hw : 1 < w) (htop : w < top), |
| ∀ (xi : ℝ) (hxi : 0 < xi) (C : ℝ),1 ≤ C → Real.log B ≤ 2*Real.log w → |
| ∀ (aStar alpha beta : ℝ),0 ≤ aStar → aStar ≤ 1 → ∀ Y : ℕ,0 < Y → ∀ z : Node, |
| z.gap = Real.log (Y : ℝ)/Real.log w-aStar+2 → z.side = .even → 199/100 ≤ z.ratio → |
| Consistent z → z.cutoff = B → z.closed = true → w^B = top → |
| ∀ (residue : ℕ → ℕ) (m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ), |
| f ∈ selections (globalBins w top xi) m → |
| descendingWord f ∈ geometricWords hw htop hxi C B K (2*Kstar+5) alpha beta z → |
| eps < |wordCountError Y (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) residue (SmallSieveFinite.smallEuler ⌊w⌋₊) (descendingWord f)| → |
| ∃ j : Fin (binCount w top xi),boundedEdgeBin w top xi Y m (2*Kstar+3) (2*Kstar+3) j ∧ |
| ∀ i : Fin (binCount w top xi),m i = 1 → j < i → |
| WitnessingBadEdge Y (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) residue |
| (eps/(4*(⌈2*Kstar+10⌉₊ : ℝ))) (SmallSieveFinite.smallEuler ⌊w⌋₊) |
| (pickedPrime f i) (selectionProduct (parentCofactorSelection f i j)) (pickedPrime f j) := by |
| obtain ⟨Kstar,B0,w0,hKs,hB0,hw0,hcore⟩ := geometric_recent_witness K eps hK heps |
| refine ⟨Kstar,B0,w0,hKs,hB0,hw0,?_⟩ |
| intro B w top hB hw0' hw htop xi hxi C hC hcomp aStar alpha beta ha0 ha1 Y hY z hroot |
| heven h199 hz hcut hclosed hpower residue m f hf hgeom hbad |
| have hh := hcore B w top hB hw0' hw htop xi hxi C hC hcomp aStar alpha beta ha0 ha1 Y hY z hroot |
| heven h199 hz hcut hclosed hpower residue (descendingWord f) hgeom hbad |
| simpa only [SoftWitnessCapture.word_multiplicity_descending hw htop hxi m f hf, |
| SoftWitnessCapture.word_selection_descending hw htop hxi m f hf] using hh |
| |
| end ErdosVarianceEligible |
| |
| |
| |
| open _root_.Filter |
| namespace ErdosVarianceEligible |
| open NumberTheoryLean FinitePathGeometry PrimeHistories PrimeBinMembership StrongReferenceTransport |
| SafeSubsetBoxGeometry BoundedEdgeBins ParentCofactorChoices ParentTailPartition |
| RawBoxRationalWitness SingletonBinSelection TwoPrimeObservableSum ActualCountErrorEdges |
| ErdosCofactorChoices ErdosSubsetWord ErdosVarianceWeighted ErdosVarianceEffective ErdosInverseBoxHeight |
| ErdosInverseCells ErdosInverseAlignment ErdosInverseBoxApplication ErdosInverseSampleCost ErdosInverseCounts |
| LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinPartition |
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
| |
| theorem outside_exceptions_eligible_word (C M : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M) : |
| ∀ᶠ top : ℝ in atTop, |
| ∀ xi : ℝ,∀ hxi : 0 < xi,xi ≤ 1 → 2*C*xi ≤ 1 → |
| ∀ (hw : 1 < sourceW top) (htop : sourceW top < top) (K aStar X : ℝ),0 ≤ aStar → |
| ∀ z : Node,0 < sourceY top → |
| z.gap = Real.log (sourceY top : ℝ)/Real.log (sourceW top)-aStar+2 → |
| Valid z.side z.ratio → Consistent z → StrongState z → z.closed = true → (sourceW top)^z.cutoff = top → |
| ∀ m : Fin (binCount (sourceW top) top xi) → ℕ,SafeAnchor hw htop hxi C (sourceB top) K z m → |
| ∀ f ∈ selections (globalBins (sourceW top) top xi) m, |
| ∀ i : Fin (binCount (sourceW top) top xi),m i = 1 → |
| ∀ J : Finset (Fin (binCount (sourceW top) top xi)), |
| (∀ j ∈ J,boundedEdgeBin (sourceW top) top xi (sourceY top) m M X j ∧ j < i) → |
| ∀ (residue : ℕ → ℕ) (delta cp sigma Cs eps : ℝ) (F : ℕ), |
| (∀ j ∈ J,SourceInverseConclusion (sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊sourceW top⌋₊) |
| residue delta (SmallSieveFinite.smallEuler ⌊sourceW top⌋₊) |
| (cofactorScale top xi (parentCofactorMultiplicity m i j)) (lower (sourceW top) top xi i) (sourceZ top) cp sigma F |
| (globalBins (sourceW top) top xi i) |
| (cofactorChoices (globalBins (sourceW top) top xi) (parentCofactorMultiplicity m i j)) (globalBins (sourceW top) top xi j)) → |
| (∃ j ∈ J,WitnessingBadEdge (sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊sourceW top⌋₊) residue delta |
| (SmallSieveFinite.smallEuler ⌊sourceW top⌋₊) (pickedPrime f i) |
| (selectionProduct (parentCofactorSelection f i j)) (pickedPrime f j)) → |
| eps < |wordCountError (sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊sourceW top⌋₊) residue |
| (SmallSieveFinite.smallEuler ⌊sourceW top⌋₊) (descendingWord f)| → |
| (¬∃ j ∈ J,actualWitness (sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊sourceW top⌋₊) residue delta |
| (SmallSieveFinite.smallEuler ⌊sourceW top⌋₊) (globalBins (sourceW top) top xi i) |
| (cofactorScale top xi (parentCofactorMultiplicity m i j)) (lower (sourceW top) top xi i) (sourceZ top) cp |
| (pickedPrime f i,selectionProduct (parentCofactorSelection f i j)) (pickedPrime f j)) → |
| let Q := rawBoxList (globalBins (sourceW top) top xi) i J residue |
| (fun j => cofactorScale top xi (parentCofactorMultiplicity m i j)) (lower (sourceW top) top xi i) (sourceZ top) cp |
| ¬hardSelection Q top (lower (sourceW top) top xi i) (width (sourceW top) top xi i) residue i Cs eps f → |
| eligibleWordWitness (sourceY top) (sourceW top) Cs residue Q (descendingWord f) := by |
| filter_upwards [structured_endpoint_heights C M hC hM, |
| ErdosInverseEuler.sourceW_tendsto_atTop.eventually_ge_atTop 2, |
| Real.tendsto_log_atTop.eventually JacobsthalSourceScale.source_scale_eventually] |
| with top hheight hw2 hscale |
| intro xi hxi hxi1 hDelta hw htop K aStar X ha z hY hroot hs hz hg hclosed hcap m hanchor |
| f hf i hi J hJ residue delta cp sigma Cs eps F hInv hbadEdge hbadWord hnoInv |
| dsimp only |
| intro hnoHard |
| let P := globalBins (sourceW top) top xi |
| let S := fun j => cofactorScale top xi (parentCofactorMultiplicity m i j) |
| let R := lower (sourceW top) top xi i |
| let Q := rawBoxList P i J residue S R (sourceZ top) cp |
| have hj : ∀ j ∈ J,m j = 1 := fun j hj => bounded_edge_singleton m j (hJ j hj).1 |
| obtain ⟨j,hjJ,r,hrQ,hrLocal,halign,hstructure⟩ := raw_box_witness P m i J f hf hi hj |
| (sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊sourceW top⌋₊) residue delta |
| (SmallSieveFinite.smallEuler ⌊sourceW top⌋₊) S R (sourceZ top) cp sigma F hInv hbadEdge hnoInv |
| have hbounds := hheight xi hxi hxi1 hDelta hw htop K aStar X ha z hY hroot hs hz hg hclosed hcap |
| m hanchor f hf i j (hJ j hjJ).2 hi (hJ j hjJ).1 residue r cp hrLocal hstructure |
| have hcomp : Real.log (sourceB top) ≤ 2*Real.log (sourceW top) := hscale.2.2.2.2 |
| have hdata := full_endpoint_data hw htop hxi hC ha hcomp (sourceY top) hY z hroot hs hz hg hclosed hcap |
| m hanchor f hf i hi |
| have hR : 0 < R := (zero_lt_one.trans hw).trans_le (bin_source_bounds (zero_lt_one.trans hw) htop hxi i).1 |
| have hwidth : 0 ≤ width (sourceW top) top xi i := (effectiveWidth_pos (zero_lt_one.trans hw) htop hxi).le |
| have hRp := ((ErdosInversePrimeBin.mem_primeBin hR.le hwidth (pickedPrime f i)).mp hdata.2.2.1).2.1.le |
| have hbad := word_error_bad_aligning_prime hw htop hxi (sourceY top) m f hf i hi residue r eps halign hbadWord |
| have hsoft := endpoint_soft_conclusion Q top R (width (sourceW top) top xi i) residue f i r Cs eps |
| hrQ hdata.1 hdata.2.1 hR hRp hw2 hdata.2.2.2 hbounds.1 hbounds.2 hbad hnoHard |
| exact eligible_word_witness_of_endpoint hw htop hxi (sourceY top) m f hf i hi residue Q r hrQ hsoft |
| |
| end ErdosVarianceEligible |
| |
| |
| end Erdos970 |
| |
| end OAI |
| |
