| import OAI.NumberTheory.Jacobsthal.Partitions.GeometricTotalBoxMass |
| import OAI.NumberTheory.Jacobsthal.Probability.ActualUnlistedEvent |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.GlobalCompactTransfer |
| open _root_.Filter FinitePathGeometry PrimeHistories PrimeBinMembership SourceStopPredicate |
| open JacobsthalSourceScale ActualRegularBoxes ActualBinOwners UniformGeometricRegularMass |
| open GeometricTotalBoxMass GeometricBoxImages GeometricRegularWords |
| open ExpandedCompactPartition CountErrorClassification RegularStoppedEvent PaperRegularStopping |
| open EligibleBoxLists ActualUnlistedEvent FilteredBoxWordMass CanonicalSubsetBox |
| open StoppedCountVertex StoppedCountAdapters StoppedTraceSets |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosCofactorChoices ErdosSubsetWord ErdosVarianceEligible |
| open ErdosPrimeInputs.PrimePrefixMass |
| open scoped Topology |
| attribute [local instance] Classical.propDecidable |
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| theorem paper_compact_from_local_bounds (K : ℝ) (hK : 3 ≤ K) : |
| ∃ Clen₀ Cgeom A : ℝ,1 ≤ Clen₀ ∧ 0 < Cgeom ∧ 0 < A ∧ |
| ∀ Clen : ℝ,Clen₀ ≤ Clen → ∀ geoErr : ℝ,0 < geoErr → |
| ∃ alpha : ℝ,0 < alpha ∧ alpha ≤ 1/400 ∧ |
| ∀ L Cs eta : ℝ,0 ≤ L → 0 ≤ Cs → 0 < eta → eta < 1/2 → |
| ∀ NQ : ℕ,∀ capErr : ℝ,0 < capErr → ∀ xi : ℝ,∀ hxi : 0 < xi, |
| xi ≤ 1 → 20*Clen*xi ≤ 1 → |
| ∃ rho : ℝ,10 < rho ∧ ∀ Kstop : ℝ,0 < Kstop → |
| ∀ᶠ top : ℝ in atTop,∃ hw : 1 < sourceW (Real.log top),∃ htop : sourceW (Real.log top) < top, |
| ∀ Y : ℕ,Y ≤ ⌊top^2/(Real.log top)^2⌋₊ → ∀ (z : Node) (a : ℕ → ℕ), |
| z.side=.even → 199/100 ≤ z.ratio → z.ratio ≤ 23/10 → Consistent z → |
| z.cutoff=sourceB (Real.log top) → z.closed=true → |
| ∀ eps tau : ℝ,0 ≤ tau → |
| ∀ Qraw : (Fin (binCount (sourceW (Real.log top)) top xi) → ℕ) → Finset ℚ, |
| let M := geometricBoxes hw htop hxi Clen (sourceB (Real.log top)) K L alpha (1/100) z |
| (∀ m∈M,(Qraw m).card ≤ NQ) → |
| (∀ m∈M,(∑ f∈(selections (globalBins (sourceW (Real.log top)) top xi) m).filter |
| (unwitnessedSelection hw htop hxi Y Cs Clen (sourceB (Real.log top)) K L alpha (1/100) eps a z (Qraw m)), |
| (selectionProduct f:ℝ)⁻¹) ≤ tau*selectionMass (globalBins (sourceW (Real.log top)) top xi) m) → |
| let w := sourceW (Real.log top) |
| let B := sourceB (Real.log top) |
| let small := LargePrimeDeletion.cutoffPrimes ⌊w⌋₊ |
| let V0 := SmallSieveFinite.smallEuler ⌊w⌋₊ |
| let stop := stopCandidate Y w Cs eta Clen B xi (Kstop*(Real.log w)^2) (rho*Kstop*(Real.log w)^2) |
| (lower w top xi) (width w top xi) (label (zero_lt_one.trans hw) htop hxi) a z |
| B^2*(∑ ps∈badCompactWords w K eps Y small a V0 z |
| (expanded w stop (sourcePrimeSet w top).card (rootVertex z ∅ (sourcePrimeSet w top) ((Y:ℝ)*V0))),prefixWeight ps) |
| ≤ Cgeom*(2*Clen*xi)+geoErr+A*tau+capErr := by |
| obtain ⟨A,BA,WA,hA,hBA,hWA,hTotal⟩ := uniform_geometric_box_mass K hK |
| obtain ⟨Clen₀,Cgeom,hClen₀,hCgeom,hGeometry⟩ := geometric_defect_uniform_in_L K hK 2 (by norm_num) |
| (1/400) (by norm_num) (by norm_num) |
| refine ⟨Clen₀,Cgeom,A,hClen₀,hCgeom,hA,?_⟩ |
| intro Clen hClen geoErr hgeoErr |
| have hClen1 : 1 ≤ Clen := hClen₀.trans hClen |
| obtain ⟨alpha,halpha,halphaUpper,hL⟩ := hGeometry Clen hClen geoErr hgeoErr |
| refine ⟨alpha,halpha,halphaUpper,?_⟩ |
| intro L Cs eta hL0 hCs heta0 heta NQ capErr hcapErr xi hxi hxi1 h20 |
| obtain ⟨BG,WG,hBG,hWG,hGeometry⟩ := hL L hL0 |
| have hbeta : 4*(1/400:ℝ)=1/100 := by norm_num |
| rw [hbeta] at hGeometry |
| have hDelta : 2*Clen*xi ≤ 1 := by |
| have hp : 0 ≤ Clen*xi := mul_nonneg (by linarith) hxi.le |
| nlinarith |
| obtain ⟨rho,hrho,hCapture⟩ := paper_regular_uncaptured_mass hK hClen1 hCs heta0 heta hxi hxi1 hDelta NQ capErr hcapErr |
| refine ⟨rho,hrho,?_⟩ |
| intro Kstop hKstop |
| have hBT := sourceB_tendsto.comp Real.tendsto_log_atTop |
| have hWT := sourceW_tendsto.comp Real.tendsto_log_atTop |
| filter_upwards [hCapture Kstop hKstop,hBT.eventually_ge_atTop (max BG BA), |
| hWT.eventually_ge_atTop (max WG WA),Real.tendsto_log_atTop.eventually source_scale_eventually] |
| with top hCapture hB hW hscale |
| dsimp only [Function.comp_def] at hB hW |
| obtain ⟨hw,htop,hCapture⟩ := hCapture |
| refine ⟨hw,htop,?_⟩ |
| intro Y hY z a hi h199 h23 hz hcut hclosed eps tau htau Qraw |
| dsimp only |
| intro hcard hLocal |
| let w := sourceW (Real.log top) |
| let B := sourceB (Real.log top) |
| let P := sourcePrimeSet w top |
| let small := LargePrimeDeletion.cutoffPrimes ⌊w⌋₊ |
| let V0 := SmallSieveFinite.smallEuler ⌊w⌋₊ |
| let v := rootVertex z ∅ P ((Y:ℝ)*V0) |
| let M := geometricBoxes hw htop hxi Clen B K L alpha (1/100) z |
| let Q := eligibleLists (Y:ℝ) w Cs M Qraw |
| let b₀ := Kstop*(Real.log w)^2 |
| let b₁ := rho*Kstop*(Real.log w)^2 |
| let stop := stopCandidate Y w Cs eta Clen B xi b₀ b₁ (lower w top xi) (width w top xi) |
| (label (zero_lt_one.trans hw) htop hxi) a z |
| let F := badCompactWords w K eps Y small a V0 z (expanded w stop P.card v) |
| have hpower : w^B=top := (sourceW_pow_sourceB hw).trans (Real.exp_log ((zero_lt_one.trans hw).trans htop)) |
| have hRef := bad_compact_reference_data w top K eps Y a z stop P.card V0 |
| have hGeom := hGeometry B w top ((le_max_left _ _).trans hB) ((le_max_left _ _).trans hW) |
| hw htop xi hxi hxi1 h20 hscale.2.2.2.2 z hi h199 h23 hz hcut hclosed hpower F hRef.1 hRef.2 |
| have hTot := hTotal B w top ((le_max_right _ _).trans hB) ((le_max_right _ _).trans hW) |
| hw htop xi Clen hxi hClen1 hscale.2.2.2.2 hDelta z hi h199 h23 hz hcut hclosed hpower L alpha (1/100) |
| have hCard := eligibleLists_card (Y:ℝ) w Cs M Qraw NQ hcard |
| have hCap := hCapture Y hY z a hi h199 h23 hz hcut hclosed Q |
| (fun m _hm => hCard m) (fun m _hm q hq => eligibleLists_eligible (Y:ℝ) w Cs M Qraw m q hq) |
| have hBox : ∀ m∈M,selectedWordMass (globalBins w top xi) m |
| (unlistedGeometric hw htop hxi Cs Clen B K L alpha (1/100) a z Q F) ≤ tau*selectionMass (globalBins w top xi) m := by |
| intro m hm |
| exact (selected_unlisted_le_unwitnessed_mass (Cs:=Cs) (C:=Clen) (B:=B) (K:=K) (L:=L) |
| (alpha:=alpha) (beta:=1/100) (eps:=eps) hw htop hxi Y M Qraw m hm a z (expanded w stop P.card v)).trans (hLocal m hm) |
| have hUnlisted := unlisted_box_transfer hw htop hxi htau a z Q F hTot hBox |
| have hSplit := expanded_geometric_mass_split hw htop hxi Y Cs eta Clen B K L alpha (1/100) b₀ b₁ a z |
| ∅ ((Y:ℝ)*V0) P.card Q F (fun _ hp => (Finset.mem_filter.mp hp).1) |
| have hScaled := mul_le_mul_of_nonneg_left hSplit (sq_nonneg B) |
| change B^2*(∑ ps∈F,prefixWeight ps) ≤ Cgeom*(2*Clen*xi)+geoErr+A*tau+capErr |
| nlinarith |
| end NumberTheoryLean.GlobalCompactTransfer |
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| namespace NumberTheoryLean.SourceCompactErrorMass |
| open _root_.Filter FinitePathGeometry PrimeHistories PrimeBinMembership SourceStopPredicate |
| open GlobalCompactTransfer GeometricBoxImages GeometricRegularWords CountErrorClassification |
| open StoppedCountVertex StoppedCountAdapters StoppedTraceSets CanonicalSubsetBox |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosInverseBoxHeight ErdosCofactorChoices ErdosSubsetWord ErdosVarianceEligible ErdosVarianceFactory |
| open ErdosPrimeInputs.PrimePrefixMass |
| open scoped Topology |
| attribute [local instance] Classical.propDecidable |
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| theorem source_compact_error_mass (K eps budget aStar Cmin : ℝ) |
| (hK : 3 ≤ K) (heps : 0 < eps) (hbudget : 0 < budget) (ha : 0 < aStar) (ha1 : aStar ≤ 1) : |
| ∃ Clen Cs xi₀ : ℝ,1 ≤ Clen ∧ 0 < Cs ∧ Cmin ≤ Cs ∧ 0 < xi₀ ∧ xi₀ ≤ 1 ∧ |
| ∀ eta : ℝ,0 < eta → eta < 1/2 → ∀ xi : ℝ,∀ hxi : 0 < xi,xi ≤ xi₀ → |
| ∃ rho : ℝ,10 < rho ∧ ∀ Kstop : ℝ,0 < Kstop → |
| ∀ᶠ top : ℝ in atTop,∃ hw : 1 < sourceW top,∃ htop : sourceW top < top, |
| ∀ (z : Node) (a : ℕ → ℕ),z.side=.even → 199/100 ≤ z.ratio → z.ratio ≤ 23/10 → |
| Consistent z → z.cutoff=sourceB top → z.closed=true → |
| z.gap=Real.log (sourceY top:ℝ)/Real.log (sourceW top)-aStar+2 → |
| let w := sourceW top |
| let B := sourceB top |
| let Y := sourceY top |
| let small := LargePrimeDeletion.cutoffPrimes ⌊w⌋₊ |
| let V0 := SmallSieveFinite.smallEuler ⌊w⌋₊ |
| let stop := stopCandidate Y w Cs eta Clen B xi (Kstop*(Real.log w)^2) (rho*Kstop*(Real.log w)^2) |
| (lower w top xi) (width w top xi) (label (zero_lt_one.trans hw) htop hxi) a z |
| B^2*(∑ ps∈badCompactWords w K eps Y small a V0 z |
| (expanded w stop (sourcePrimeSet w top).card (rootVertex z ∅ (sourcePrimeSet w top) ((Y:ℝ)*V0))),prefixWeight ps) ≤ budget := by |
| obtain ⟨Clen,Cgeom,A,hClen,hCgeom,hA,hGlobal⟩ := paper_compact_from_local_bounds K hK |
| have hClen0 : 0 < Clen := by linarith |
| have hquarter : 0 < budget/4 := by positivity |
| obtain ⟨alpha,halpha,_halphaUpper,hGlobal⟩ := hGlobal Clen (le_refl _) (budget/4) hquarter |
| let tau := budget/(4*A) |
| have htau : 0 < tau := by dsimp [tau]; positivity |
| obtain ⟨Kstar,hKstar,NQ,Cs,hCs,hCsMin,_hCsK,xiF,hxiF,hxiF1,hFactory⟩ := |
| source_raw_box_factory Clen K aStar alpha eps tau Cmin hClen hK ha ha1 halpha heps htau |
| let xi₀ := min xiF (min (1/(20*Clen)) (budget/(8*Clen*Cgeom))) |
| have hxi₀ : 0 < xi₀ := lt_min hxiF (lt_min (by positivity) (by positivity)) |
| have hxi₀1 : xi₀ ≤ 1 := (min_le_left _ _).trans hxiF1 |
| refine ⟨Clen,Cs,xi₀,hClen,hCs,hCsMin,hxi₀,hxi₀1,?_⟩ |
| intro eta heta0 heta xi hxi hx |
| have hxF : xi ≤ xiF := hx.trans (min_le_left _ _) |
| have hxi1 : xi ≤ 1 := hx.trans hxi₀1 |
| have h20 : 20*Clen*xi ≤ 1 := by |
| have hh := (le_div_iff₀ (show 0 < 20*Clen by positivity)).mp |
| ((hx.trans (min_le_right _ _)).trans (min_le_left _ _)) |
| nlinarith |
| have hGeomBudget : Cgeom*(2*Clen*xi) ≤ budget/4 := by |
| have hh := (le_div_iff₀ (show 0 < 8*Clen*Cgeom by positivity)).mp |
| ((hx.trans (min_le_right _ _)).trans (min_le_right _ _)) |
| nlinarith |
| have hL : 0 ≤ 2*Kstar+5 := by linarith |
| obtain ⟨rho,hrho,hGlobal⟩ := hGlobal (2*Kstar+5) Cs eta hL hCs.le heta0 heta NQ (budget/4) hquarter xi hxi hxi1 h20 |
| refine ⟨rho,hrho,?_⟩ |
| intro Kstop hKstop |
| filter_upwards [hGlobal Kstop hKstop,hFactory xi hxi hxF] with top hGlobal hFactory |
| obtain ⟨hw,htop,hGlobal⟩ := hGlobal |
| obtain ⟨_hwF,_htopF,hFactory⟩ := hFactory |
| refine ⟨hw,htop,?_⟩ |
| intro z a hi h199 h23 hz hcut hclosed hroot |
| dsimp only |
| let M := geometricBoxes hw htop hxi Clen (sourceB top) K (2*Kstar+5) alpha (1/100) z |
| have hLists (m : Fin (binCount (sourceW top) top xi) → ℕ) (hm : m∈M) := |
| hFactory z hi h199 hz hcut hclosed hroot a m hm |
| have hAll : ∀ m : Fin (binCount (sourceW top) top xi) → ℕ,∃ Q : Finset ℚ, |
| Q.card ≤ NQ ∧ (m∈M → |
| (∑ f∈(selections (globalBins (sourceW top) top xi) m).filter |
| (unwitnessedSelection hw htop hxi (sourceY top) Cs Clen (sourceB top) K (2*Kstar+5) alpha (1/100) eps a z Q), |
| (selectionProduct f:ℝ)⁻¹) ≤ tau*selectionMass (globalBins (sourceW top) top xi) m) := by |
| intro m |
| by_cases hm : m∈M |
| · obtain ⟨Q,hcard,hMass⟩ := hLists m hm |
| exact ⟨Q,hcard,fun _ => hMass⟩ |
| · exact ⟨∅,by simp,fun hm' => False.elim (hm hm')⟩ |
| choose Qraw hCard hLocal using hAll |
| have hBound := hGlobal (sourceY top) (le_refl _) z a hi h199 h23 hz hcut hclosed eps tau htau.le Qraw |
| (fun m _hm => hCard m) hLocal |
| have hTau : A*tau=budget/4 := by dsimp [tau]; field_simp [hA.ne'] |
| rw [hTau] at hBound |
| dsimp only at hBound |
| exact hBound.trans (by nlinarith) |
| end NumberTheoryLean.SourceCompactErrorMass |
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| end Erdos970 |
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| end OAI |
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