| import OAI.NumberTheory.Jacobsthal.Probability.RegularStoppedEvent |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.RegularStoppedMass |
| open FinitePathGeometry PrimeHistories PrimeBinMembership SourceStopPredicate ActualWordSelection ActualRegularBoxes |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinPartition |
| open ActualRegularAdmission ActualPrefixClearance RegularBadTagWords SourceMarkedStopping |
| open StoppedTraceSets StoppedCountAdapters PriorPrimeWindow MissingWindowPrimeMass |
| open StrongReferenceTransport StrongSourceFamilies ActualPrimeHigh ReferenceAdmission |
| open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail |
| open ErdosPrimeInputs.HarmonicPrimeMeasure ErdosInverseAlignment |
| open RegularStoppedEvent |
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| attribute [local instance] Classical.propDecidable |
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| theorem uncaptured_mass_le_exceptions {w top xi Cs eta Clen B K b₀ b₁ : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hC : 0 ≤ Clen) |
| (hcomp : Real.log B ≤ 2*Real.log w) (hmesh : 2*(xi/Real.log w) ≤ 6*(2*Clen*xi)) |
| (hb₀ : 0 < b₀) (hCs : Cs < 2*b₀) (hsearch : b₁ < w^((1/4:ℝ))) |
| (hh : xi/Real.log w ≤ b₀) (hsmall : 2*Clen*xi+(206/100)*(xi/Real.log w) ≤ (4/100)*b₀) |
| (Y : ℕ) (a : ℕ → ℕ) (z : Node) (hs : Valid z.side z.ratio) (hz : Consistent z) |
| (hg : StrongState z) (hclosed : z.closed=true) (hcap : w^z.cutoff=top) (hroot : b₁/2 < z.cutoff) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) : |
| (∑ ps ∈ uncapturedRegular hw htop hxi Y Cs eta Clen B K b₀ b₁ a z Q,prefixWeight ps) ≤ |
| (∑ ps ∈ regularBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Clen B K z Q (sourceClass a),prefixWeight ps) + |
| (∑ ps ∈ missingWindowPrefixes w 1 K b₀ b₁ z,prefixWeight ps) := by |
| have hsub := uncaptured_subset_exceptions (eta := eta) (K := K) hw htop hxi hC hcomp hmesh hb₀ hCs hsearch hh hsmall |
| Y a z hs hz hg hclosed hcap hroot Q |
| apply (Finset.sum_le_sum_of_subset_of_nonneg hsub (fun ps _ _ => prefixWeight_nonneg ps)).trans |
| have he : |
| (∑ ps ∈ regularBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Clen B K z Q (sourceClass a) ∪ |
| missingWindowPrefixes w 1 K b₀ b₁ z,prefixWeight ps) + |
| (∑ ps ∈ regularBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Clen B K z Q (sourceClass a) ∩ |
| missingWindowPrefixes w 1 K b₀ b₁ z,prefixWeight ps) = _ := Finset.sum_union_inter |
| have hn : 0 ≤ ∑ ps ∈ regularBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Clen B K z Q (sourceClass a) ∩ |
| missingWindowPrefixes w 1 K b₀ b₁ z,prefixWeight ps := Finset.sum_nonneg (fun ps _ => prefixWeight_nonneg ps) |
| linarith |
| end NumberTheoryLean.RegularStoppedMass |
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| namespace NumberTheoryLean.StoppedGeometryInvariance |
| open FinitePathGeometry StoppedCountVertex StoppedCountAdapters StoppedEvaluation StoppedTraceSets |
| open RegularStoppedEvent SourceStopPredicate LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels |
| open LogarithmicBinPartition |
| open PrimeHistories |
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| attribute [local instance] Classical.propDecidable |
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| theorem traces_eq_of_geometry (w : ℝ) (stop : List ℕ → Prop) (n : ℕ) (v v' : Vertex) |
| (hn : v.node=v'.node) (hp : v.past=v'.past) (ha : v.available=v'.available) : |
| traces w stop n v=traces w stop n v' := by |
| induction n generalizing v v' with |
| | zero => |
| simp only [traces] |
| apply ite_cond_congr |
| simp only [lowerStop,hn,hp] |
| | succ n ih => |
| have hstop : lowerStop stop v ↔ lowerStop stop v' := by simp only [lowerStop,hn,hp] |
| have hinc : includedPrimes w v=includedPrimes w v' := by simp only [includedPrimes,hn,ha] |
| have hchild : ∀ p,traces w stop n (child w v p)=traces w stop n (child w v' p) := by |
| intro p |
| exact ih (child w v p) (child w v' p) (by simp only [child,hn]) |
| (by simp only [child,hp]) (by simp only [child,ha]) |
| simp only [traces,hstop,hinc,hchild] |
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| theorem root_stopped_independent (w : ℝ) (stop : List ℕ → Prop) (n : ℕ) (z : Node) |
| (P H H' : Finset ℕ) (mu mu' : ℝ) : |
| stopped w stop n (rootVertex z H P mu)=stopped w stop n (rootVertex z H' P mu') := |
| congrArg Prod.snd (traces_eq_of_geometry w stop n _ _ rfl rfl rfl) |
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| theorem hasSourceStop_iff_original_root {w top xi : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (Y : ℕ) (Cs eta Clen B b₀ b₁ : ℝ) (a : ℕ → ℕ) (z : Node) (ps : List ℕ) |
| (H : Finset ℕ) (mu : ℝ) : |
| hasSourceStop hw htop hxi Y Cs eta Clen B b₀ b₁ a z ps ↔ |
| ∃ pre tail : List ℕ,ps=pre++tail ∧ pre ∈ stopped w |
| (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) |
| (sourcePrimeSet w top).card (rootVertex z H (sourcePrimeSet w top) mu) := by |
| unfold hasSourceStop |
| rw [root_stopped_independent w _ _ z (sourcePrimeSet w top) ∅ H 1 mu] |
| end NumberTheoryLean.StoppedGeometryInvariance |
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| namespace NumberTheoryLean.ExpandedNoStop |
| open StoppedCountVertex StoppedEvaluation StoppedVertexHistory StoppedTraceSets FirstStopSemantics |
| open FinitePathGeometry StoppedCountAdapters RegularStoppedEvent StoppedGeometryInvariance |
| open SourceStopPredicate LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinPartition |
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| attribute [local instance] Classical.propDecidable |
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| theorem expanded_no_lower_stop (w : ℝ) (stop : List ℕ → Prop) (n : ℕ) (v : Vertex) |
| {ps : List ℕ} (hp : ps ∈ expanded w stop n v) : |
| ∀ pre tail : List ℕ,ps=pre++tail → ¬lowerStop stop (after w v pre) := by |
| induction n generalizing v ps with |
| | zero => |
| unfold expanded at hp |
| rw [traces] at hp |
| split_ifs at hp with hs |
| · exact False.elim (Finset.notMem_empty _ hp) |
| · have he : ps=[] := Finset.mem_singleton.mp hp |
| subst ps |
| intro pre tail heq |
| have hpre := (List.append_eq_nil_iff.mp heq.symm).1 |
| simpa only [hpre,after] using hs |
| | succ n ih => |
| unfold expanded at hp |
| rw [traces] at hp |
| split_ifs at hp with hs |
| · exact False.elim (Finset.notMem_empty _ hp) |
| · rcases Finset.mem_insert.mp hp with he | hp |
| · subst ps |
| intro pre tail heq |
| have hpre := (List.append_eq_nil_iff.mp heq.symm).1 |
| simpa only [hpre,after] using hs |
| · obtain ⟨p,_hinc,hps⟩ := Finset.mem_biUnion.mp hp |
| obtain ⟨qs,hqs,rfl⟩ := Finset.mem_image.mp hps |
| intro pre tail heq |
| cases pre with |
| | nil => exact hs |
| | cons q pre => |
| rw [List.cons_append] at heq |
| obtain ⟨he,htail⟩ := List.cons.inj heq |
| subst q |
| exact ih (child w v p) hqs pre tail htail |
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| theorem expanded_no_stopped_prefix (w : ℝ) (stop : List ℕ → Prop) (n N : ℕ) (v : Vertex) |
| {ps : List ℕ} (hp : ps ∈ expanded w stop n v) : |
| ¬∃ pre tail : List ℕ,ps=pre++tail ∧ pre∈stopped w stop N v := by |
| rintro ⟨pre,tail,he,hstop⟩ |
| exact expanded_no_lower_stop w stop n v hp pre tail he (stopped_first w stop N v hstop).1 |
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| theorem expanded_no_source_stop {w top xi : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (Y : ℕ) (Cs eta Clen B b₀ b₁ : ℝ) (a : ℕ → ℕ) (z : PrimeHistories.Node) |
| (H : Finset ℕ) (mu : ℝ) (n : ℕ) {ps : List ℕ} |
| (hp : ps∈expanded w |
| (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) |
| n (rootVertex z H (sourcePrimeSet w top) mu)) : |
| ¬hasSourceStop hw htop hxi Y Cs eta Clen B b₀ b₁ a z ps := by |
| rw [hasSourceStop_iff_original_root hw htop hxi Y Cs eta Clen B b₀ b₁ a z ps H mu] |
| exact expanded_no_stopped_prefix w _ n _ _ hp |
| end NumberTheoryLean.ExpandedNoStop |
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| namespace NumberTheoryLean.PaperRegularStopping |
| open _root_.Filter FinitePathGeometry PrimeHistories PrimeBinMembership SourceStopPredicate SourceNodeCoordinates |
| open JacobsthalSourceScale PaperMissingPrimeWindow ActualRegularBoxes ActualPrefixClearance |
| open RegularStoppedEvent RegularStoppedMass PaperRegularTagCapture MovingStopMargins |
| open ActualBinOwners StrongReferenceTransport LogarithmicBinScale |
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| attribute [local instance] Classical.propDecidable |
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| theorem paper_regular_uncaptured_mass {R Clen Cs eta xi : ℝ} (hR : 3 ≤ R) (hC : 1 ≤ Clen) (hCs : 0 ≤ Cs) |
| (heta0 : 0 < eta) (heta : eta < 1/2) (hxi : 0 < xi) (hxi1 : xi ≤ 1) |
| (hDelta : 2*Clen*xi ≤ 1) (H : ℕ) (epsilon : ℝ) (hepsilon : 0 < epsilon) : |
| ∃ rho : ℝ,10 < rho ∧ ∀ K : ℝ,0 < K → |
| ∀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 → |
| ∀ Q : (Fin (binCount (sourceW (Real.log top)) top xi) → ℕ) → Finset ℚ, |
| (∀ mult ∈ regularBoxes hw htop hxi Clen (sourceB (Real.log top)) R z,(Q mult).card ≤ H) → |
| (∀ mult ∈ regularBoxes hw htop hxi Clen (sourceB (Real.log top)) R z, |
| ∀ q ∈ Q mult,eligible Y (sourceW (Real.log top)) Cs q) → |
| (sourceB (Real.log top))^2 * |
| (∑ ps ∈ uncapturedRegular hw htop hxi Y Cs eta Clen (sourceB (Real.log top)) R |
| (K*(Real.log (sourceW (Real.log top)))^2) |
| (rho*K*(Real.log (sourceW (Real.log top)))^2) a z Q, |
| ErdosPrimeInputs.PrimePrefixMass.prefixWeight ps) ≤ epsilon := by |
| obtain ⟨rho,hrho,hK⟩ := paper_missing_prime_window_mass R (epsilon/2) hR (by positivity) |
| refine ⟨rho,hrho,?_⟩ |
| intro K hKpos |
| obtain ⟨L₀,_hL₀,hmiss⟩ := hK K hKpos |
| have hwT := sourceW_tendsto.comp Real.tendsto_log_atTop |
| have hBT := sourceB_tendsto.comp Real.tendsto_log_atTop |
| filter_upwards [paper_regular_bad_tag_mass hR hC hCs heta0 heta hxi hxi1 hDelta H (epsilon/2) (by positivity), |
| hwT.eventually (moving_stop_margins Cs Clen xi hxi (by linarith : 0 < rho) hKpos), |
| Real.tendsto_log_atTop.eventually source_scale_eventually, |
| Real.tendsto_log_atTop.eventually sourceW_le_sourceB_eventually, |
| hBT.eventually_ge_atTop 3,Real.tendsto_log_atTop.eventually_ge_atTop L₀, |
| eventually_gt_atTop (1:ℝ)] with top htag hmargin hscale hWB hB3 hL htop1 |
| obtain ⟨hw,htop,htag⟩ := htag |
| obtain ⟨_hw,hlog,hb₀,hCsWin,hsearch,hh,hsmall,hbelow⟩ := hmargin |
| refine ⟨hw,htop,?_⟩ |
| intro Y hY z a hi h199 h23 hz hcut hclosed Q hcard helig |
| have hs := (source_node_bounds hscale.2.1 z hi h199 h23 hz hcut).1 |
| have hg := source_strong_state hB3 z hi h199 hz hcut |
| have hcap : (sourceW (Real.log top))^z.cutoff=top := by |
| rw [hcut] |
| exact (sourceW_pow_sourceB hw).trans (Real.exp_log (zero_lt_one.trans htop1)) |
| have hroot : (rho*K*(Real.log (sourceW (Real.log top)))^2)/2 < z.cutoff := by |
| rw [hcut] |
| exact hbelow _ hWB |
| have hm := uncaptured_mass_le_exceptions (eta := eta) (K := R) hw htop hxi (by linarith : 0 ≤ Clen) hscale.2.2.2.2 |
| (source_mesh_clearance_condition hlog hxi.le hC) hb₀ hCsWin hsearch hh hsmall |
| Y a z hs hz hg hclosed hcap hroot Q |
| have htagmass := htag Y hY z (sourceClass a) hi h199 h23 hz hcut hclosed Q hcard helig |
| have hmissmass := hmiss (Real.log top) hL 1 (by norm_num) (by norm_num) z hi h199 h23 hz hcut |
| have hm' := mul_le_mul_of_nonneg_left hm (sq_nonneg (sourceB (Real.log top))) |
| rw [mul_add] at hm' |
| simp only [Function.comp_def] at hm' |
| linarith |
| end NumberTheoryLean.PaperRegularStopping |
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| namespace NumberTheoryLean.ExpandedHardCapture |
| open FinitePathGeometry PrimeHistories SourceStopPredicate ActualRegularBoxes ActualWordSelection |
| open StoppedCountAdapters StoppedTraceSets RegularStoppedEvent ExpandedNoStop |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosPrimeInputs.HarmonicPrimeMeasure ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail |
| open ErdosInverseAlignment |
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| attribute [local instance] Classical.propDecidable |
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| theorem expanded_listed_subset_uncaptured {w top xi : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (Y : ℕ) (Cs eta Clen B K b₀ b₁ : ℝ) (a : ℕ → ℕ) (z : Node) |
| (H : Finset ℕ) (mu : ℝ) (n : ℕ) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (F : Finset (List ℕ)) |
| (hexp : F ⊆ expanded w |
| (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) |
| n (rootVertex z H (sourcePrimeSet w top) mu)) |
| (hreg : F ⊆ regularWords hw htop hxi Clen B K z) |
| (halign : ∀ ps∈F,∃ q∈Q (wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) ps), |
| ∀ p∈ps,Cs < primeExponent w p → aligns (sourceClass a) q p) : |
| F ⊆ uncapturedRegular hw htop hxi Y Cs eta Clen B K b₀ b₁ a z Q := by |
| intro ps hp |
| exact Finset.mem_filter.mpr ⟨hreg hp,halign ps hp, |
| expanded_no_source_stop hw htop hxi Y Cs eta Clen B b₀ b₁ a z H mu n (hexp hp)⟩ |
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| theorem expanded_listed_mass_le {w top xi : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (Y : ℕ) (Cs eta Clen B K b₀ b₁ : ℝ) (a : ℕ → ℕ) (z : Node) |
| (H : Finset ℕ) (mu : ℝ) (n : ℕ) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (F : Finset (List ℕ)) |
| (hexp : F ⊆ expanded w |
| (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) |
| n (rootVertex z H (sourcePrimeSet w top) mu)) |
| (hreg : F ⊆ regularWords hw htop hxi Clen B K z) |
| (halign : ∀ ps∈F,∃ q∈Q (wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) ps), |
| ∀ p∈ps,Cs < primeExponent w p → aligns (sourceClass a) q p) : |
| (∑ ps∈F,prefixWeight ps) ≤ |
| ∑ ps∈uncapturedRegular hw htop hxi Y Cs eta Clen B K b₀ b₁ a z Q,prefixWeight ps := |
| Finset.sum_le_sum_of_subset_of_nonneg |
| (expanded_listed_subset_uncaptured hw htop hxi Y Cs eta Clen B K b₀ b₁ a z H mu n Q F hexp hreg halign) |
| (fun ps _ _ => prefixWeight_nonneg ps) |
| end NumberTheoryLean.ExpandedHardCapture |
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| end Erdos970 |
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| end OAI |
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