import Erdos1135.ND.Discrepancy.A5FlatNoHitRate import Erdos1135.ND.Probability.UniformFloorPerturbation import Erdos1135.Tao.Section5.TimeBoundedNoHit /-! # Scheduled no-hit rate for the uniform ND source block The ordinary ND-M2 escape row bounds sources that never reach the passage threshold. A logarithmic-time theorem needs the larger event of sources that do not reach it by the literal Section 5 schedule. The same valuation certificate proves the same flat-source rate once the time witness is retained. -/ namespace Erdos1135 namespace ND open Filter open scoped Topology noncomputable section /-- Uniform-block probability of failing to hit `floor x` by its literal Section 5 schedule. -/ noncomputable def uniformScheduledNoHitProbability (x y : ℝ) (hwindow : (oddBlock y).Nonempty) : ℝ := Tao.pmfProb (uniformOddBlockPMF y hwindow) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 (Nat.floor x) (Tao.taoSection5N0 (Nat.floor x))} /-- Scheduled no-hit event on the natural A5 source carrier. -/ def ndA5NoHitWithinEvent (B : ℕ) : Set Tao.TaoOddNat := {N | ¬ Tao.syracuseHitsAtMostWithin N.1 B (Tao.taoSection5N0 B)} private theorem pmfOuterMass_toReal_mono_time {ι : Type*} (p : PMF ι) {A D : Set ι} (hAD : A ⊆ D) : (p.toOuterMeasure A).toReal ≤ (p.toOuterMeasure D).toReal := by have hfinite : p.toOuterMeasure D ≠ ⊤ := by apply ne_of_lt calc p.toOuterMeasure D ≤ p.toOuterMeasure Set.univ := p.toOuterMeasure.mono (Set.subset_univ D) _ = 1 := (p.toOuterMeasure_apply_eq_one_iff Set.univ).2 (Set.subset_univ _) _ < ⊤ := ENNReal.one_lt_top exact ENNReal.toReal_mono hfinite (p.toOuterMeasure.mono hAD) /-- On one exact harmonic band, scheduled failure maps to the same strict-low valuation event used by the eventual no-hit estimate. -/ theorem ndA5HarmonicBandNoHitWithinMass_le_error {B : ℕ} (descent : Tao.TaoSection5DescentScaleFacts B) {branch : Tao.TaoSection5SourceBranch} {j : ℕ} (hB : 1 ≤ B) (hcount : 0 < ndA5BandCount B branch) (hj : j < ndA5BandCount B branch) : ((ndA5HarmonicBandPMF B branch j hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal ≤ Tao.taoSection5NoHitError B := by let μ := ndA5HarmonicBandPMF B branch j hB hcount let n := Tao.taoSection5N0 B let E := Tao.taoLowValuationWeightEvent n have hsupport : μ.toOuterMeasure (ndA5NoHitWithinEvent B ∩ {N | N.1 ∈ ndA5OddBand B branch j}) = μ.toOuterMeasure (ndA5NoHitWithinEvent B) := by simpa only [μ] using (ndA5HarmonicBandPMF_toOuterMeasure_inter_band hB hcount (ndA5NoHitWithinEvent B)) have hmap : (μ.toOuterMeasure (ndA5NoHitWithinEvent B)).toReal ≤ ((Tao.taoProp19ActualValuationLaw μ n).toOuterMeasure E).toReal := by calc (μ.toOuterMeasure (ndA5NoHitWithinEvent B)).toReal = (μ.toOuterMeasure (ndA5NoHitWithinEvent B ∩ {N | N.1 ∈ ndA5OddBand B branch j})).toReal := congrArg ENNReal.toReal hsupport.symm _ ≤ (μ.toOuterMeasure {N | Tao.syracuseValuationPNatList n N.1 N.2 ∈ E}).toReal := by apply pmfOuterMass_toReal_mono_time rintro N ⟨hno, hband⟩ apply descent.actualValuations_mem_low_of_not_hitsWithin (branch := branch) · exact ndA5OddBand_mem_taoSection5SourceWindow hB hj hband · exact hno _ = ((Tao.taoProp19ActualValuationLaw μ n).toOuterMeasure E).toReal := by unfold Tao.taoProp19ActualValuationLaw rw [PMF.toOuterMeasure_map_apply] apply congrArg ENNReal.toReal apply congrArg μ.toOuterMeasure rfl have hM : 1 ≤ Tao.taoSection5NPrime B := by have hn0 := descent.five_le_n0 unfold Tao.taoSection5NPrime omega have hvaluation := taoProp19ValuationTV_ndA5HarmonicBandPMF_le hB j hcount hM unfold Tao.taoProp19ValuationTV at hvaluation have hcompare : ((Tao.taoProp19ActualValuationLaw μ n).toOuterMeasure E).toReal ≤ ((Tao.geom2PNatListPMF n).toOuterMeasure E).toReal + 4 * (2 : ℝ) ^ (-((1 / 128 : ℝ) * (n : ℝ))) := by apply Tao.pmfOuterMass_le_add_of_taoPMFFullL1_le simpa only [μ, n] using hvaluation have hideal := Tao.geom2PNatListPMF_lowWeightEvent_le_exp n calc ((ndA5HarmonicBandPMF B branch j hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal ≤ ((Tao.taoProp19ActualValuationLaw μ n).toOuterMeasure E).toReal := by simpa only [μ] using hmap _ ≤ ((Tao.geom2PNatListPMF n).toOuterMeasure E).toReal + 4 * (2 : ℝ) ^ (-((1 / 128 : ℝ) * (n : ℝ))) := hcompare _ ≤ Real.exp (-((n : ℝ) / 3200)) + 4 * (2 : ℝ) ^ (-((1 / 128 : ℝ) * (n : ℝ))) := by exact add_le_add (by simpa only [E] using hideal) le_rfl _ = Tao.taoSection5NoHitError B := by rfl theorem ndA5FlatBandNoHitWithinMass_le_exp_mul_error {B : ℕ} (descent : Tao.TaoSection5DescentScaleFacts B) {branch : Tao.TaoSection5SourceBranch} {j : ℕ} (hB : 1 ≤ B) (hcount : 0 < ndA5BandCount B branch) (hj : j < ndA5BandCount B branch) : ((ndA5FlatBandPMF B branch j hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal ≤ Real.exp (ndA5BandBeta B branch) * Tao.taoSection5NoHitError B := by exact (ndA5FlatBandPMF_eventMass_le_exp_mul_harmonic hB hcount (ndA5NoHitWithinEvent B)).trans (mul_le_mul_of_nonneg_left (ndA5HarmonicBandNoHitWithinMass_le_error descent hB hcount hj) (Real.exp_pos _).le) theorem uniformScheduledNoHitProbability_natCast_sourceY_eq_flatSourceMass {B : ℕ} (hB : 1 ≤ B) (branch : Tao.TaoSection5SourceBranch) (hcount : 0 < ndA5BandCount B branch) : uniformScheduledNoHitProbability (B : ℝ) (Tao.taoSection5SourceY B branch) (ndA5OddBlock_sourceY_nonempty hB branch hcount) = ((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal := by unfold uniformScheduledNoHitProbability ndA5FlatSourcePMF rw [Tao.pmfProb_eq_toOuterMeasure_toReal, PMF.toOuterMeasure_map_apply] apply congrArg ENNReal.toReal apply congrArg (uniformOddBlockPMF (Tao.taoSection5SourceY B branch) (ndA5OddBlock_sourceY_nonempty hB branch hcount)).toOuterMeasure ext N simp only [Set.mem_setOf_eq, Set.mem_preimage, ndA5NoHitWithinEvent, Nat.floor_natCast] rfl /-- Both natural flat-source branches retain the existing `44 B^-1/32000` rate for failure by the literal schedule. -/ theorem eventually_ndA5FlatSourceNoHitWithinMass_le_forty_four : ∀ᶠ B : ℕ in atTop, ∀ (branch : Tao.TaoSection5SourceBranch) (hB : 1 ≤ B) (hcount : 0 < ndA5BandCount B branch), ((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal ≤ 44 * (B : ℝ) ^ (-(1 / 32000 : ℝ)) := by have hlog : Tendsto (fun B : ℕ => Real.log (B : ℝ)) atTop atTop := Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop filter_upwards [Tao.eventually_taoSection5DescentScaleFacts, hlog.eventually_ge_atTop (300000 : ℝ)] with B descent hlogB intro branch hB hcount let R := (B : ℝ) ^ (-(1 / 32000 : ℝ)) have hR : 0 ≤ R := Real.rpow_nonneg (Nat.cast_nonneg B) _ have hfortyR : 0 ≤ (40 : ℝ) * R := mul_nonneg (by norm_num) hR have hexp : Real.exp (ndA5BandBeta B branch) ≤ 4 := by calc Real.exp (ndA5BandBeta B branch) ≤ Real.exp (2 * Real.log 2) := (Real.exp_lt_exp.mpr (ndA5BandBeta_lt_two_mul_logTwo (branch := branch) hB hlogB)).le _ = 4 := by rw [two_mul, Real.exp_add, Real.exp_log (by norm_num : (0 : ℝ) < 2)] norm_num have hband (j : ℕ) (hj : j < ndA5BandCount B branch) : ((ndA5FlatBandPMF B branch j hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal ≤ 40 * R := by calc ((ndA5FlatBandPMF B branch j hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal ≤ Real.exp (ndA5BandBeta B branch) * Tao.taoSection5NoHitError B := ndA5FlatBandNoHitWithinMass_le_exp_mul_error descent hB hcount hj _ ≤ Real.exp (ndA5BandBeta B branch) * (10 * R) := mul_le_mul_of_nonneg_left (by simpa only [R] using Tao.taoSection5NoHitError_le_ten_mul_threshold_rpow hB) (Real.exp_pos _).le _ ≤ 4 * (10 * R) := mul_le_mul_of_nonneg_right hexp (by positivity) _ = 40 * R := by ring have hbands : (∑ j ∈ Finset.range (ndA5BandCount B branch), ndA5FlatBandWeight B branch j * ((ndA5FlatBandPMF B branch j hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal) ≤ ∑ j ∈ Finset.range (ndA5BandCount B branch), ndA5FlatBandWeight B branch j * (40 * R) := by apply Finset.sum_le_sum intro j hj exact mul_le_mul_of_nonneg_left (hband j (Finset.mem_range.mp hj)) (ndA5FlatBandWeight_nonneg B branch j) have hsumWeight : (∑ j ∈ Finset.range (ndA5BandCount B branch), ndA5FlatBandWeight B branch j) ≤ 1 := by have hnormalize := sum_ndA5FlatBandWeight_add_topWeight_eq_one hB branch hcount have htopNonneg := ndA5FlatTopWeight_nonneg B branch linarith have hpow : (B : ℝ) ^ (-1 : ℝ) ≤ R := by exact Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast hB : (1 : ℝ) ≤ B) (by norm_num) have htop : ndA5FlatTopRestrictedMass B branch (ndA5NoHitWithinEvent B) ≤ 4 * R := by calc ndA5FlatTopRestrictedMass B branch (ndA5NoHitWithinEvent B) ≤ ndA5FlatTopWeight B branch := ndA5FlatTopRestrictedMass_le_weight hB hcount _ _ ≤ 4 / (B : ℝ) := ndA5FlatTopWeight_le_four_div hB branch hcount _ = 4 * (B : ℝ) ^ (-1 : ℝ) := by rw [Real.rpow_neg_one] ring _ ≤ 4 * R := mul_le_mul_of_nonneg_left hpow (by norm_num) calc ((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal = (∑ j ∈ Finset.range (ndA5BandCount B branch), ndA5FlatBandWeight B branch j * ((ndA5FlatBandPMF B branch j hB hcount).toOuterMeasure (ndA5NoHitWithinEvent B)).toReal) + ndA5FlatTopRestrictedMass B branch (ndA5NoHitWithinEvent B) := ndA5FlatSourcePMF_eventMass_eq_band_mixture_add_top hB hcount (ndA5NoHitWithinEvent B) _ ≤ (∑ j ∈ Finset.range (ndA5BandCount B branch), ndA5FlatBandWeight B branch j * (40 * R)) + 4 * R := add_le_add hbands htop _ = (∑ j ∈ Finset.range (ndA5BandCount B branch), ndA5FlatBandWeight B branch j) * (40 * R) + 4 * R := by rw [Finset.sum_mul] _ ≤ 1 * (40 * R) + 4 * R := by exact add_le_add_left (mul_le_mul_of_nonneg_right hsumWeight hfortyR) (4 * R) _ = 44 * R := by ring _ = 44 * (B : ℝ) ^ (-(1 / 32000 : ℝ)) := by rfl theorem eventually_uniformScheduledNoHitProbability_natCast_sourceY_le : ∀ᶠ B : ℕ in atTop, ∀ (branch : Tao.TaoSection5SourceBranch) (hB : 1 ≤ B) (hcount : 0 < ndA5BandCount B branch), uniformScheduledNoHitProbability (B : ℝ) (Tao.taoSection5SourceY B branch) (ndA5OddBlock_sourceY_nonempty hB branch hcount) ≤ 44 * (B : ℝ) ^ (-(1 / 32000 : ℝ)) := by filter_upwards [eventually_ndA5FlatSourceNoHitWithinMass_le_forty_four] with B hnoHit intro branch hB hcount rw [uniformScheduledNoHitProbability_natCast_sourceY_eq_flatSourceMass hB branch hcount] exact hnoHit branch hB hcount /-- Uniform floor-to-real source transport applies directly because both carriers use the same scheduled event at `floor x`. -/ theorem eventually_uniformScheduledNoHitProbability_floor_transport_le : ∀ᶠ x : ℝ in atTop, ∀ (branch : Tao.TaoSection5SourceBranch) (hFloor : (oddBlock (Tao.taoSection5SourceY (Nat.floor x) branch)).Nonempty) (hReal : (oddBlock (transportSourceY x branch)).Nonempty), uniformScheduledNoHitProbability x (transportSourceY x branch) hReal ≤ uniformScheduledNoHitProbability x (Tao.taoSection5SourceY (Nat.floor x) branch) hFloor + 96 * x ^ (-(99 / 100 : ℝ)) := by filter_upwards [eventually_abs_pmfProb_uniformOddBlock_floor_transport_sub_le] with x hsource intro branch hFloor hReal have hdisc := hsource branch hFloor hReal {N : ℕ | ¬ Tao.syracuseHitsAtMostWithin N (Nat.floor x) (Tao.taoSection5N0 (Nat.floor x))} unfold uniformScheduledNoHitProbability have hdisc' : |Tao.pmfProb (uniformOddBlockPMF (Tao.taoSection5SourceY (Nat.floor x) branch) hFloor) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 (Nat.floor x) (Tao.taoSection5N0 (Nat.floor x))} - Tao.pmfProb (uniformOddBlockPMF (transportSourceY x branch) hReal) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 (Nat.floor x) (Tao.taoSection5N0 (Nat.floor x))}| ≤ 96 * x ^ (-(99 / 100 : ℝ)) := by simpa only [Set.mem_setOf_eq] using hdisc linarith [neg_le_abs (Tao.pmfProb (uniformOddBlockPMF (Tao.taoSection5SourceY (Nat.floor x) branch) hFloor) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 (Nat.floor x) (Tao.taoSection5N0 (Nat.floor x))} - Tao.pmfProb (uniformOddBlockPMF (transportSourceY x branch) hReal) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 (Nat.floor x) (Tao.taoSection5N0 (Nat.floor x))})] private theorem logTime_half_le_floor_cast {x : ℝ} (hx : 2 ≤ x) : x / 2 ≤ ((Nat.floor x : ℕ) : ℝ) := by have hfloor := Nat.lt_floor_add_one x nlinarith private theorem logTime_floor_power_le_two_mul {x : ℝ} (hx : 2 ≤ x) : ((Nat.floor x : ℕ) : ℝ) ^ (-(1 / 32000 : ℝ)) ≤ 2 * x ^ (-(1 / 32000 : ℝ)) := by have hxpos : 0 < x := by linarith have hhalfPos : 0 < x / 2 := div_pos hxpos (by norm_num) have hbase := Real.rpow_le_rpow_of_nonpos hhalfPos (logTime_half_le_floor_cast hx) (show -(1 / 32000 : ℝ) ≤ 0 by norm_num) have htwo : (2 : ℝ) ^ (1 / 32000 : ℝ) ≤ 2 := by calc (2 : ℝ) ^ (1 / 32000 : ℝ) ≤ (2 : ℝ) ^ (1 : ℝ) := Real.rpow_le_rpow_of_exponent_le (by norm_num) (by norm_num) _ = 2 := by norm_num calc ((Nat.floor x : ℕ) : ℝ) ^ (-(1 / 32000 : ℝ)) ≤ (x / 2) ^ (-(1 / 32000 : ℝ)) := hbase _ = (2 : ℝ) ^ (1 / 32000 : ℝ) * x ^ (-(1 / 32000 : ℝ)) := by rw [Real.div_rpow hxpos.le (by norm_num : (0 : ℝ) ≤ 2), Real.rpow_neg (by norm_num : (0 : ℝ) ≤ 2)] simp only [div_eq_mul_inv, inv_inv] ring _ ≤ 2 * x ^ (-(1 / 32000 : ℝ)) := mul_le_mul_of_nonneg_right htwo (Real.rpow_nonneg hxpos.le _) /-- On the actual real source block, scheduled top-passage failure has a uniform power rate. -/ theorem eventually_uniformScheduledNoHitProbability_transportSourceY_le : ∀ᶠ x : ℝ in atTop, ∀ (branch : Tao.TaoSection5SourceBranch) (hReal : (oddBlock (transportSourceY x branch)).Nonempty), uniformScheduledNoHitProbability x (transportSourceY x branch) hReal ≤ 184 * x ^ (-(1 / 32000 : ℝ)) := by have hFloorTendsto : Tendsto (fun x : ℝ => Nat.floor x) atTop atTop := tendsto_nat_floor_atTop have hFloorCastTendsto : Tendsto (fun x : ℝ => ((Nat.floor x : ℕ) : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop.comp hFloorTendsto have hLogFloorTendsto : Tendsto (fun x : ℝ => Real.log ((Nat.floor x : ℕ) : ℝ)) atTop atTop := Real.tendsto_log_atTop.comp hFloorCastTendsto filter_upwards [eventually_uniformScheduledNoHitProbability_floor_transport_le, hFloorTendsto.eventually eventually_uniformScheduledNoHitProbability_natCast_sourceY_le, hLogFloorTendsto.eventually_ge_atTop (300000 : ℝ), eventually_ge_atTop (2 : ℝ)] with x htransport hnatural hlogFloor hx intro branch hReal let B := Nat.floor x have hB : 1 ≤ B := Tao.one_le_floor_of_one_le (by linarith) have hcountThree : 3 ≤ ndA5BandCount B branch := ndA5BandCount_ge_three_of_guards hB (by simpa only [B] using hlogFloor) have hcount : 0 < ndA5BandCount B branch := by omega let hFloor := ndA5OddBlock_sourceY_nonempty hB branch hcount have htransport' := htransport branch hFloor hReal have hnatural' := hnatural branch hB hcount have hfloorPow := logTime_floor_power_le_two_mul hx have hxOne : 1 ≤ x := by linarith have htransportPow : x ^ (-(99 / 100 : ℝ)) ≤ x ^ (-(1 / 32000 : ℝ)) := Real.rpow_le_rpow_of_exponent_le hxOne (by norm_num) calc uniformScheduledNoHitProbability x (transportSourceY x branch) hReal ≤ uniformScheduledNoHitProbability x (Tao.taoSection5SourceY B branch) hFloor + 96 * x ^ (-(99 / 100 : ℝ)) := by simpa only [B, hFloor] using htransport' _ = uniformScheduledNoHitProbability (B : ℝ) (Tao.taoSection5SourceY B branch) hFloor + 96 * x ^ (-(99 / 100 : ℝ)) := by unfold uniformScheduledNoHitProbability simp only [B, Nat.floor_natCast] _ ≤ 44 * (B : ℝ) ^ (-(1 / 32000 : ℝ)) + 96 * x ^ (-(99 / 100 : ℝ)) := add_le_add hnatural' le_rfl _ ≤ 44 * (2 * x ^ (-(1 / 32000 : ℝ))) + 96 * x ^ (-(1 / 32000 : ℝ)) := add_le_add (mul_le_mul_of_nonneg_left hfloorPow (by norm_num)) (mul_le_mul_of_nonneg_left htransportPow (by norm_num)) _ = 184 * x ^ (-(1 / 32000 : ℝ)) := by ring end end ND end Erdos1135