import Erdos1135.Tao.Section5.NoHitRate import Erdos1135.Tao.Syracuse.TimeBounded /-! # Section 5 scheduled no-hit estimate The existing no-hit proof is driven by deterministic descent at the literal time `taoSection5N0 B`. This leaf retains that witness and proves the same polynomial error for failure to hit by the scheduled time. -/ namespace Erdos1135 namespace Tao noncomputable section open Filter /-- Probability that a source in one odd logarithmic window has not hit `B` within `T` Syracuse steps. -/ noncomputable def syracuseNoHitWithinWindowProb (B T lo hi : ℕ) (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) : ℝ := pmfProb (oddLogWindowPMF lo hi hmass) {N : {n : ℕ // n ∈ oddLogWindow lo hi} | ¬ syracuseHitsAtMostWithin (N : ℕ) B T} theorem syracuseNoHitWithinWindowProb_eq_logFinsetProb (B T lo hi : ℕ) (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) : syracuseNoHitWithinWindowProb B T lo hi hmass = logFinsetProb (oddLogWindow lo hi) {N : ℕ | ¬ syracuseHitsAtMostWithin N B T} := by unfold syracuseNoHitWithinWindowProb simpa using pmfProb_oddLogWindowPMF lo hi hmass {N : ℕ | ¬ syracuseHitsAtMostWithin N B T} theorem syracuseNoHitWithinWindowProb_le_one (B T lo hi : ℕ) (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) : syracuseNoHitWithinWindowProb B T lo hi hmass ≤ 1 := by unfold syracuseNoHitWithinWindowProb calc pmfProb (oddLogWindowPMF lo hi hmass) {N : {N : ℕ // N ∈ oddLogWindow lo hi} | ¬ syracuseHitsAtMostWithin (N : ℕ) B T} ≤ pmfProb (oddLogWindowPMF lo hi hmass) Set.univ := pmfProb_mono _ (Set.subset_univ _) _ = 1 := pmfProb_univ _ /-- A pointwise scheduled-failure-to-valuation-event implication transports the finite source probability into the actual valuation-law event mass. -/ theorem syracuseNoHitWithinWindowProb_le_actualValuationLaw_event (B T lo hi n : ℕ) (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) (E : Set (List ℕ+)) (hpoint : ∀ N : {N : ℕ // N ∈ oddLogWindow lo hi}, ¬ syracuseHitsAtMostWithin (N : ℕ) B T → syracuseValuationPNatList n (oddLogWindowValueToOddNat N).1 (oddLogWindowValueToOddNat N).2 ∈ E) : syracuseNoHitWithinWindowProb B T lo hi hmass ≤ ((taoProp19ActualValuationLaw (oddLogWindowOddNatPMF lo hi hmass) n).toOuterMeasure E).toReal := by unfold syracuseNoHitWithinWindowProb calc pmfProb (oddLogWindowPMF lo hi hmass) {N : {N : ℕ // N ∈ oddLogWindow lo hi} | ¬ syracuseHitsAtMostWithin (N : ℕ) B T} ≤ pmfProb (oddLogWindowPMF lo hi hmass) {N : {N : ℕ // N ∈ oddLogWindow lo hi} | syracuseValuationPNatList n (oddLogWindowValueToOddNat N).1 (oddLogWindowValueToOddNat N).2 ∈ E} := by exact pmfProb_mono (oddLogWindowPMF lo hi hmass) hpoint _ = ((taoProp19ActualValuationLaw (oddLogWindowOddNatPMF lo hi hmass) n).toOuterMeasure E).toReal := by rw [pmfProb_eq_toOuterMeasure_toReal] unfold taoProp19ActualValuationLaw oddLogWindowOddNatPMF rw [PMF.toOuterMeasure_map_apply, PMF.toOuterMeasure_map_apply] apply congrArg ENNReal.toReal apply congrArg (oddLogWindowPMF lo hi hmass).toOuterMeasure ext N rfl /-- Failure to hit by the literal Section 5 time forces the same strict low-valuation event used by the eventual no-hit proof. -/ theorem TaoSection5DescentScaleFacts.actualValuations_mem_low_of_not_hitsWithin {B : ℕ} (facts : TaoSection5DescentScaleFacts B) {N : {N : ℕ // Odd N}} {branch : TaoSection5SourceBranch} (hmem : N.1 ∈ oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch)) (hno : ¬ syracuseHitsAtMostWithin N.1 B (taoSection5N0 B)) : syracuseValuationPNatList (taoSection5N0 B) N.1 N.2 ∈ taoLowValuationWeightEvent (taoSection5N0 B) := by change 10 * taoTupleWeight (syracuseValuationPNatList (taoSection5N0 B) N.1 N.2) < 19 * taoSection5N0 B by_contra hlow have hweight : 19 * taoSection5N0 B ≤ 10 * taoTupleWeight (syracuseValuationPNatList (taoSection5N0 B) N.1 N.2) := by omega apply hno exact ⟨taoSection5N0 B, le_rfl, facts.iterate_le_threshold hmem hweight⟩ /-- The native Section 5 comparison gives the original error bound for failure to hit by the scheduled time. -/ theorem taoSection5_noHitWithinScheduleWindowProb_le_error_of_inputs {B : ℕ} (descent : TaoSection5DescentScaleFacts B) (prop19 : TaoSection5LogWindowProp19Output B) (branch : TaoSection5SourceBranch) (hmass : 0 < logFinsetMass (oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch))) : syracuseNoHitWithinWindowProb B (taoSection5N0 B) (taoSection5SourceLo B branch) (taoSection5SourceHi B branch) hmass ≤ taoSection5NoHitError B := by have hmassEq : hmass = prop19.schedule.mass_pos branch := Subsingleton.elim _ _ subst hmass let lo := taoSection5SourceLo B branch let hi := taoSection5SourceHi B branch let n := taoSection5N0 B let source := oddLogWindowOddNatPMF lo hi (prop19.schedule.mass_pos branch) let E := taoLowValuationWeightEvent n have hmap : syracuseNoHitWithinWindowProb B n lo hi (prop19.schedule.mass_pos branch) ≤ ((taoProp19ActualValuationLaw source n).toOuterMeasure E).toReal := by apply syracuseNoHitWithinWindowProb_le_actualValuationLaw_event intro N hno apply descent.actualValuations_mem_low_of_not_hitsWithin (N := oddLogWindowValueToOddNat N) · exact N.property · simpa only [n, lo, hi] using hno have hvaluation := prop19.valuationBound branch unfold taoProp19ValuationTV at hvaluation have hcompare : ((taoProp19ActualValuationLaw source n).toOuterMeasure E).toReal ≤ ((geom2PNatListPMF n).toOuterMeasure E).toReal + 4 * (2 : ℝ) ^ (-((1 / 128 : ℝ) * (n : ℝ))) := by apply pmfOuterMass_le_add_of_taoPMFFullL1_le simpa only [source, n, lo, hi] using hvaluation have hideal := geom2PNatListPMF_lowWeightEvent_le_exp n calc syracuseNoHitWithinWindowProb B n lo hi (prop19.schedule.mass_pos branch) ≤ ((taoProp19ActualValuationLaw source n).toOuterMeasure E).toReal := hmap _ ≤ ((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 _ = taoSection5NoHitError B := by rfl /-- Scheduled failure has the same explicit polynomial rate as the checked eventual no-hit output. -/ theorem TaoSection5NoHitRateOutput.noHitWithinSchedule_le {B : ℕ} (rate : TaoSection5NoHitRateOutput B) (branch : TaoSection5SourceBranch) (hmass : 0 < logFinsetMass (oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch))) : syracuseNoHitWithinWindowProb B (taoSection5N0 B) (taoSection5SourceLo B branch) (taoSection5SourceHi B branch) hmass ≤ 10 * (B : ℝ) ^ (-(1 / 32000 : ℝ)) := by exact (taoSection5_noHitWithinScheduleWindowProb_le_error_of_inputs rate.output.descent rate.output.prop19 branch hmass).trans rate.error_le /-- Both canonical source branches satisfy the scheduled polynomial no-hit rate for all sufficiently large thresholds. -/ theorem eventually_taoSection5_noHitWithinSchedule_le : ∀ᶠ B : ℕ in atTop, ∀ branch : TaoSection5SourceBranch, ∀ hmass : 0 < logFinsetMass (oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch)), syracuseNoHitWithinWindowProb B (taoSection5N0 B) (taoSection5SourceLo B branch) (taoSection5SourceHi B branch) hmass ≤ 10 * (B : ℝ) ^ (-(1 / 32000 : ℝ)) := by filter_upwards [eventually_taoSection5NoHitRateOutput] with B rate intro branch hmass exact rate.noHitWithinSchedule_le branch hmass end end Tao end Erdos1135