import Erdos1135.ND.Discrepancy.A5ReferenceRealFloorPass /-! # A5 real harmonic cross and escape rows This leaf keeps the two LOG-BLOCK inputs separate. The cross-branch passage row transports the checked natural harmonic comparison through the two real source windows. The escape row transports the genuine harmonic no-hit event directly and never passes through the artificial no-hit totalizer. -/ namespace Erdos1135 namespace ND open Filter open scoped Topology noncomputable section /-- The checked native Section 5 harmonic no-hit packet is definitionally the ND logarithmic odd-block law and has the explicit coefficient ten. -/ theorem pmfProb_logOddBlockPMF_natCast_sourceY_noHit_le_ten {B : Nat} (rate : Tao.TaoSection5NoHitRateOutput B) (branch : Tao.TaoSection5SourceBranch) (hmass : 0 < Tao.logFinsetMass (oddBlock (Tao.taoSection5SourceY B branch))) : Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hmass) {N | ¬ Tao.syracuseHitsAtMost N.1 B} <= 10 * (B : Real) ^ (-(1 / 32000 : Real)) := by have hmassTao : 0 < Tao.logFinsetMass (Tao.oddLogWindow (Tao.taoSection5SourceLo B branch) (Tao.taoSection5SourceHi B branch)) := by simpa [oddBlock, Tao.taoNyOddWindow, Tao.taoSection5SourceLo, Tao.taoSection5SourceHi, alpha] using hmass calc Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hmass) {N | ¬ Tao.syracuseHitsAtMost N.1 B} = Tao.syracuseNoHitWindowProb B (Tao.taoSection5SourceLo B branch) (Tao.taoSection5SourceHi B branch) hmassTao := by unfold logOddBlockPMF Tao.syracuseNoHitWindowProb Tao.oddLogWindowPMF Tao.syracuseNoHitAtMost oddBlock Tao.taoNyOddWindow Tao.taoSection5SourceLo Tao.taoSection5SourceHi rfl _ <= Tao.taoSection5NoHitError B := rate.output.noHit_le branch hmassTao _ <= 10 * (B : Real) ^ (-(1 / 32000 : Real)) := rate.error_le /-- The floor harmonic source has positive mass under the common real/floor normalizer packet. -/ theorem logFinsetMass_oddBlock_floorSourceY_pos {x : Real} (facts : Tao.TaoProp111RealFloorWindowMassFacts x) (branch : Tao.TaoSection5SourceBranch) : 0 < Tao.logFinsetMass (oddBlock (Tao.taoSection5SourceY (Nat.floor x) branch)) := by have hlower : 0 < Real.log ((Nat.floor x : Nat) : Real) / 8000 := div_pos facts.log_floor_pos (by norm_num) cases branch with | alpha => exact hlower.trans_le (by simpa [oddBlock, Tao.taoSection5SourceY, alpha] using facts.alpha_floor) | alphaSq => exact hlower.trans_le (by simpa [oddBlock, Tao.taoSection5SourceY, alpha] using facts.alphaSq_floor) /-- Genuine harmonic escape on either real source branch. The natural Section 5 power rate pays one scalar harmonic floor-source error and no Hahn factor. -/ theorem eventually_pmfProb_logOddBlock_transportSourceY_noHit_le : ∀ᶠ x : Real in atTop, ∀ (hx : 1 <= x) (branch : Tao.TaoSection5SourceBranch) (hReal : 0 < Tao.logFinsetMass (oddBlock (transportSourceY x branch))), Tao.pmfProb (logOddBlockPMF (transportSourceY x branch) hReal) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x} <= 10 * ((Nat.floor x : Nat) : Real) ^ (-(1 / 32000 : Real)) + 96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real)) := by have hFloorTendsto : Tendsto (fun x : Real => Nat.floor x) atTop atTop := tendsto_nat_floor_atTop filter_upwards [Tao.eventually_taoProp111RealFloorWindowMassFacts, hFloorTendsto.eventually Tao.eventually_taoSection5NoHitRateOutput] with x facts rate intro hx branch hReal let B := Nat.floor x let hFloor := logFinsetMass_oddBlock_floorSourceY_pos facts branch have hNatural : Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMost N.1 B} <= 10 * (B : Real) ^ (-(1 / 32000 : Real)) := by simpa only [B, hFloor] using pmfProb_logOddBlockPMF_natCast_sourceY_noHit_le_ten rate branch hFloor have hFloorEvent : Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x} = Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMost N.1 B} := by congr 1 ext N simp only [Set.mem_setOf_eq] rw [Tao.syracuseHitsAtMostReal_iff_floor (zero_le_one.trans hx)] have hdisc : |Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x} - Tao.pmfProb (logOddBlockPMF (transportSourceY x branch) hReal) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x}| <= 96000 / ((B : Real) * Real.log (B : Real)) := by unfold logOddBlockPMF change |Tao.pmfProb (Tao.logFinsetPMF (oddBlock (Tao.taoSection5SourceY B branch)) hFloor) {N | N.1 ∈ {n : Nat | ¬ Tao.syracuseHitsAtMostReal n x}} - Tao.pmfProb (Tao.logFinsetPMF (oddBlock (transportSourceY x branch)) hReal) {N | N.1 ∈ {n : Nat | ¬ Tao.syracuseHitsAtMostReal n x}}| <= _ rw [Tao.pmfProb_logFinsetPMF, Tao.pmfProb_logFinsetPMF] simpa only [B] using abs_logFinsetProb_oddBlock_floor_transport_sub_le facts branch {N : Nat | ¬ Tao.syracuseHitsAtMostReal N x} calc Tao.pmfProb (logOddBlockPMF (transportSourceY x branch) hReal) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x} <= Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x} + |Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x} - Tao.pmfProb (logOddBlockPMF (transportSourceY x branch) hReal) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x}| := by linarith [neg_le_abs (Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x} - Tao.pmfProb (logOddBlockPMF (transportSourceY x branch) hReal) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x})] _ <= Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMostReal N.1 x} + 96000 / ((B : Real) * Real.log (B : Real)) := add_le_add le_rfl hdisc _ = Tao.pmfProb (logOddBlockPMF (Tao.taoSection5SourceY B branch) hFloor) {N | ¬ Tao.syracuseHitsAtMost N.1 B} + 96000 / ((B : Real) * Real.log (B : Real)) := by rw [hFloorEvent] _ <= 10 * (B : Real) ^ (-(1 / 32000 : Real)) + 96000 / ((B : Real) * Real.log (B : Real)) := add_le_add hNatural le_rfl /-- Any two real harmonic source branches have passage laws within the raw cross ledger `32*R_B + 4*deltaH`. No uniform source error occurs. -/ theorem eventually_ndA5RealHarmonicBranchesPassTV_le {c kappa d : Real} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∀ᶠ x : Real in atTop, ∀ (hx : 1 <= x) (branch1 branch2 : Tao.TaoSection5SourceBranch) (hRealMass1 : 0 < Tao.logFinsetMass (oddBlock (transportSourceY x branch1))) (hRealMass2 : 0 < Tao.logFinsetMass (oddBlock (transportSourceY x branch2))), let B := Nat.floor x Tao.taoTV (logPassLaw x (transportSourceY x branch1) hx hRealMass1) (logPassLaw x (transportSourceY x branch2) hx hRealMass2) <= 32 * ndA5ReferenceInteriorScalarRate B d + 4 * (96000 / ((B : Real) * Real.log (B : Real))) := by have hFloorTendsto : Tendsto (fun x : Real => Nat.floor x) atTop atTop := tendsto_nat_floor_atTop have hFloorCastTendsto : Tendsto (fun x : Real => ((Nat.floor x : Nat) : Real)) atTop atTop := tendsto_natCast_atTop_atTop.comp hFloorTendsto have hLogFloorTendsto : Tendsto (fun x : Real => Real.log ((Nat.floor x : Nat) : Real)) atTop atTop := Real.tendsto_log_atTop.comp hFloorCastTendsto filter_upwards [Tao.eventually_taoProp111RealFloorWindowMassFacts, hFloorTendsto.eventually (eventually_ndA5HarmonicBranchesTotalizedPassTV_le_thirty_two hPhase hd hd20 hdk), hLogFloorTendsto.eventually_ge_atTop (300000 : Real)] with x facts hNaturalTV hlog intro hx branch1 branch2 hRealMass1 hRealMass2 dsimp only let B := Nat.floor x have hB : 1 <= B := Tao.one_le_floor_of_one_le hx have hcount1 : 0 < ndA5BandCount B branch1 := by have : 3 <= ndA5BandCount B branch1 := ndA5BandCount_ge_three_of_guards hB (by simpa only [B] using hlog) omega have hcount2 : 0 < ndA5BandCount B branch2 := by have : 3 <= ndA5BandCount B branch2 := ndA5BandCount_ge_three_of_guards hB (by simpa only [B] using hlog) omega let hFloorMass1 := logFinsetMass_ndA5OddBlock_sourceY_pos hB branch1 hcount1 let hFloorMass2 := logFinsetMass_ndA5OddBlock_sourceY_pos hB branch2 hcount2 let hReal1 := logPassLaw x (transportSourceY x branch1) hx hRealMass1 let hFloor1 := logPassLaw x (Tao.taoSection5SourceY B branch1) hx hFloorMass1 let hFloor2 := logPassLaw x (Tao.taoSection5SourceY B branch2) hx hFloorMass2 let hReal2 := logPassLaw x (transportSourceY x branch2) hx hRealMass2 have hhFloorReal1 : Tao.taoTV hFloor1 hReal1 <= 2 * (96000 / ((B : Real) * Real.log (B : Real))) := by simpa only [hFloor1, hReal1, B, hFloorMass1, hRealMass1] using taoTV_logPassLaw_floor_transport_le facts branch1 hx hFloorMass1 hRealMass1 have hhRealFloor1 : Tao.taoTV hReal1 hFloor1 <= 2 * (96000 / ((B : Real) * Real.log (B : Real))) := by rw [Tao.taoTV_comm] exact hhFloorReal1 have hFloorMiddle : Tao.taoTV hFloor1 hFloor2 <= 32 * ndA5ReferenceInteriorScalarRate B d := by have hhFloor1 : hFloor1 = ndTotalizedPassLaw (Nat.floor x) (Tao.one_le_floor_of_one_le hx) (ndA5HarmonicSourcePMF (Nat.floor x) branch1 (Tao.one_le_floor_of_one_le hx) hcount1) := by simpa only [hFloor1, hFloorMass1, B] using logPassLaw_floorSource_eq_ndTotalizedHarmonic hx branch1 hcount1 have hhFloor2 : hFloor2 = ndTotalizedPassLaw (Nat.floor x) (Tao.one_le_floor_of_one_le hx) (ndA5HarmonicSourcePMF (Nat.floor x) branch2 (Tao.one_le_floor_of_one_le hx) hcount2) := by simpa only [hFloor2, hFloorMass2, B] using logPassLaw_floorSource_eq_ndTotalizedHarmonic hx branch2 hcount2 rw [hhFloor1, hhFloor2] simpa only [B] using hNaturalTV branch1 branch2 (Tao.one_le_floor_of_one_le hx) hcount1 hcount2 have hhFloorReal2 : Tao.taoTV hFloor2 hReal2 <= 2 * (96000 / ((B : Real) * Real.log (B : Real))) := by simpa only [hFloor2, hReal2, B, hFloorMass2, hRealMass2] using taoTV_logPassLaw_floor_transport_le facts branch2 hx hFloorMass2 hRealMass2 change Tao.taoTV hReal1 hReal2 <= _ calc Tao.taoTV hReal1 hReal2 <= Tao.taoTV hReal1 hFloor1 + Tao.taoTV hFloor1 hReal2 := Tao.taoTV_triangle hReal1 hFloor1 hReal2 _ <= Tao.taoTV hReal1 hFloor1 + (Tao.taoTV hFloor1 hFloor2 + Tao.taoTV hFloor2 hReal2) := add_le_add le_rfl (Tao.taoTV_triangle hFloor1 hFloor2 hReal2) _ <= 2 * (96000 / ((B : Real) * Real.log (B : Real))) + (32 * ndA5ReferenceInteriorScalarRate B d + 2 * (96000 / ((B : Real) * Real.log (B : Real)))) := add_le_add hhRealFloor1 (add_le_add hFloorMiddle hhFloorReal2) _ = 32 * ndA5ReferenceInteriorScalarRate B d + 4 * (96000 / ((B : Real) * Real.log (B : Real))) := by ring /-- Ordered LOG-BLOCK full-L1 row, from the `x^alpha` harmonic source to the `x^(alpha^2)` harmonic source. -/ theorem eventually_ndA5RealCrossHarmonicPassTV_le {c kappa d : Real} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∀ᶠ x : Real in atTop, ∀ (hx : 1 <= x), ∀ (hAlpha : 0 < Tao.logFinsetMass (oddBlock (transportSourceY x .alpha))) (hAlphaSq : 0 < Tao.logFinsetMass (oddBlock (transportSourceY x .alphaSq))), let B := Nat.floor x Tao.taoTV (logPassLaw x (transportSourceY x .alpha) hx hAlpha) (logPassLaw x (transportSourceY x .alphaSq) hx hAlphaSq) <= 32 * ndA5ReferenceInteriorScalarRate B d + 4 * (96000 / ((B : Real) * Real.log (B : Real))) := by filter_upwards [eventually_ndA5RealHarmonicBranchesPassTV_le hPhase hd hd20 hdk] with x hcross intro hx hAlpha hAlphaSq simpa using hcross hx .alpha .alphaSq hAlpha hAlphaSq /-- The ordered LOG-BLOCK row controls every output event with the same raw ledger. Project full L1 is used directly, with no additional factor. -/ theorem eventually_abs_pmfProb_ndA5RealCrossHarmonicPass_sub_le {c kappa d : Real} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∀ᶠ x : Real in atTop, ∀ (hx : 1 <= x) (E : Set {M : Nat // M <= Nat.floor x}), ∀ (hAlpha : 0 < Tao.logFinsetMass (oddBlock (transportSourceY x .alpha))) (hAlphaSq : 0 < Tao.logFinsetMass (oddBlock (transportSourceY x .alphaSq))), let B := Nat.floor x |Tao.pmfProb (logPassLaw x (transportSourceY x .alpha) hx hAlpha) E - Tao.pmfProb (logPassLaw x (transportSourceY x .alphaSq) hx hAlphaSq) E| <= 32 * ndA5ReferenceInteriorScalarRate B d + 4 * (96000 / ((B : Real) * Real.log (B : Real))) := by filter_upwards [eventually_ndA5RealCrossHarmonicPassTV_le hPhase hd hd20 hdk] with x hcross intro hx E hAlpha hAlphaSq have hTV := hcross hx hAlpha hAlphaSq exact (Tao.abs_pmfProb_sub_le_taoTV _ _ E).trans hTV end end ND end Erdos1135