import Erdos1135.ND.LogTime.Count import Erdos1135.ND.LogTime.A5Section3Time import Erdos1135.ND.RhinUnconditional import Erdos1135.ND.Discrepancy.A5ReferenceND31Main /-! # Main logarithmic-time ND31 regime This assembly keeps the caller-selected Rhin/PhaseGap passage exponent all the way through the time-carrying Section 3 packet, and combines the three bounded-time top-block charges without weakening that exponent. -/ namespace Erdos1135 namespace ND open Filter open scoped Topology noncomputable section theorem syracuseNoHitWithinWindowProb_taoNy_eq_pmfProb_logOddBlockPMF (B T : ℕ) (y : ℝ) (hmassTao : 0 < Tao.logFinsetMass (Tao.taoNyOddWindow y Tao.taoAlpha)) (hmassND : 0 < Tao.logFinsetMass (oddBlock y)) : Tao.syracuseNoHitWithinWindowProb B T (Tao.taoNyLo y) (Tao.taoNyHi y Tao.taoAlpha) hmassTao = Tao.pmfProb (logOddBlockPMF y hmassND) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 B T} := by unfold Tao.syracuseNoHitWithinWindowProb Tao.oddLogWindowPMF logOddBlockPMF oddBlock Tao.taoNyOddWindow rfl /-- The powered main regime preserves every caller-selected exponent strictly below the Rhin cap. Scheduled no-hit is absorbed from its genuine power rate; the A5 passage term and the final target rate retain the literal `d`. -/ theorem ndRhinND31LogTimeMainRegime_sameD (d : ℝ) (hd : 0 < d) (hdCap : d < 5 / 143) : NDM2Bounds (1 / 32000 : ℝ) d ∧ ∃ Nstar : ℕ, 2 ≤ Nstar ∧ ∃ Cmain : ℝ, 0 ≤ Cmain ∧ ∀ (N0 : ℕ), Nstar ≤ N0 → ∀ (X : ℝ), (N0 : ℝ) < X → oddSyracuseLogTimeBadRatio ndSyracuseLogTimeConstant N0 X ≤ Cmain * (Real.log (N0 : ℝ)) ^ (-d) := by rcases existsPhaseGapRhin with ⟨cPhase, hPhase⟩ have hd20 : d < 1 / 20 := hdCap.trans (by norm_num) have hdk : d < 1 / (2 * (143 / 10 : ℝ)) := by calc d < 5 / 143 := hdCap _ = 1 / (2 * (143 / 10 : ℝ)) := by norm_num rcases exists_ndA5ReferenceSection3AmbientTimeTerminal hPhase hd hd20 hdk with ⟨Csec, Btime, hCsec, hBtimeTwo, htime⟩ rcases ndRhinRate_sameD d hd hdCap with ⟨hM2, _h31Old, _h16Old, _h13Old⟩ refine ⟨hM2, ?_⟩ rcases hM2 with ⟨_CHit, CTr, x0, _hCHit, hCTr, hx0, hM2bounds⟩ rcases eventually_atTop.1 (eventually_rpow_neg_one_div_32000_le_log_rpow_neg d) with ⟨Qabs, hQabs⟩ rcases eventually_atTop.1 eventually_uniformScheduledNoHitProbability_transportSourceY_le with ⟨Qsched, hQsched⟩ let r : ℝ := Tao.taoSection3AmbientRatio let e : ℝ := r ^ 2 let E : ℝ := e ^ (-d) let Cprob : ℝ := (184 + CTr) * E + Csec let Cmain : ℝ := 2 + 2 * Cprob have he : 0 < e := by dsimp only [e, r] exact pow_pos Tao.taoSection3AmbientRatio_pos 2 have hE : 0 ≤ E := Real.rpow_nonneg he.le _ have hCprob : 0 ≤ Cprob := by dsimp only [Cprob] exact add_nonneg (mul_nonneg (add_nonneg (by norm_num) hCTr) hE) hCsec have hCmain : 0 ≤ Cmain := by dsimp only [Cmain] positivity have hpow : Tendsto (fun N : ℕ => (N : ℝ) ^ e) atTop atTop := by simpa only [Function.comp_def] using (tendsto_rpow_atTop he).comp tendsto_natCast_atTop_atTop have hlarge : ∀ᶠ N : ℕ in atTop, 2 ≤ N ∧ Btime ≤ N ∧ Real.exp 1 ≤ (N : ℝ) ∧ max x0 (max Qabs Qsched) ≤ (N : ℝ) ^ e := by filter_upwards [eventually_ge_atTop (2 : ℕ), eventually_ge_atTop Btime, tendsto_natCast_atTop_atTop.eventually_ge_atTop (Real.exp 1), hpow.eventually_ge_atTop (max x0 (max Qabs Qsched))] with N hN htimeN hexpN hcutN exact ⟨hN, htimeN, hexpN, hcutN⟩ rcases eventually_atTop.1 hlarge with ⟨Nstar, hNstar⟩ refine ⟨Nstar, (hNstar Nstar le_rfl).1, Cmain, hCmain, ?_⟩ intro N0 hNstarN0 X hN0X rcases hNstar N0 hNstarN0 with ⟨hN0Two, hBtimeN0, hExpN0, hcutN0⟩ have hN0OneNat : 1 ≤ N0 := by omega have hN0One : (1 : ℝ) ≤ (N0 : ℝ) := by exact_mod_cast hN0OneNat have hXOne : 1 ≤ X := hN0One.trans hN0X.le have hXpos : 0 < X := zero_lt_one.trans_le hXOne have hN0nonneg : (0 : ℝ) ≤ (N0 : ℝ) := Nat.cast_nonneg N0 have hpowStrict : (N0 : ℝ) ^ e < X ^ e := Real.rpow_lt_rpow hN0nonneg hN0X he let q : ℝ := Tao.taoSection3AmbientBoundary X 2 let y : ℝ := transportSourceY q .alpha have hqEq : q = X ^ e := by rfl have hcutQ : max x0 (max Qabs Qsched) ≤ q := by rw [hqEq] exact hcutN0.trans hpowStrict.le have hqx0 : x0 ≤ q := (le_max_left _ _).trans hcutQ have hqQabs : Qabs ≤ q := (le_max_left Qabs Qsched).trans ((le_max_right x0 (max Qabs Qsched)).trans hcutQ) have hqQsched : Qsched ≤ q := (le_max_right Qabs Qsched).trans ((le_max_right x0 (max Qabs Qsched)).trans hcutQ) have hqTwo : 2 ≤ q := hx0.trans hqx0 have hq : 1 ≤ q := one_le_two.trans hqTwo have heOne : e ≤ 1 := by dsimp only [e, r] nlinarith [Tao.taoSection3AmbientRatio_pos, Tao.taoSection3AmbientRatio_lt_one] have hqX : q ≤ X := by rw [hqEq] calc X ^ e ≤ X ^ (1 : ℝ) := Real.rpow_le_rpow_of_exponent_le hXOne heOne _ = X := by simp have hQabsX : Qabs ≤ X := hqQabs.trans hqX have hqy : q ≤ y := by dsimp only [y, transportSourceY] exact Real.self_le_rpow_of_one_le hq Tao.taoAlpha_one_lt.le have hyTwo : 2 ≤ y := hqTwo.trans hqy have hy : 1 ≤ y := one_le_two.trans hyTwo have hX0 : 0 ≤ X := zero_le_one.trans hXOne have hyBoundary : y = Tao.taoSection3AmbientBoundary X 1 := by dsimp only [y, q, transportSourceY, alpha] simpa only [Nat.reduceAdd] using Tao.taoSection3AmbientBoundary_succ_rpow hX0 1 have hupper : y ^ alpha = X := by calc y ^ alpha = (Tao.taoSection3AmbientBoundary X 1) ^ Tao.taoAlpha := by rw [hyBoundary] _ = Tao.taoSection3AmbientBoundary X 0 := by simpa only [Nat.reduceAdd] using Tao.taoSection3AmbientBoundary_succ_rpow hX0 0 _ = X := by simp [Tao.taoSection3AmbientBoundary] have hM2q := hM2bounds q hqx0 hq .alpha change ∃ hwindow : (oddBlock y).Nonempty, ∃ hmass : 0 < Tao.logFinsetMass (oddBlock y), uniformNoHitProbability q y hwindow ≤ _CHit * q ^ (-(1 / 32000 : ℝ)) ∧ passFullL1 q y hq hwindow hmass ≤ CTr * (Real.log q) ^ (-d) at hM2q rcases hM2q with ⟨hwindow, hmass, _hOldHit, hPass⟩ rcases htime hBtimeN0 hN0X with ⟨T, hT, hmassTime, hTimeSource⟩ have hSource : Tao.pmfProb (logOddBlockPMF y hmass) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 N0 T} ≤ Csec * (Real.log (N0 : ℝ)) ^ (-d) := by have hmassTimeY : 0 < Tao.logFinsetMass (Tao.taoNyOddWindow y Tao.taoAlpha) := by simpa only [hyBoundary] using hmassTime have hTimeSourceY : Tao.syracuseNoHitWithinWindowProb N0 T (Tao.taoNyLo y) (Tao.taoNyHi y Tao.taoAlpha) hmassTimeY ≤ Csec * (Real.log (N0 : ℝ)) ^ (-d) := by simpa only [hyBoundary] using hTimeSource have heq := syracuseNoHitWithinWindowProb_taoNy_eq_pmfProb_logOddBlockPMF N0 T y hmassTimeY hmass rw [heq] at hTimeSourceY exact hTimeSourceY have hScheduled := hQsched q hqQsched .alpha hwindow have hclock : ∀ N : ℕ, Nat.ceil y ≤ N → ((Tao.taoSection5N0 (Nat.floor q) + T : ℕ) : ℝ) ≤ ndSyracuseLogTimeConstant * Real.log (N : ℝ) := by intro N hNy have hNy' : Nat.ceil (Tao.taoSection3AmbientBoundary X 1) ≤ N := by simpa only [hyBoundary] using hNy simpa only [q] using (ndTopPassageBudget_cast_le_pointwiseLog hXOne hNy' hT) have hCount := oddSyracuseLogTimeBadRatio_le_ceil_div_add_two_mul_timeTerms hN0OneNat hq hy hXOne hupper hwindow hmass hclock let R : ℝ := (Real.log (N0 : ℝ)) ^ (-d) have hN0pos : 0 < (N0 : ℝ) := by positivity have hN0OneStrict : (1 : ℝ) < (N0 : ℝ) := by exact_mod_cast (show 1 < N0 by omega) have hlogN0Pos : 0 < Real.log (N0 : ℝ) := Real.log_pos hN0OneStrict have hlogN0One : 1 ≤ Real.log (N0 : ℝ) := (Real.le_log_iff_exp_le hN0pos).2 hExpN0 have hlogNX : Real.log (N0 : ℝ) < Real.log X := Real.log_lt_log hN0pos hN0X have hlogXPos : 0 < Real.log X := hlogN0Pos.trans hlogNX have hXRate : (Real.log X) ^ (-d) ≤ R := by dsimp only [R] exact Real.rpow_le_rpow_of_nonpos hlogN0Pos hlogNX.le (neg_nonpos.mpr hd.le) have hBottomPower : (Nat.ceil y : ℝ) / X ≤ 2 * X ^ (-(1 / 32000 : ℝ)) := by rw [hyBoundary] exact ceil_taoSection3AmbientBoundary_one_div_le_real_power hXOne have hBottom : (Nat.ceil y : ℝ) / X ≤ 2 * R := by calc (Nat.ceil y : ℝ) / X ≤ 2 * X ^ (-(1 / 32000 : ℝ)) := hBottomPower _ ≤ 2 * (Real.log X) ^ (-d) := mul_le_mul_of_nonneg_left (hQabs X hQabsX) (by norm_num) _ ≤ 2 * R := mul_le_mul_of_nonneg_left hXRate (by norm_num) have hlogqPos : 0 < Real.log q := Real.log_pos (one_lt_two.trans_le hqTwo) have hlogqEq : Real.log q = e * Real.log X := by rw [hqEq] exact Real.log_rpow hXpos e have hlogqRate : (Real.log q) ^ (-d) = E * (Real.log X) ^ (-d) := by rw [hlogqEq] dsimp only [E] exact Real.mul_rpow he.le hlogXPos.le have hlogqRateLe : (Real.log q) ^ (-d) ≤ E * R := by rw [hlogqRate] exact mul_le_mul_of_nonneg_left hXRate hE have hScheduledRate : uniformScheduledNoHitProbability q y hwindow ≤ 184 * E * R := by calc uniformScheduledNoHitProbability q y hwindow ≤ 184 * q ^ (-(1 / 32000 : ℝ)) := hScheduled _ ≤ 184 * (Real.log q) ^ (-d) := mul_le_mul_of_nonneg_left (hQabs q hqQabs) (by norm_num) _ ≤ 184 * (E * R) := mul_le_mul_of_nonneg_left hlogqRateLe (by norm_num) _ = 184 * E * R := by ring have hPassRate : passFullL1 q y hq hwindow hmass ≤ CTr * E * R := by calc passFullL1 q y hq hwindow hmass ≤ CTr * (Real.log q) ^ (-d) := hPass _ ≤ CTr * (E * R) := mul_le_mul_of_nonneg_left hlogqRateLe hCTr _ = CTr * E * R := by ring have hSourceRate : Tao.pmfProb (logOddBlockPMF y hmass) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 N0 T} ≤ Csec * R := by simpa only [R] using hSource calc oddSyracuseLogTimeBadRatio ndSyracuseLogTimeConstant N0 X ≤ (Nat.ceil y : ℝ) / X + 2 * (uniformScheduledNoHitProbability q y hwindow + passFullL1 q y hq hwindow hmass + Tao.pmfProb (logOddBlockPMF y hmass) {N | ¬ Tao.syracuseHitsAtMostWithin N.1 N0 T}) := hCount _ ≤ 2 * R + 2 * (184 * E * R + CTr * E * R + Csec * R) := by gcongr _ = Cmain * R := by dsimp only [Cmain, Cprob] ring _ = Cmain * (Real.log (N0 : ℝ)) ^ (-d) := by rfl /-- Compatibility form of the original existential main-regime endpoint. The witness `1/40` is now selected only at this wrapper; the producer above retains every admissible caller-selected exponent. -/ theorem exists_ndRhinND31LogTimeMainRegime : ∃ d : ℝ, 0 < d ∧ NDM2Bounds (1 / 32000 : ℝ) d ∧ ∃ Nstar : ℕ, 2 ≤ Nstar ∧ ∃ Cmain : ℝ, 0 ≤ Cmain ∧ ∀ (N0 : ℕ), Nstar ≤ N0 → ∀ (X : ℝ), (N0 : ℝ) < X → oddSyracuseLogTimeBadRatio ndSyracuseLogTimeConstant N0 X ≤ Cmain * (Real.log (N0 : ℝ)) ^ (-d) := by rcases ndRhinND31LogTimeMainRegime_sameD (1 / 40 : ℝ) (by norm_num) (by norm_num) with ⟨hM2, hMain⟩ exact ⟨1 / 40, by norm_num, hM2, hMain⟩ end end ND end Erdos1135