import Erdos1135.Tao.Fourier.Prop114CanonicalAssembly import Erdos1135.Tao.Section5.BranchPassTV import Erdos1135.Tao.Section5.NoHitEndpoint /-! # Section 5 Natural Proposition 1.11 Rate This leaf specializes the checked fine-scale mixing theorem at exponent one, absorbs the canonical branch passage comparison into a logarithmic rate, and patches the finite threshold prefix with the universal full-L1 bound. It then combines that pass-TV rate with the checked canonical no-hit rate. -/ namespace Erdos1135 namespace Tao open Filter open scoped Topology noncomputable section private theorem natCast_log_nonneg_prop111Rate (B : ℕ) : 0 ≤ Real.log (B : ℝ) := by by_cases hB : B = 0 · simp [hB] · exact Real.log_nonneg (by exact_mod_cast (Nat.one_le_iff_ne_zero.mpr hB)) /-- The floor defining the Section 5 step-back time eventually retains at least half of its real main term. -/ theorem eventually_log_div_two_hundred_thousand_le_taoSection5M0 : ∀ᶠ B : ℕ in atTop, Real.log B / 200000 ≤ (taoSection5M0 B : 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 [hlog.eventually_ge_atTop (200000 : ℝ)] with B hlarge have hfloor : Real.log B / 100000 - 1 < (taoSection5M0 B : ℝ) := by have h := Nat.lt_floor_add_one (Real.log B / 100000) rw [← taoSection5M0_eq_floor_log B] at h have h' : Real.log B / 100000 < (taoSection5M0 B : ℝ) + 1 := by exact_mod_cast h linarith nlinarith /-- Every fixed exponent-one mixing constant is eventually absorbed by the repaired interior rate. -/ theorem eventually_mix_div_taoSection5M0_le_powerInteriorDelta {Cmix : ℝ} (hCmix : 0 ≤ Cmix) : ∀ᶠ B : ℕ in atTop, Cmix / (taoSection5M0 B : ℝ) ≤ taoSection5PowerInteriorDelta B := by have hlog : Tendsto (fun B : ℕ => Real.log (B : ℝ)) atTop atTop := Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop have hgrowth : Tendsto (fun B : ℕ => Real.rpow (Real.log B) (9 / 10 : ℝ)) atTop atTop := (tendsto_rpow_atTop (by norm_num : (0 : ℝ) < 9 / 10)).comp hlog filter_upwards [eventually_log_div_two_hundred_thousand_le_taoSection5M0, hgrowth.eventually_ge_atTop (200000 * Cmix), hlog.eventually_ge_atTop (1 : ℝ)] with B hm0Lower hgrowthB hlogOne let L := Real.log B have hLpos : 0 < L := zero_lt_one.trans_le hlogOne have hm0BasePos : 0 < L / 200000 := div_pos hLpos (by norm_num) have hdenominator : Cmix / (taoSection5M0 B : ℝ) ≤ Cmix / (L / 200000) := div_le_div_of_nonneg_left hCmix hm0BasePos (by simpa only [L] using hm0Lower) have hfactor : Real.rpow L (-1 / 10 : ℝ) * L = Real.rpow L (9 / 10 : ℝ) := by calc Real.rpow L (-1 / 10 : ℝ) * L = Real.rpow L (-1 / 10 : ℝ) * Real.rpow L (1 : ℝ) := by congr 1 exact (Real.rpow_one L).symm _ = Real.rpow L ((-1 / 10 : ℝ) + 1) := (Real.rpow_add hLpos (-1 / 10 : ℝ) 1).symm _ = Real.rpow L (9 / 10 : ℝ) := by norm_num have hbaseRate : Cmix / (L / 200000) ≤ Real.rpow L (-1 / 10 : ℝ) := by apply (div_le_iff₀ hm0BasePos).2 have hgrowthB' : 200000 * Cmix ≤ Real.rpow L (9 / 10 : ℝ) := by simpa only [L] using hgrowthB rw [← hfactor] at hgrowthB' nlinarith exact hdenominator.trans (by simpa only [L, taoSection5PowerInteriorDelta] using hbaseRate) /-- Once the mixing quotient is at the repaired-interior scale, all four raw passage-TV errors are bounded by the single exact coefficient `3744366`. -/ theorem taoSection5_passWindowTV_rawError_le_log_rate {B : ℕ} {Cmix : ℝ} (hlogOne : 1 ≤ Real.log B) (hmixRate : Cmix / (taoSection5M0 B : ℝ) ≤ taoSection5PowerInteriorDelta B) : 64184 * taoSection5PowerInteriorDelta B + 180 * (Cmix / (taoSection5M0 B : ℝ)) + 3680000 * Real.rpow (Real.log B) (-1 / 5 : ℝ) + 2 * (B : ℝ) ^ (-(4 / 5 : ℝ)) ≤ 3744366 * taoSection5PowerInteriorDelta B := by let L := Real.log B let rate := Real.rpow L (-1 / 10 : ℝ) have hLpos : 0 < L := by dsimp only [L] exact zero_lt_one.trans_le hlogOne have hlogRate : Real.rpow L (-1 / 5 : ℝ) ≤ rate := by dsimp only [rate] exact Real.rpow_le_rpow_of_exponent_le (by simpa only [L] using hlogOne) (by norm_num) have hBgt : (1 : ℝ) < (B : ℝ) := (Real.log_pos_iff (Nat.cast_nonneg B)).mp (by simpa only [L] using hLpos) have hBpos : (0 : ℝ) < (B : ℝ) := zero_lt_one.trans hBgt have hlogLeB : L ≤ (B : ℝ) := by have h := Real.log_le_sub_one_of_pos hBpos dsimp only [L] linarith have hthresholdExponent : (B : ℝ) ^ (-(4 / 5 : ℝ)) ≤ (B : ℝ) ^ (-1 / 10 : ℝ) := Real.rpow_le_rpow_of_exponent_le hBgt.le (by norm_num) have hthresholdBase : (B : ℝ) ^ (-1 / 10 : ℝ) ≤ rate := by dsimp only [rate] exact Real.rpow_le_rpow_of_nonpos hLpos hlogLeB (by norm_num) have hthresholdRate : (B : ℝ) ^ (-(4 / 5 : ℝ)) ≤ rate := hthresholdExponent.trans hthresholdBase have hdelta : taoSection5PowerInteriorDelta B = rate := by rfl rw [hdelta] at hmixRate ⊢ calc 64184 * rate + 180 * (Cmix / (taoSection5M0 B : ℝ)) + 3680000 * Real.rpow (Real.log B) (-1 / 5 : ℝ) + 2 * (B : ℝ) ^ (-(4 / 5 : ℝ)) ≤ 64184 * rate + 180 * rate + 3680000 * rate + 2 * rate := by gcongr _ = 3744366 * rate := by ring /-- The canonical alpha/alpha-squared passage laws satisfy the unconditional natural-threshold logarithmic rate for all sufficiently large thresholds. -/ theorem eventually_taoSection5_passWindowTV_alpha_alphaSq_le_log_rate : ∀ᶠ B : ℕ in atTop, ∀ (hB : 1 ≤ B) (hmassAlpha : 0 < logFinsetMass (oddLogWindow (taoSection5SourceLo B .alpha) (taoSection5SourceHi B .alpha))) (hmassAlphaSq : 0 < logFinsetMass (oddLogWindow (taoSection5SourceLo B .alphaSq) (taoSection5SourceHi B .alphaSq))), syracusePassWindowTV B (taoSection5SourceLo B .alpha) (taoSection5SourceHi B .alpha) (taoSection5SourceLo B .alphaSq) (taoSection5SourceHi B .alphaSq) hB hmassAlpha hmassAlphaSq ≤ 3744366 * taoSection5PowerInteriorDelta B := by obtain ⟨Cmix, hCmix, hmix⟩ := TaoProp114FineScaleMixingStatement.bound taoProp114FineScaleMixing (A := 1) (by norm_num) filter_upwards [eventually_taoSection5PassEventPartitionFacts, eventually_taoSection5AffineSourceScaleFacts, eventually_taoSection5PowerInteriorBoundaryMassFacts, eventually_taoSection5PassTypicalFailureFacts, eventually_taoSection5_passTimeRatio_le_nine, eventually_mix_div_taoSection5M0_le_powerInteriorDelta hCmix] with B partitionFacts affineFacts boundaryFacts failureFacts hratio hmixRate intro hB hmassAlpha hmassAlphaSq have hraw := taoSection5_passWindowTV_alpha_alphaSq_le_capped_of_mixing (A := 1) (Cmix := Cmix) partitionFacts affineFacts boundaryFacts failureFacts hmix hCmix (hratio .alpha) (hratio .alphaSq) have hlogOne : 1 ≤ Real.log B := by linarith [boundaryFacts.log_large] calc syracusePassWindowTV B (taoSection5SourceLo B .alpha) (taoSection5SourceHi B .alpha) (taoSection5SourceLo B .alphaSq) (taoSection5SourceHi B .alphaSq) hB hmassAlpha hmassAlphaSq ≤ 64184 * taoSection5PowerInteriorDelta B + 180 * (Cmix / (taoSection5M0 B : ℝ)) + 3680000 * Real.rpow (Real.log B) (-1 / 5 : ℝ) + 2 * (B : ℝ) ^ (-(4 / 5 : ℝ)) := by simpa only [pow_one] using hraw _ ≤ 3744366 * taoSection5PowerInteriorDelta B := taoSection5_passWindowTV_rawError_le_log_rate hlogOne hmixRate /-- All-threshold rate statement for the canonical pair of Section 5 passage laws. -/ def TaoSection5CanonicalPassTVRateStatement : Prop := ∃ tvErr : ℕ → ℝ, ∃ C c : ℝ, 0 ≤ C ∧ 0 < c ∧ (∀ᶠ B : ℕ in atTop, 0 ≤ tvErr B ∧ tvErr B ≤ C * (Real.log (B : ℝ)) ^ (-c)) ∧ ∀ B : ℕ, ∀ (hB : 1 ≤ B), ∀ (hmassAlpha : 0 < logFinsetMass (oddLogWindow (taoSection5SourceLo B .alpha) (taoSection5SourceHi B .alpha))) (hmassAlphaSq : 0 < logFinsetMass (oddLogWindow (taoSection5SourceLo B .alphaSq) (taoSection5SourceHi B .alphaSq))), syracusePassWindowTV B (taoSection5SourceLo B .alpha) (taoSection5SourceHi B .alpha) (taoSection5SourceLo B .alphaSq) (taoSection5SourceHi B .alphaSq) hB hmassAlpha hmassAlphaSq ≤ tvErr B /-- The finite prefix is capped by `2`; above one cutoff the error is exactly the checked `3744366 * log(B)^(-1/10)` rate. -/ theorem taoSection5CanonicalPassTVRateStatement_checked : TaoSection5CanonicalPassTVRateStatement := by rcases eventually_atTop.1 eventually_taoSection5_passWindowTV_alpha_alphaSq_le_log_rate with ⟨B0, hB0⟩ let tvErr : ℕ → ℝ := fun B => if B0 ≤ B then 3744366 * taoSection5PowerInteriorDelta B else 2 refine ⟨tvErr, 3744366, (1 / 10 : ℝ), by norm_num, by norm_num, ?_, ?_⟩ · filter_upwards [eventually_ge_atTop B0] with B hlarge rw [show tvErr B = 3744366 * taoSection5PowerInteriorDelta B by simp [tvErr, hlarge]] constructor · exact mul_nonneg (by norm_num) (Real.rpow_nonneg (natCast_log_nonneg_prop111Rate B) _) · simp only [taoSection5PowerInteriorDelta] norm_num · intro B hB hmassAlpha hmassAlphaSq by_cases hlarge : B0 ≤ B · rw [show tvErr B = 3744366 * taoSection5PowerInteriorDelta B by simp [tvErr, hlarge]] exact hB0 B hlarge hB hmassAlpha hmassAlphaSq · rw [show tvErr B = 2 by simp [tvErr, hlarge]] unfold syracusePassWindowTV exact taoTV_le_two _ _ /-- Pass-TV rate over the endpoint relation consumed by the concrete natural Proposition 1.11 socket. -/ def TaoSection5CanonicalPairPassTVRateStatement : Prop := ∃ tvErr : ℕ → ℝ, ∃ C c : ℝ, 0 ≤ C ∧ 0 < c ∧ (∀ᶠ B : ℕ in atTop, 0 ≤ tvErr B ∧ tvErr B ≤ C * (Real.log (B : ℝ)) ^ (-c)) ∧ ∀ B lo1 hi1 lo2 hi2 : ℕ, (hB : 1 ≤ B) → TaoProp111WindowPair B lo1 hi1 lo2 hi2 → ∀ (hmass1 : 0 < logFinsetMass (oddLogWindow lo1 hi1)) (hmass2 : 0 < logFinsetMass (oddLogWindow lo2 hi2)), syracusePassWindowTV B lo1 hi1 lo2 hi2 hB hmass1 hmass2 ≤ tvErr B theorem TaoSection5CanonicalPassTVRateStatement.to_pair (h : TaoSection5CanonicalPassTVRateStatement) : TaoSection5CanonicalPairPassTVRateStatement := by rcases h with ⟨tvErr, C, c, hC, hc, hrate, hbound⟩ refine ⟨tvErr, C, c, hC, hc, hrate, ?_⟩ intro B lo1 hi1 lo2 hi2 hB hpair hmass1 hmass2 rcases hpair with ⟨rfl, rfl, rfl, rfl⟩ simpa only [taoSection5SourceLo_alpha_eq_prop111Y1, taoSection5SourceHi_alpha_eq_prop111Y1, taoSection5SourceLo_alphaSq_eq_prop111Y2, taoSection5SourceHi_alphaSq_eq_prop111Y2] using hbound B hB hmass1 hmass2 theorem taoSection5CanonicalPairPassTVRateStatement_checked : TaoSection5CanonicalPairPassTVRateStatement := TaoSection5CanonicalPassTVRateStatement.to_pair taoSection5CanonicalPassTVRateStatement_checked /-- Two rate packets with the same endpoint relation combine after weakening to the minimum exponent and maximum coefficient. -/ theorem taoProp111ConcreteFirstPassageStabilizationRate_of_pair_rates (hnoHit : TaoSection5CanonicalPairNoHitRateStatement) (hpassTV : TaoSection5CanonicalPairPassTVRateStatement) : TaoProp111ConcreteFirstPassageStabilizationRateStatement := by rcases hnoHit with ⟨noHitErr, CnoHit, cnoHit, hCnoHit, hcnoHit, hnoHitRate, hnoHitBound⟩ rcases hpassTV with ⟨tvErr, Ctv, ctv, hCtv, hctv, htvRate, htvBound⟩ let C := max CnoHit Ctv let c := min cnoHit ctv refine ⟨noHitErr, tvErr, C, c, ?_, ?_, ?_, ?_⟩ · exact hCnoHit.trans (le_max_left _ _) · exact lt_min hcnoHit hctv · have hlog : Tendsto (fun B : ℕ => Real.log (B : ℝ)) atTop atTop := Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop filter_upwards [hnoHitRate, htvRate, eventually_ge_atTop 1, hlog.eventually_ge_atTop (1 : ℝ)] with B hno htv hB hlogOne have hBOne : (1 : ℝ) ≤ (B : ℝ) := by exact_mod_cast hB have hnoPow : (B : ℝ) ^ (-cnoHit) ≤ (B : ℝ) ^ (-c) := Real.rpow_le_rpow_of_exponent_le hBOne (neg_le_neg (min_le_left cnoHit ctv)) have htvPow : (Real.log (B : ℝ)) ^ (-ctv) ≤ (Real.log (B : ℝ)) ^ (-c) := Real.rpow_le_rpow_of_exponent_le hlogOne (neg_le_neg (min_le_right cnoHit ctv)) refine ⟨hno.1, htv.1, ?_, ?_⟩ · calc noHitErr B ≤ CnoHit * (B : ℝ) ^ (-cnoHit) := hno.2 _ ≤ CnoHit * (B : ℝ) ^ (-c) := mul_le_mul_of_nonneg_left hnoPow hCnoHit _ ≤ C * (B : ℝ) ^ (-c) := mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.rpow_nonneg (Nat.cast_nonneg B) _) · calc tvErr B ≤ Ctv * (Real.log (B : ℝ)) ^ (-ctv) := htv.2 _ ≤ Ctv * (Real.log (B : ℝ)) ^ (-c) := mul_le_mul_of_nonneg_left htvPow hCtv _ ≤ C * (Real.log (B : ℝ)) ^ (-c) := mul_le_mul_of_nonneg_right (le_max_right _ _) (Real.rpow_nonneg (natCast_log_nonneg_prop111Rate B) _) · intro B lo1 hi1 lo2 hi2 hB hpair hmass1 hmass2 rcases hnoHitBound B lo1 hi1 lo2 hi2 hpair hmass1 hmass2 with ⟨hno1, hno2⟩ exact ⟨hno1, hno2, htvBound B lo1 hi1 lo2 hi2 hB hpair hmass1 hmass2⟩ /-- Checked rate-bearing natural-threshold Proposition 1.11. -/ theorem taoProp111ConcreteFirstPassageStabilizationRate_checked : TaoProp111ConcreteFirstPassageStabilizationRateStatement := taoProp111ConcreteFirstPassageStabilizationRate_of_pair_rates taoSection5CanonicalPairNoHitRateStatement_checked taoSection5CanonicalPairPassTVRateStatement_checked end end Tao end Erdos1135