import Erdos1135.ND.Discrepancy.A5ReferenceND31FullCount import Erdos1135.ND.Discrepancy.A5ReferenceNDM2 import Erdos1135.ND.Discrepancy.A5ReferenceAllSourceTStarL import Erdos1135.ND.Discrepancy.A5ReferenceSection3GeometricBudget import Erdos1135.Tao.Section3.AmbientScalePilot /-! # Powered main regime for ND-3.1 This leaf assembles the checked full-count splice at the second descending ambient boundary. One powered natural cutoff simultaneously clears the ND-M2 threshold, the power-to-log threshold, and the source-support guard. The zero and bounded-target cases remain outside this main-regime module. -/ namespace Erdos1135 namespace ND open Filter open scoped Topology noncomputable section /-- The strict prefix below the first descending ambient boundary costs a genuine power of the ambient endpoint. -/ theorem ceil_taoSection3AmbientBoundary_one_div_le_real_power {X : Real} (hX : 1 <= X) : (Nat.ceil (Tao.taoSection3AmbientBoundary X 1) : Real) / X <= 2 * X ^ (-(1 / 32000 : Real)) := by have hXpos : 0 < X := zero_lt_one.trans_le hX have hyOne : 1 <= Tao.taoSection3AmbientBoundary X 1 := Tao.taoSection3AmbientBoundary_one_le hX 1 have hceil : (Nat.ceil (Tao.taoSection3AmbientBoundary X 1) : Real) < Tao.taoSection3AmbientBoundary X 1 + 1 := Nat.ceil_lt_add_one (Tao.taoSection3AmbientBoundary_nonneg hXpos.le 1) have hceilTwo : (Nat.ceil (Tao.taoSection3AmbientBoundary X 1) : Real) <= 2 * Tao.taoSection3AmbientBoundary X 1 := by linarith have hratio : Tao.taoSection3AmbientBoundary X 1 / X = X ^ (Tao.taoSection3AmbientRatio - 1) := by rw [Tao.taoSection3AmbientBoundary] simp only [pow_one] exact (Real.rpow_sub_one hXpos.ne' Tao.taoSection3AmbientRatio).symm have hexponent : Tao.taoSection3AmbientRatio - 1 <= -(1 / 32000 : Real) := by norm_num [Tao.taoSection3AmbientRatio, Tao.taoAlpha] calc (Nat.ceil (Tao.taoSection3AmbientBoundary X 1) : Real) / X <= (2 * Tao.taoSection3AmbientBoundary X 1) / X := div_le_div_of_nonneg_right hceilTwo hXpos.le _ = 2 * X ^ (Tao.taoSection3AmbientRatio - 1) := by rw [mul_div_assoc, hratio] _ <= 2 * X ^ (-(1 / 32000 : Real)) := mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hX hexponent) (by norm_num) /-- Above one powered natural cutoff, the second descending ambient boundary feeds ND-M2 and the first boundary feeds the exact full-count block. The constant and cutoff are both selected before the target and counting endpoint. -/ theorem exists_ndA5ReferenceND31MainRegime {c kappa d : Real} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∃ Nstar : Nat, 2 <= Nstar ∧ ∃ Cmain : Real, 0 <= Cmain ∧ ∀ (N0 : Nat), Nstar <= N0 -> ∀ (X : Real), (N0 : Real) < X -> oddSyracuseBadRatio N0 X <= Cmain * (Real.log (N0 : Real)) ^ (-d) := by rcases ndA5ReferenceNDM2Bounds_sameD hPhase hd hd20 hdk with ⟨CHit, CTr, x0, hCHit, hCTr, hx0, hM2⟩ rcases exists_ndA5ReferenceAllSourceTStarL hPhase hd hd20 hdk with ⟨CL, hCL, hAllSource⟩ rcases eventually_atTop.1 (eventually_rpow_neg_one_div_32000_le_log_rpow_neg d) with ⟨Qabs, hQabs⟩ let r : Real := Tao.taoSection3AmbientRatio let e : Real := r ^ 2 let E : Real := e ^ (-d) let Cprob : Real := (CHit + CTr) * E + CL let Cmain : Real := 2 + 2 * Cprob have he : 0 < e := by dsimp only [e, r] exact pow_pos Tao.taoSection3AmbientRatio_pos 2 have hE : 0 <= E := by exact Real.rpow_nonneg he.le _ have hCprob : 0 <= Cprob := by dsimp only [Cprob] exact add_nonneg (mul_nonneg (add_nonneg hCHit hCTr) hE) hCL have hCmain : 0 <= Cmain := by dsimp only [Cmain] positivity have hpow : Tendsto (fun N : Nat => (N : Real) ^ e) atTop atTop := by simpa only [Function.comp_def] using (tendsto_rpow_atTop he).comp tendsto_natCast_atTop_atTop have hlarge : ∀ᶠ N : Nat in atTop, 2 <= N ∧ max x0 Qabs <= (N : Real) ^ e := by filter_upwards [eventually_ge_atTop (2 : Nat), hpow.eventually_ge_atTop (max x0 Qabs)] with N hN hcut exact ⟨hN, hcut⟩ 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, hcutN0⟩ have hN0OneNat : 1 <= N0 := by omega have hN0One : (1 : Real) <= (N0 : Real) := 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 : Real) <= (N0 : Real) := Nat.cast_nonneg N0 have hpowStrict : (N0 : Real) ^ e < X ^ e := Real.rpow_lt_rpow hN0nonneg hN0X he let q : Real := Tao.taoSection3AmbientBoundary X 2 let y : Real := transportSourceY q .alpha have hqEq : q = X ^ e := by rfl have hcutQ : max x0 Qabs <= q := by rw [hqEq] exact hcutN0.trans hpowStrict.le have hqx0 : x0 <= q := (le_max_left _ _).trans hcutQ have hqQabs : Qabs <= q := (le_max_right _ _).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 : Real) := 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 := hM2 q hqx0 hq .alpha change ∃ hwindow : (oddBlock y).Nonempty, ∃ hmass : 0 < Tao.logFinsetMass (oddBlock y), uniformNoHitProbability q y hwindow <= CHit * q ^ (-(1 / 32000 : Real)) ∧ passFullL1 q y hq hwindow hmass <= CTr * (Real.log q) ^ (-d) at hM2q rcases hM2q with ⟨hwindow, hmass, hHit, hPass⟩ have hHarmonic := hAllSource N0 hN0Two y hyTwo hmass have hCount := oddSyracuseBadRatio_le_ceil_div_add_two_mul_fixedEventTerms hN0OneNat hq hy hXOne hupper hwindow hmass let R : Real := (Real.log (N0 : Real)) ^ (-d) have hN0OneStrict : (1 : Real) < (N0 : Real) := by exact_mod_cast (show 1 < N0 by omega) have hlogN0Pos : 0 < Real.log (N0 : Real) := Real.log_pos hN0OneStrict have hlogNX : Real.log (N0 : Real) < Real.log X := Real.log_lt_log (by positivity) 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 : Real) / X <= 2 * X ^ (-(1 / 32000 : Real)) := by rw [hyBoundary] exact ceil_taoSection3AmbientBoundary_one_div_le_real_power hXOne have hBottom : (Nat.ceil y : Real) / X <= 2 * R := by calc (Nat.ceil y : Real) / X <= 2 * X ^ (-(1 / 32000 : Real)) := 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 hqpos : 0 < q := zero_lt_one.trans_le hq 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 hAbsorbQ := hQabs q hqQabs have hHitRate : uniformNoHitProbability q y hwindow <= CHit * E * R := by calc uniformNoHitProbability q y hwindow <= CHit * q ^ (-(1 / 32000 : Real)) := hHit _ <= CHit * (Real.log q) ^ (-d) := mul_le_mul_of_nonneg_left hAbsorbQ hCHit _ <= CHit * (E * R) := mul_le_mul_of_nonneg_left hlogqRateLe hCHit _ = CHit * 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 hTerms : uniformNoHitProbability q y hwindow + passFullL1 q y hq hwindow hmass + Tao.pmfProb (logOddBlockPMF y hmass) {N | N0 < syracuseOrbitMin N.1} <= Cprob * R := by calc uniformNoHitProbability q y hwindow + passFullL1 q y hq hwindow hmass + Tao.pmfProb (logOddBlockPMF y hmass) {N | N0 < syracuseOrbitMin N.1} <= CHit * E * R + CTr * E * R + CL * R := add_le_add (add_le_add hHitRate hPassRate) hHarmonic _ = Cprob * R := by dsimp only [Cprob] ring calc oddSyracuseBadRatio N0 X <= (Nat.ceil y : Real) / X + 2 * (uniformNoHitProbability q y hwindow + passFullL1 q y hq hwindow hmass + Tao.pmfProb (logOddBlockPMF y hmass) {N | N0 < syracuseOrbitMin N.1}) := hCount _ <= 2 * R + 2 * (Cprob * R) := add_le_add hBottom (mul_le_mul_of_nonneg_left hTerms (by norm_num)) _ = Cmain * R := by dsimp only [Cmain] ring _ = Cmain * (Real.log (N0 : Real)) ^ (-d) := by rfl end end ND end Erdos1135