import Erdos1135.ND.Discrepancy.A5ReferenceSection3Facade /-! # Actual-real rates for the A5 Section 3 facade This leaf converts the capped harmonic errors from their natural floor argument to separate rates in the actual real threshold. The escape row keeps its power rate `1 / 32000`, while the ordered passage row keeps the symbolic logarithmic exponent `d`; no common or minimum exponent is formed. -/ namespace Erdos1135 namespace ND open Filter open scoped Topology noncomputable section private theorem half_le_natFloor_cast {x : Real} (hx : 2 <= x) : x / 2 <= ((Nat.floor x : Nat) : Real) := by have hfloor := Nat.lt_floor_add_one x nlinarith private theorem half_log_le_log_natFloor_cast {x : Real} (hx : 2 <= x) (hlog : 2 <= Real.log x) : Real.log x / 2 <= Real.log ((Nat.floor x : Nat) : Real) := by have hxpos : 0 < x := by linarith have hxhalfPos : 0 < x / 2 := div_pos hxpos (by norm_num) have hhalf := half_le_natFloor_cast hx have hfloorPos : 0 < ((Nat.floor x : Nat) : Real) := hxhalfPos.trans_le hhalf have hmono := Real.strictMonoOn_log.monotoneOn hxhalfPos hfloorPos hhalf rw [Real.log_div hxpos.ne' (by norm_num : (2 : Real) ≠ 0)] at hmono have hlogTwo : Real.log (2 : Real) < 1 := by have h := Real.log_lt_sub_one_of_pos (show (0 : Real) < 2 by norm_num) (show (2 : Real) ≠ 1 by norm_num) norm_num at h exact h linarith private theorem rpow_neg_le_two_rpow_mul_of_half_le {a b c e : Real} (ha : 0 < a) (haOne : 1 <= a) (hab : a / 2 <= b) (hc : 0 <= c) (hec : e <= c) : b ^ (-c) <= (2 : Real) ^ c * a ^ (-e) := by have hhalfPos : 0 < a / 2 := div_pos ha (by norm_num) have hbase : b ^ (-c) <= (a / 2) ^ (-c) := Real.rpow_le_rpow_of_nonpos hhalfPos hab (neg_nonpos.mpr hc) have hfactor : (a / 2) ^ (-c) = (2 : Real) ^ c * a ^ (-c) := by rw [Real.div_rpow ha.le (by norm_num : (0 : Real) <= 2), Real.rpow_neg (by norm_num : (0 : Real) <= 2)] simp only [div_eq_mul_inv, inv_inv] ring have hexponent : a ^ (-c) <= a ^ (-e) := Real.rpow_le_rpow_of_exponent_le haOne (neg_le_neg hec) calc b ^ (-c) <= (a / 2) ^ (-c) := hbase _ = (2 : Real) ^ c * a ^ (-c) := hfactor _ <= (2 : Real) ^ c * a ^ (-e) := mul_le_mul_of_nonneg_left hexponent (Real.rpow_nonneg (by norm_num) _) private theorem section3DeltaH_floor_le_inv_x {x : Real} (hx : 2 <= x) (hlogFloor : 1 <= Real.log ((Nat.floor x : Nat) : Real)) : 96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real)) <= 192000 / x := by have hxpos : 0 < x := by linarith have hhalf := half_le_natFloor_cast hx have hfloorNonneg : 0 <= ((Nat.floor x : Nat) : Real) := Nat.cast_nonneg _ have hfloorLeProd : ((Nat.floor x : Nat) : Real) <= ((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real) := by simpa only [mul_one] using mul_le_mul_of_nonneg_left hlogFloor hfloorNonneg have hden : x / 2 <= ((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real) := hhalf.trans hfloorLeProd calc 96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real)) <= 96000 / (x / 2) := div_le_div_of_nonneg_left (by norm_num) (div_pos hxpos (by norm_num)) hden _ = 192000 / x := by field_simp [hxpos.ne'] norm_num private theorem section3DeltaH_floor_le_real_power {x e : Real} (hx : 2 <= x) (hlogFloor : 1 <= Real.log ((Nat.floor x : Nat) : Real)) (heOne : e <= 1) : 96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real)) <= 192000 * x ^ (-e) := by have hxOne : 1 <= x := by linarith have hpow : x ^ (-1 : Real) <= x ^ (-e) := Real.rpow_le_rpow_of_exponent_le hxOne (neg_le_neg heOne) calc 96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real)) <= 192000 / x := section3DeltaH_floor_le_inv_x hx hlogFloor _ = 192000 * x ^ (-1 : Real) := by rw [Real.rpow_neg_one] ring _ <= 192000 * x ^ (-e) := mul_le_mul_of_nonneg_left hpow (by norm_num) private theorem four_mul_section3DeltaH_floor_le_real_log_power {x d : Real} (hx : 2 <= x) (hlog : 1 <= Real.log x) (hlogFloor : 1 <= Real.log ((Nat.floor x : Nat) : Real)) (hdOne : d <= 1) : 4 * (96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real))) <= 768000 * (Real.log x) ^ (-d) := by have hxpos : 0 < x := by linarith have hlogPos : 0 < Real.log x := zero_lt_one.trans_le hlog have hlogLeX : Real.log x <= x := by have h := Real.log_le_sub_one_of_pos hxpos linarith have hinv : 1 / x <= 1 / Real.log x := one_div_le_one_div_of_le hlogPos hlogLeX have hlogPow : (Real.log x) ^ (-1 : Real) <= (Real.log x) ^ (-d) := Real.rpow_le_rpow_of_exponent_le hlog (neg_le_neg hdOne) calc 4 * (96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real))) <= 4 * (192000 / x) := mul_le_mul_of_nonneg_left (section3DeltaH_floor_le_inv_x hx hlogFloor) (by norm_num) _ = 768000 * (1 / x) := by ring _ <= 768000 * (1 / Real.log x) := mul_le_mul_of_nonneg_left hinv (by norm_num) _ = 768000 * (Real.log x) ^ (-1 : Real) := by rw [Real.rpow_neg_one] ring _ <= 768000 * (Real.log x) ^ (-d) := mul_le_mul_of_nonneg_left hlogPow (by norm_num) /-- The capped harmonic no-hit error has its genuine power rate in the actual real threshold. The fixed finite prefix is eventually crossed for every cutoff `B0`. -/ theorem eventually_ndA5ReferenceSection3NoHitError_floor_le_real_power (B0 : Nat) : ∀ᶠ x : Real in atTop, ndA5ReferenceSection3NoHitError B0 (Nat.floor x) <= 192020 * x ^ (-(1 / 32000 : Real)) := by filter_upwards [tendsto_nat_floor_atTop.eventually_ge_atTop B0, eventually_ge_atTop (2 : Real), Real.tendsto_log_atTop.eventually_ge_atTop (2 : Real)] with x hcut hx hlog have hxpos : 0 < x := by linarith have hxOne : 1 <= x := by linarith have hhalf := half_le_natFloor_cast hx have hlogHalf := half_log_le_log_natFloor_cast hx hlog have hlogFloor : 1 <= Real.log ((Nat.floor x : Nat) : Real) := by linarith have hpowBase : ((Nat.floor x : Nat) : Real) ^ (-(1 / 32000 : Real)) <= (2 : Real) ^ (1 / 32000 : Real) * x ^ (-(1 / 32000 : Real)) := rpow_neg_le_two_rpow_mul_of_half_le hxpos hxOne hhalf (by norm_num) (le_refl _) have htwo : (2 : Real) ^ (1 / 32000 : Real) <= 2 := by calc (2 : Real) ^ (1 / 32000 : Real) <= (2 : Real) ^ (1 : Real) := Real.rpow_le_rpow_of_exponent_le (by norm_num) (by norm_num) _ = 2 := by norm_num have hpow : ((Nat.floor x : Nat) : Real) ^ (-(1 / 32000 : Real)) <= 2 * x ^ (-(1 / 32000 : Real)) := hpowBase.trans (mul_le_mul_of_nonneg_right htwo (Real.rpow_nonneg hxpos.le _)) have hdelta := section3DeltaH_floor_le_real_power hx hlogFloor (show (1 / 32000 : Real) <= 1 by norm_num) rw [ndA5ReferenceSection3NoHitError, if_pos hcut] unfold ndA5ReferenceSection3RawNoHitError calc 10 * ((Nat.floor x : Nat) : Real) ^ (-(1 / 32000 : Real)) + 96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real)) <= 10 * (2 * x ^ (-(1 / 32000 : Real))) + 192000 * x ^ (-(1 / 32000 : Real)) := add_le_add (mul_le_mul_of_nonneg_left hpow (by norm_num)) hdelta _ = 192020 * x ^ (-(1 / 32000 : Real)) := by ring /-- The capped ordered passage error has its genuine logarithmic rate in the actual real threshold, with the full-L1 coefficient `768064`. -/ theorem eventually_ndA5ReferenceSection3PassTVError_floor_le_real_log_power (B0 : Nat) {d : Real} (hd : 0 < d) (hd20 : d < 1 / 20) : ∀ᶠ x : Real in atTop, ndA5ReferenceSection3PassTVError B0 d (Nat.floor x) <= 768064 * (Real.log x) ^ (-d) := by have hd0 : 0 <= d := hd.le have hdOne : d <= 1 := by linarith [hd20] filter_upwards [tendsto_nat_floor_atTop.eventually_ge_atTop B0, eventually_ge_atTop (2 : Real), Real.tendsto_log_atTop.eventually_ge_atTop (2 : Real)] with x hcut hx hlog have hlogPos : 0 < Real.log x := by linarith have hlogOne : 1 <= Real.log x := by linarith have hlogHalf := half_log_le_log_natFloor_cast hx hlog have hlogFloor : 1 <= Real.log ((Nat.floor x : Nat) : Real) := by linarith have hpowBase : Real.log ((Nat.floor x : Nat) : Real) ^ (-d) <= (2 : Real) ^ d * (Real.log x) ^ (-d) := rpow_neg_le_two_rpow_mul_of_half_le hlogPos hlogOne hlogHalf hd0 (le_refl _) have htwo : (2 : Real) ^ d <= 2 := by calc (2 : Real) ^ d <= (2 : Real) ^ (1 : Real) := Real.rpow_le_rpow_of_exponent_le (by norm_num) hdOne _ = 2 := by norm_num have hpow : Real.log ((Nat.floor x : Nat) : Real) ^ (-d) <= 2 * (Real.log x) ^ (-d) := hpowBase.trans (mul_le_mul_of_nonneg_right htwo (Real.rpow_nonneg hlogPos.le _)) have hdelta := four_mul_section3DeltaH_floor_le_real_log_power hx hlogOne hlogFloor hdOne rw [ndA5ReferenceSection3PassTVError, if_pos hcut] unfold ndA5ReferenceSection3RawPassTVError ndA5ReferenceInteriorScalarRate calc 32 * Real.log ((Nat.floor x : Nat) : Real) ^ (-d) + 4 * (96000 / (((Nat.floor x : Nat) : Real) * Real.log ((Nat.floor x : Nat) : Real))) <= 32 * (2 * (Real.log x) ^ (-d)) + 768000 * (Real.log x) ^ (-d) := add_le_add (mul_le_mul_of_nonneg_left hpow (by norm_num)) hdelta _ = 768064 * (Real.log x) ^ (-d) := by ring /-- One cutoff from the checked Section 3 facade simultaneously carries its global all-threshold bounds, nonnegativity, and the two separate actual-real rates. The cutoff is chosen before every real threshold and later trace parameter. -/ theorem exists_ndA5ReferenceSection3SplitRealRatePacket {c kappa d : Real} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∃ B0 : Nat, 2 <= B0 ∧ (∀ B : Nat, 0 <= ndA5ReferenceSection3NoHitError B0 B ∧ 0 <= ndA5ReferenceSection3PassTVError B0 d B) ∧ (∀ᶠ x : Real in atTop, ndA5ReferenceSection3NoHitError B0 (Nat.floor x) <= 192020 * x ^ (-(1 / 32000 : Real))) ∧ (∀ᶠ x : Real in atTop, ndA5ReferenceSection3PassTVError B0 d (Nat.floor x) <= 768064 * (Real.log x) ^ (-d)) ∧ NDA5ReferenceSection3Bounds (ndA5ReferenceSection3NoHitError B0) (ndA5ReferenceSection3PassTVError B0 d) := by rcases exists_ndA5ReferenceSection3AllThresholdBounds hPhase hd hd20 hdk with ⟨B0, hB0, hnonneg, _hraw, hbounds⟩ exact ⟨B0, hB0, hnonneg, eventually_ndA5ReferenceSection3NoHitError_floor_le_real_power B0, eventually_ndA5ReferenceSection3PassTVError_floor_le_real_log_power B0 hd hd20, hbounds⟩ end end ND end Erdos1135