import Erdos1135.Tao.Section5.ReversePrefix import Erdos1135.Tao.Section5.DescentScale /-! # Section 5 Reverse-Prefix Scalar Budget This leaf proves the branch-free eventual scalar inequality consumed by the reverse strict-prefix argument. The proof keeps the floor loss sign-safe and uses only the canonical `m0`, `n0`, and logarithmic slack scales. -/ namespace Erdos1135 namespace Tao open Filter open scoped Topology noncomputable section private theorem one_fourth_le_log_four_thirds_scalar : (1 / 4 : ℝ) ≤ Real.log (4 / 3 : ℝ) := by have h := Real.one_sub_inv_le_log_of_pos (show (0 : ℝ) < 4 / 3 by norm_num) norm_num at h ⊢ exact h private theorem log_two_pos_scalar : 0 < Real.log (2 : ℝ) := Real.log_pos (by norm_num) private theorem log_two_lt_one_scalar : Real.log (2 : ℝ) < 1 := by have h := Real.log_lt_sub_one_of_pos (show (0 : ℝ) < 2 by norm_num) (show (2 : ℝ) ≠ 1 by norm_num) norm_num at h ⊢ exact h /-- Any fixed sublinear power of `log B` is eventually bounded by a supplied positive multiple of `log B`. -/ private theorem eventually_log_rpow_le_mul_log {p c : ℝ} (hp : 0 < 1 - p) (hc : 0 < c) : ∀ᶠ B : ℕ in atTop, Real.rpow (Real.log B) p ≤ c * Real.log B := by have hlog : Tendsto (fun B : ℕ => Real.log (B : ℝ)) atTop atTop := Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop have hratio : Tendsto (fun B : ℕ => Real.rpow (Real.log B) (-(1 - p))) atTop (nhds 0) := (tendsto_rpow_neg_atTop hp).comp hlog filter_upwards [hratio.eventually_le_const hc, hlog.eventually_gt_atTop (0 : ℝ)] with B hratioB hlogB have hlogNonneg : 0 ≤ Real.log B := hlogB.le calc Real.rpow (Real.log B) p = Real.rpow (Real.log B) (-(1 - p) + 1) := by congr 1 ring _ = Real.rpow (Real.log B) (-(1 - p)) * Real.rpow (Real.log B) (1 : ℝ) := Real.rpow_add hlogB (-(1 - p)) (1 : ℝ) _ = Real.rpow (Real.log B) (-(1 - p)) * Real.log B := by congr 1 exact Real.rpow_one (Real.log B) _ ≤ c * Real.log B := mul_le_mul_of_nonneg_right hratioB hlogNonneg private theorem eventually_taoSection5ReversePrefixExponentMargin : ∀ᶠ B : ℕ in atTop, taoSection5CanonicalLostSlack B + 2 * Real.log 2 * taoSection5TypicalSlack B + Real.log 2 < (taoSection5M0 B : ℝ) * Real.log (4 / 3 : 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 (4000000 : ℝ), eventually_log_rpow_le_mul_log (p := (3 / 5 : ℝ)) (c := (1 / 8000000 : ℝ)) (by norm_num) (by norm_num), eventually_log_rpow_le_mul_log (p := (7 / 10 : ℝ)) (c := (1 / 4000000 : ℝ)) (by norm_num) (by norm_num)] with B hlogLarge htypical hlost have hlogPos : 0 < Real.log B := by nlinarith have hm0Floor : 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 nlinarith have hm0Lower : Real.log B / 200000 ≤ (taoSection5M0 B : ℝ) := by nlinarith have hm0LogLower : Real.log B / 800000 ≤ (taoSection5M0 B : ℝ) * Real.log (4 / 3 : ℝ) := by calc Real.log B / 800000 = (Real.log B / 200000) * (1 / 4 : ℝ) := by ring _ ≤ (taoSection5M0 B : ℝ) * (1 / 4 : ℝ) := mul_le_mul_of_nonneg_right hm0Lower (by norm_num) _ ≤ (taoSection5M0 B : ℝ) * Real.log (4 / 3 : ℝ) := mul_le_mul_of_nonneg_left one_fourth_le_log_four_thirds_scalar (Nat.cast_nonneg _) have htypicalNonneg : 0 ≤ taoSection5TypicalSlack B := by exact Real.rpow_nonneg hlogPos.le _ have hlogTypical : Real.log 2 * taoSection5TypicalSlack B ≤ taoSection5TypicalSlack B := by simpa only [one_mul] using mul_le_mul_of_nonneg_right log_two_lt_one_scalar.le htypicalNonneg have htypicalBudget : 2 * Real.log 2 * taoSection5TypicalSlack B ≤ Real.log B / 4000000 := by have htypical' : taoSection5TypicalSlack B ≤ Real.log B / 8000000 := by change Real.rpow (Real.log B) (3 / 5 : ℝ) ≤ Real.log B / 8000000 convert htypical using 1 <;> ring nlinarith have hlostBudget : taoSection5CanonicalLostSlack B ≤ Real.log B / 4000000 := by change Real.rpow (Real.log B) (7 / 10 : ℝ) ≤ Real.log B / 4000000 convert hlost using 1 <;> ring have hlogTwoBudget : Real.log 2 ≤ Real.log B / 4000000 := by nlinarith [log_two_lt_one_scalar] have htotal : taoSection5CanonicalLostSlack B + 2 * Real.log 2 * taoSection5TypicalSlack B + Real.log 2 < Real.log B / 800000 := by nlinarith exact htotal.trans_le hm0LogLower private theorem taoSection5ReversePrefixScalarBudget_of {B : ℕ} (facts : TaoSection5DescentScaleFacts B) (hmargin : taoSection5CanonicalLostSlack B + 2 * Real.log 2 * taoSection5TypicalSlack B + Real.log 2 < (taoSection5M0 B : ℝ) * Real.log (4 / 3 : ℝ)) : taoSection5ReversePrefixScalarBudget B := by have hBpos : (0 : ℝ) < B := by exact_mod_cast facts.one_le_B have hmainExponent : Real.log 2 + 2 * Real.log 2 * taoSection5TypicalSlack B < -taoSection5CanonicalLostSlack B + (taoSection5M0 B : ℝ) * Real.log (4 / 3 : ℝ) := by linarith have hfactor : 2 * Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) < Real.exp (-taoSection5CanonicalLostSlack B) * (4 / 3 : ℝ) ^ taoSection5M0 B := by calc 2 * Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) = Real.exp (Real.log 2) * Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) := by rw [Real.exp_log (by norm_num : (0 : ℝ) < 2)] _ = Real.exp (Real.log 2 + 2 * Real.log 2 * taoSection5TypicalSlack B) := by rw [Real.exp_add] _ < Real.exp (-taoSection5CanonicalLostSlack B + (taoSection5M0 B : ℝ) * Real.log (4 / 3 : ℝ)) := Real.exp_lt_exp.mpr hmainExponent _ = Real.exp (-taoSection5CanonicalLostSlack B) * Real.exp ((taoSection5M0 B : ℝ) * Real.log (4 / 3 : ℝ)) := by rw [Real.exp_add] _ = Real.exp (-taoSection5CanonicalLostSlack B) * (4 / 3 : ℝ) ^ taoSection5M0 B := by rw [Real.exp_nat_mul, Real.exp_log (by norm_num : (0 : ℝ) < 4 / 3)] have hfactorB : 2 * Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) * (B : ℝ) < (Real.exp (-taoSection5CanonicalLostSlack B) * (4 / 3 : ℝ) ^ taoSection5M0 B) * (B : ℝ) := mul_lt_mul_of_pos_right hfactor hBpos have hoffsetB : (3 : ℝ) ^ taoSection5N0 B ≤ (B : ℝ) := by calc (3 : ℝ) ^ taoSection5N0 B ≤ (B : ℝ) ^ (1 / 5 : ℝ) := facts.offset _ ≤ (B : ℝ) ^ (99 / 100 : ℝ) + (B : ℝ) ^ (1 / 5 : ℝ) := le_add_of_nonneg_left (Real.rpow_nonneg (Nat.cast_nonneg B) _) _ ≤ (B : ℝ) := facts.absorption have hexpOne : 1 ≤ Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) := by apply Real.one_le_exp have hslack : 0 ≤ taoSection5TypicalSlack B := by exact Real.rpow_nonneg (Real.log_nonneg (by exact_mod_cast facts.one_le_B)) _ positivity have hB_le_main : (B : ℝ) ≤ Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) * (B : ℝ) := by simpa only [one_mul] using mul_le_mul_of_nonneg_right hexpOne (Nat.cast_nonneg B) unfold taoSection5ReversePrefixScalarBudget rw [taoSection5CanonicalLostLower, taoSection5CanonicalLostScale] calc Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) * (B : ℝ) + (3 : ℝ) ^ taoSection5N0 B ≤ Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) * (B : ℝ) + (B : ℝ) := add_le_add_right hoffsetB _ _ ≤ Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) * (B : ℝ) + Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) * (B : ℝ) := add_le_add_right hB_le_main _ _ = 2 * Real.exp (2 * Real.log 2 * taoSection5TypicalSlack B) * (B : ℝ) := by ring _ < Real.exp (-taoSection5CanonicalLostSlack B) * ((4 / 3 : ℝ) ^ taoSection5M0 B * (B : ℝ)) := by simpa only [mul_assoc] using hfactorB /-- The canonical reverse-prefix scalar budget holds for all sufficiently large thresholds. -/ theorem eventually_taoSection5ReversePrefixScalarBudget : ∀ᶠ B : ℕ in atTop, taoSection5ReversePrefixScalarBudget B := by filter_upwards [eventually_taoSection5DescentScaleFacts, eventually_taoSection5ReversePrefixExponentMargin] with B facts hmargin exact taoSection5ReversePrefixScalarBudget_of facts hmargin end end Tao end Erdos1135