import Mathlib.Analysis.SpecialFunctions.Pow.Continuity import Erdos1135.Tao.Section3.AmbientScalePilot /-! # Ambient Section 3 Least Crossing This leaf selects the first descending ambient boundary whose value is at most `B^(1 / alpha)`. The weak crossing convention is intentional: equality already makes the corresponding seed window time-zero good. It can differ by one index from Tao's printed strict crossing, so all endpoint polarities are exported explicitly. -/ namespace Erdos1135 namespace Tao open Filter open scoped Topology noncomputable section /-- For an ambient top strictly above one, its descending boundaries are strictly decreasing. -/ theorem taoSection3AmbientBoundary_strictAnti {X : ℝ} (hX : 1 < X) : StrictAnti (taoSection3AmbientBoundary X) := by simpa only [taoSection3AmbientBoundary, Function.comp_apply] using (Real.strictMono_rpow_of_base_gt_one hX).comp_strictAnti (pow_right_strictAnti₀ taoSection3AmbientRatio_pos taoSection3AmbientRatio_lt_one) /-- Every positive ambient boundary sequence converges to one. -/ theorem tendsto_taoSection3AmbientBoundary_atTop {X : ℝ} (hX : 0 < X) : Tendsto (taoSection3AmbientBoundary X) atTop (nhds 1) := by have hpow : Tendsto (fun i : ℕ => taoSection3AmbientRatio ^ i) atTop (nhds 0) := tendsto_pow_atTop_nhds_zero_of_lt_one taoSection3AmbientRatio_pos.le taoSection3AmbientRatio_lt_one have hrpow := (Real.continuousAt_const_rpow (a := X) (b := 0) hX.ne').tendsto.comp hpow simpa only [taoSection3AmbientBoundary, Real.rpow_zero] using hrpow /-- A positive target above one is eventually crossed by the descending ambient boundary sequence. -/ theorem exists_taoSection3AmbientBoundary_le {B : ℕ} {X : ℝ} (hB : 2 ≤ B) (hX : 0 < X) : ∃ i : ℕ, taoSection3AmbientBoundary X i ≤ (B : ℝ) ^ taoSection3AmbientRatio := by have hBone : (1 : ℝ) < (B : ℝ) := by exact_mod_cast (show 1 < B by omega) have htarget : (1 : ℝ) < (B : ℝ) ^ taoSection3AmbientRatio := Real.one_lt_rpow hBone taoSection3AmbientRatio_pos obtain ⟨i, hi⟩ := ((tendsto_taoSection3AmbientBoundary_atTop hX).eventually_lt_const htarget).exists exact ⟨i, hi.le⟩ /-- The first weak crossing of `B^(1 / alpha)` by the descending ambient boundaries. The index is exported as `K = j + 2`, avoiding natural subtraction in the subsequent trace. All strict and weak inequalities are sharp: equality can occur at each displayed upper endpoint. -/ theorem exists_taoSection3AmbientLeastCrossing {B : ℕ} {X : ℝ} (hB : 2 ≤ B) (hBX : (B : ℝ) < X) : ∃ K j : ℕ, K = j + 2 ∧ (∀ i, i < K → (B : ℝ) ^ taoSection3AmbientRatio < taoSection3AmbientBoundary X i) ∧ (B : ℝ) ^ (taoSection3AmbientRatio ^ 2) < taoSection3AmbientBoundary X K ∧ taoSection3AmbientBoundary X K ≤ (B : ℝ) ^ taoSection3AmbientRatio ∧ (B : ℝ) ^ taoSection3AmbientRatio < taoSection3AmbientBoundary X (j + 1) ∧ taoSection3AmbientBoundary X (j + 1) ≤ (B : ℝ) ∧ (B : ℝ) ^ (taoSection3AmbientRatio ^ 3) < taoSection3AmbientBoundary X (K + 1) ∧ taoSection3AmbientBoundary X (K + 1) ≤ (B : ℝ) ^ (taoSection3AmbientRatio ^ 2) := by classical have hBcast_nonneg : (0 : ℝ) ≤ (B : ℝ) := Nat.cast_nonneg B have hBone : (1 : ℝ) < (B : ℝ) := by exact_mod_cast (show 1 < B by omega) have hXone : (1 : ℝ) < X := hBone.trans hBX have hXpos : (0 : ℝ) < X := zero_lt_one.trans hXone have hXnonneg : (0 : ℝ) ≤ X := hXpos.le let P : ℕ → Prop := fun i => taoSection3AmbientBoundary X i ≤ (B : ℝ) ^ taoSection3AmbientRatio have hex : ∃ i : ℕ, P i := exists_taoSection3AmbientBoundary_le hB hXpos let K : ℕ := Nat.find hex have hKcross : P K := Nat.find_spec hex have hbefore : ∀ i, i < K → (B : ℝ) ^ taoSection3AmbientRatio < taoSection3AmbientBoundary X i := by intro i hi exact lt_of_not_ge (Nat.find_min hex hi) have hKtwo : 2 ≤ K := by apply (Nat.le_find_iff hex 2).2 intro i hi hPi have hi_cases : i = 0 ∨ i = 1 := by omega rcases hi_cases with rfl | rfl · have htarget_lt_B : (B : ℝ) ^ taoSection3AmbientRatio < (B : ℝ) := Real.rpow_lt_self_of_one_lt hBone taoSection3AmbientRatio_lt_one have hPzero : X ≤ (B : ℝ) ^ taoSection3AmbientRatio := by simpa only [P, taoSection3AmbientBoundary, pow_zero, Real.rpow_one] using hPi linarith · have htarget_lt_source : (B : ℝ) ^ taoSection3AmbientRatio < X ^ taoSection3AmbientRatio := Real.rpow_lt_rpow hBcast_nonneg hBX taoSection3AmbientRatio_pos have hPone : X ^ taoSection3AmbientRatio ≤ (B : ℝ) ^ taoSection3AmbientRatio := by simpa only [P, taoSection3AmbientBoundary, pow_one] using hPi exact (not_le_of_gt htarget_lt_source) hPone obtain ⟨j, hKj⟩ := Nat.exists_eq_add_of_le' hKtwo have hpred_lt : j + 1 < K := by omega have hpred_lower : (B : ℝ) ^ taoSection3AmbientRatio < taoSection3AmbientBoundary X (j + 1) := hbefore (j + 1) hpred_lt have hcross_lower : (B : ℝ) ^ (taoSection3AmbientRatio ^ 2) < taoSection3AmbientBoundary X K := by have hraised := Real.rpow_lt_rpow (Real.rpow_nonneg hBcast_nonneg taoSection3AmbientRatio) hpred_lower taoSection3AmbientRatio_pos calc (B : ℝ) ^ (taoSection3AmbientRatio ^ 2) = ((B : ℝ) ^ taoSection3AmbientRatio) ^ taoSection3AmbientRatio := by rw [pow_two, Real.rpow_mul hBcast_nonneg] _ < (taoSection3AmbientBoundary X (j + 1)) ^ taoSection3AmbientRatio := hraised _ = taoSection3AmbientBoundary X (j + 2) := by symm simpa only [taoSection3AmbientRatio] using taoSection3AmbientBoundary_succ_eq_rpow_inv hXnonneg (j + 1) _ = taoSection3AmbientBoundary X K := by rw [hKj] have hpred_upper : taoSection3AmbientBoundary X (j + 1) ≤ (B : ℝ) := by have hraised := Real.rpow_le_rpow (taoSection3AmbientBoundary_nonneg hXnonneg K) hKcross taoAlpha_pos.le calc taoSection3AmbientBoundary X (j + 1) = (taoSection3AmbientBoundary X (j + 2)) ^ taoAlpha := by symm exact taoSection3AmbientBoundary_succ_rpow hXnonneg (j + 1) _ = (taoSection3AmbientBoundary X K) ^ taoAlpha := by rw [hKj] _ ≤ ((B : ℝ) ^ taoSection3AmbientRatio) ^ taoAlpha := hraised _ = (B : ℝ) := by simpa only [taoSection3AmbientRatio] using Real.rpow_inv_rpow hBcast_nonneg taoAlpha_pos.ne' have hnext_lower : (B : ℝ) ^ (taoSection3AmbientRatio ^ 3) < taoSection3AmbientBoundary X (K + 1) := by have hraised := Real.rpow_lt_rpow (Real.rpow_nonneg hBcast_nonneg (taoSection3AmbientRatio ^ 2)) hcross_lower taoSection3AmbientRatio_pos calc (B : ℝ) ^ (taoSection3AmbientRatio ^ 3) = ((B : ℝ) ^ (taoSection3AmbientRatio ^ 2)) ^ taoSection3AmbientRatio := by rw [show taoSection3AmbientRatio ^ 3 = taoSection3AmbientRatio ^ 2 * taoSection3AmbientRatio by ring, Real.rpow_mul hBcast_nonneg] _ < (taoSection3AmbientBoundary X K) ^ taoSection3AmbientRatio := hraised _ = taoSection3AmbientBoundary X (K + 1) := by symm simpa only [taoSection3AmbientRatio] using taoSection3AmbientBoundary_succ_eq_rpow_inv hXnonneg K have hnext_upper : taoSection3AmbientBoundary X (K + 1) ≤ (B : ℝ) ^ (taoSection3AmbientRatio ^ 2) := by have hraised := Real.rpow_le_rpow (taoSection3AmbientBoundary_nonneg hXnonneg K) hKcross taoSection3AmbientRatio_pos.le calc taoSection3AmbientBoundary X (K + 1) = (taoSection3AmbientBoundary X K) ^ taoSection3AmbientRatio := by simpa only [taoSection3AmbientRatio] using taoSection3AmbientBoundary_succ_eq_rpow_inv hXnonneg K _ ≤ ((B : ℝ) ^ taoSection3AmbientRatio) ^ taoSection3AmbientRatio := hraised _ = (B : ℝ) ^ (taoSection3AmbientRatio ^ 2) := by rw [pow_two, Real.rpow_mul hBcast_nonneg] exact ⟨K, j, hKj, hbefore, hcross_lower, hKcross, hpred_lower, hpred_upper, hnext_lower, hnext_upper⟩ /-- Consumer-facing compression of the least-crossing packet. The subsequent trace only needs this subtraction-free `j`-indexed cut, not the private `Nat.find` characterization. -/ theorem exists_taoSection3AmbientTraceCut {B : ℕ} {X : ℝ} (hB : 2 ≤ B) (hBX : (B : ℝ) < X) : ∃ j : ℕ, (B : ℝ) < taoSection3AmbientBoundary X j ∧ (B : ℝ) ^ taoSection3AmbientRatio < taoSection3AmbientBoundary X (j + 1) ∧ taoSection3AmbientBoundary X (j + 1) ≤ (B : ℝ) ∧ (B : ℝ) ^ (taoSection3AmbientRatio ^ 2) < taoSection3AmbientBoundary X (j + 2) ∧ taoSection3AmbientBoundary X (j + 2) ≤ (B : ℝ) ^ taoSection3AmbientRatio ∧ (B : ℝ) ^ (taoSection3AmbientRatio ^ 3) < taoSection3AmbientBoundary X (j + 3) ∧ taoSection3AmbientBoundary X (j + 3) ≤ (B : ℝ) ^ (taoSection3AmbientRatio ^ 2) := by rcases exists_taoSection3AmbientLeastCrossing hB hBX with ⟨K, j, hKj, _hbefore, hcrossLower, hcrossUpper, hpredLower, hpredUpper, hnextLower, hnextUpper⟩ have hBcast_nonneg : (0 : ℝ) ≤ (B : ℝ) := Nat.cast_nonneg B have hXnonneg : (0 : ℝ) ≤ X := hBcast_nonneg.trans hBX.le have hleft : (B : ℝ) < taoSection3AmbientBoundary X j := by have hraised := Real.rpow_lt_rpow (Real.rpow_nonneg hBcast_nonneg taoSection3AmbientRatio) hpredLower taoAlpha_pos calc (B : ℝ) = ((B : ℝ) ^ taoSection3AmbientRatio) ^ taoAlpha := by symm simpa only [taoSection3AmbientRatio] using Real.rpow_inv_rpow hBcast_nonneg taoAlpha_pos.ne' _ < (taoSection3AmbientBoundary X (j + 1)) ^ taoAlpha := hraised _ = taoSection3AmbientBoundary X j := taoSection3AmbientBoundary_succ_rpow hXnonneg j refine ⟨j, hleft, hpredLower, hpredUpper, ?_, ?_, ?_, ?_⟩ · simpa only [hKj] using hcrossLower · simpa only [hKj] using hcrossUpper · simpa only [hKj, Nat.add_assoc, Nat.reduceAdd] using hnextLower · simpa only [hKj, Nat.add_assoc, Nat.reduceAdd] using hnextUpper end end Tao end Erdos1135