import Erdos1135.Tao.Section6.GlobalTypicalEvent import Erdos1135.Tao.Section6.HeadGateIndex import Erdos1135.Tao.Section6.HeadGateLocalization /-! # Section 6 First-Crossing Partition This deterministic leaf extracts Tao's unique first prefix crossing from the repaired global event. For each fixed `CA ≥ 17`, sufficiently large globally typical sources produce a unique pair `k (a : ℕ)) k (by simpa using hk) have hpos : 0 < ((full[k] : ℕ+) : ℕ) := full[k].property have hweight : taoTupleWeight (full.take (k + 1)) = taoTupleWeight (full.take k) + ((full[k] : ℕ+) : ℕ) := by simpa [taoTupleWeight, ← List.map_eq_flatMap, List.map_take] using hsum omega /-- Prefix weights are monotone in the prefix length. -/ theorem taoTupleWeight_take_mono (full : List ℕ+) {i j : ℕ} (hij : i ≤ j) : taoTupleWeight (full.take i) ≤ taoTupleWeight (full.take j) := by have h := taoTupleWeight_take_le (full.take j) i simpa [List.take_take, Nat.min_eq_left hij] using h /-- Equality at the lower endpoint belongs to the later crossing, as required by Tao's weak/strict convention. -/ theorem taoSection6PrefixCrossing_at_prefix_weight (full : List ℕ+) {k : ℕ} (hk : k < full.length) : taoSection6PrefixCrossing (((taoTupleWeight (full.take k) : ℕ) : ℝ)) full k := by refine ⟨hk, le_rfl, ?_⟩ exact_mod_cast taoTupleWeight_take_lt_take_succ full hk /-- `k=0` crossing canary, including a one-coordinate source. -/ theorem taoSection6PrefixCrossing_zero (full : List ℕ+) (hfull : full ≠ []) : taoSection6PrefixCrossing 0 full 0 := by have hlength : 0 < full.length := by cases full with | nil => exact (hfull rfl).elim | cons a as => simp simpa [taoTupleWeight] using taoSection6PrefixCrossing_at_prefix_weight full hlength /-- Terminal crossing canary `k=length-1`, hence later tail length zero. -/ theorem taoSection6PrefixCrossing_terminal_at_prefix_weight (full : List ℕ+) (hfull : full ≠ []) : taoSection6PrefixCrossing (((taoTupleWeight (full.take (full.length - 1)) : ℕ) : ℝ)) full (full.length - 1) := by have hlength : 0 < full.length := by cases full with | nil => exact (hfull rfl).elim | cons a as => simp apply taoSection6PrefixCrossing_at_prefix_weight omega /-- Two crossings of the same source and threshold have the same index. This does not require separate endpoint hypotheses once both crossings exist. -/ theorem taoSection6PrefixCrossing.eq {q : ℝ} {full : List ℕ+} {k₁ k₂ : ℕ} (h₁ : taoSection6PrefixCrossing q full k₁) (h₂ : taoSection6PrefixCrossing q full k₂) : k₁ = k₂ := by by_cases hlt : k₁ < k₂ · have hmono : taoTupleWeight (full.take (k₁ + 1)) ≤ taoTupleWeight (full.take k₂) := taoTupleWeight_take_mono full (Nat.succ_le_of_lt hlt) have hmono_real : ((taoTupleWeight (full.take (k₁ + 1)) : ℕ) : ℝ) ≤ ((taoTupleWeight (full.take k₂) : ℕ) : ℝ) := by exact_mod_cast hmono linarith [h₁.2.2, h₂.2.1] · by_cases hgt : k₂ < k₁ · have hmono : taoTupleWeight (full.take (k₂ + 1)) ≤ taoTupleWeight (full.take k₁) := taoTupleWeight_take_mono full (Nat.succ_le_of_lt hgt) have hmono_real : ((taoTupleWeight (full.take (k₂ + 1)) : ℕ) : ℝ) ≤ ((taoTupleWeight (full.take k₁) : ℕ) : ℝ) := by exact_mod_cast hmono linarith [h₂.2.2, h₁.2.1] · omega /-- A nonnegative real threshold below total weight has one unique prefix crossing. The proof selects the least prefix whose successor exceeds `q`. -/ theorem existsUnique_taoSection6PrefixCrossing {q : ℝ} {full : List ℕ+} (hq0 : 0 ≤ q) (hqfull : q < ((taoTupleWeight full : ℕ) : ℝ)) : ∃! k : ℕ, taoSection6PrefixCrossing q full k := by classical have hfull : full ≠ [] := by intro hnil have hbad : q < 0 := by simpa [hnil, taoTupleWeight] using hqfull exact (not_lt_of_ge hq0) hbad have hlength : 0 < full.length := by cases full with | nil => exact (hfull rfl).elim | cons a as => simp let P : ℕ → Prop := fun k => k < full.length ∧ q < ((taoTupleWeight (full.take (k + 1)) : ℕ) : ℝ) have hex : ∃ k, P k := by refine ⟨full.length - 1, ?_, ?_⟩ · exact Nat.sub_lt hlength (by omega) · have hlast : full.length - 1 + 1 = full.length := by omega simpa [P, hlast] using hqfull let K := Nat.find hex have hKspec : P K := Nat.find_spec hex have hKlower : ((taoTupleWeight (full.take K) : ℕ) : ℝ) ≤ q := by by_cases hK0 : K = 0 · rw [hK0] simpa [taoTupleWeight] using hq0 · by_contra hnot have hlt : q < ((taoTupleWeight (full.take K) : ℕ) : ℝ) := lt_of_not_ge hnot have hpred_lt : K - 1 < K := Nat.sub_lt (Nat.pos_of_ne_zero hK0) (by omega) have hpred : P (K - 1) := by refine ⟨hpred_lt.trans hKspec.1, ?_⟩ have hsucc : K - 1 + 1 = K := by omega simpa [hsucc] using hlt exact (Nat.find_min hex hpred_lt) hpred refine ⟨K, ⟨hKspec.1, hKlower, hKspec.2⟩, ?_⟩ intro k hk by_cases hkK : k < K · exact ((Nat.find_min hex hkK) ⟨hk.1, hk.2.2⟩).elim · by_cases hKk : K < k · have hmono : taoTupleWeight (full.take (K + 1)) ≤ taoTupleWeight (full.take k) := taoTupleWeight_take_mono full (Nat.succ_le_of_lt hKk) have hmono_real : ((taoTupleWeight (full.take (K + 1)) : ℕ) : ℝ) ≤ ((taoTupleWeight (full.take k) : ℕ) : ℝ) := by exact_mod_cast hmono linarith [hKspec.2, hk.2.1] · omega /-- Elementary lower bound for Tao's logarithmic slope. -/ theorem taoSection6_one_lt_log_three_div_log_two : (1 : ℝ) < Real.log 3 / Real.log 2 := by apply (lt_div_iff₀ taoCor63_log_two_pos).2 simpa using (Real.log_lt_log (by norm_num : (0 : ℝ) < 2) (by norm_num : (2 : ℝ) < 3)) /-- Pointwise global crossing once the logarithmic error has entered a fixed linear budget. -/ theorem taoSection6GlobalTypical_starQ_crossing_of_log_budget {CA : ℝ} {n : ℕ} (hCA : 17 ≤ CA) (hn : 2 ≤ n) (hbudget : ((5 / 2 : ℝ) * CA ^ 2 + CA) * Real.log (n : ℝ) ≤ (3 / 10 : ℝ) * (n : ℝ)) {full : List ℕ+} (hglobal : taoSection6GlobalTypical CA n full) : 0 ≤ taoCor63StarQ CA n ∧ taoCor63StarQ CA n < ((taoTupleWeight full : ℕ) : ℝ) := by have hnreal : (0 : ℝ) < (n : ℝ) := by positivity have hlog : 0 < Real.log (n : ℝ) := Real.log_pos (by exact_mod_cast (show 1 < n by omega)) have hlog0 : 0 ≤ Real.log (n : ℝ) := hlog.le have hCA0 : 0 ≤ CA := by linarith have hcoeff : CA ^ 2 ≤ (5 / 2 : ℝ) * CA ^ 2 + CA := by nlinarith [sq_nonneg CA] have hQbudget : CA ^ 2 * Real.log (n : ℝ) ≤ (n : ℝ) := by have hmul := mul_le_mul_of_nonneg_right hcoeff hlog0 nlinarith [hmul, hbudget] have hratio_lower := mul_lt_mul_of_pos_left taoSection6_one_lt_log_three_div_log_two hnreal have hQ0 : 0 ≤ taoCor63StarQ CA n := by unfold taoCor63StarQ nlinarith have hx : 0 ≤ (n : ℝ) * Real.log (n : ℝ) := by positivity have hepsilon : 0 < (1 : ℝ) / (5 * Real.log (n : ℝ)) := by positivity have hyoung := young_sqrt_le_epsilon (A := CA) (ε := (1 : ℝ) / (5 * Real.log (n : ℝ))) (x := (n : ℝ) * Real.log (n : ℝ)) hepsilon hx have hyoung' : CA * Real.sqrt ((n : ℝ) * Real.log (n : ℝ)) ≤ (n : ℝ) / 10 + (5 / 2 : ℝ) * CA ^ 2 * Real.log (n : ℝ) := by calc CA * Real.sqrt ((n : ℝ) * Real.log (n : ℝ)) ≤ (((1 : ℝ) / (5 * Real.log (n : ℝ))) / 2) * ((n : ℝ) * Real.log (n : ℝ)) + CA ^ 2 / (2 * ((1 : ℝ) / (5 * Real.log (n : ℝ)))) := hyoung _ = (n : ℝ) / 10 + (5 / 2 : ℝ) * CA ^ 2 * Real.log (n : ℝ) := by field_simp [hlog.ne'] ring have herror : CA * (Real.sqrt ((n : ℝ) * Real.log (n : ℝ)) + Real.log (n : ℝ)) ≤ (2 / 5 : ℝ) * (n : ℝ) := by nlinarith [hyoung', hbudget] have htotal := taoCor63StarInclusiveTypical_total_lower (taoSection6GlobalTypical_typical hglobal) (by rw [taoSection6GlobalTypical_length hglobal] omega) rw [taoSection6GlobalTypical_length hglobal] at htotal unfold taoCor63PrefixError at htotal have hweight : (8 / 5 : ℝ) * (n : ℝ) ≤ ((taoTupleWeight full : ℕ) : ℝ) := by nlinarith [htotal, herror] have hratio_upper := mul_lt_mul_of_pos_left taoSection6_log_three_div_log_two_lt_eight_fifths hnreal have hQupper : taoCor63StarQ CA n < (8 / 5 : ℝ) * (n : ℝ) := by have hsub : 0 ≤ CA ^ 2 * Real.log (n : ℝ) := mul_nonneg (sq_nonneg _) hlog0 unfold taoCor63StarQ nlinarith exact ⟨hQ0, hQupper.trans_le hweight⟩ /-- For fixed `CA ≥ 17`, every sufficiently large globally typical source has its strict first crossing inside the source horizon. -/ theorem exists_taoSection6GlobalTypical_starQ_crossing (CA : ℝ) (hCA : 17 ≤ CA) : ∃ Ncross : ℕ, ∀ n : ℕ, Ncross ≤ n → ∀ full : List ℕ+, taoSection6GlobalTypical CA n full → 0 ≤ taoCor63StarQ CA n ∧ taoCor63StarQ CA n < ((taoTupleWeight full : ℕ) : ℝ) := by let D : ℝ := (5 / 2 : ℝ) * CA ^ 2 + CA have hD : 0 < D := by dsimp [D] nlinarith [sq_nonneg CA] obtain ⟨Nbudget, hNbudget⟩ := exists_taoSection6_const_mul_log_le D (3 / 10 : ℝ) hD (by norm_num) refine ⟨max Nbudget 2, ?_⟩ intro n hn full hglobal apply taoSection6GlobalTypical_starQ_crossing_of_log_budget hCA · exact (le_max_right _ _).trans hn · simpa [D] using hNbudget n ((le_max_left _ _).trans hn) · exact hglobal /-- A global crossing restricted to its successor prefix is exactly the local head gate, with `l` equal to total head weight. -/ theorem taoSection6HeadGate_of_global_prefixCrossing {CA : ℝ} {n k : ℕ} {full : List ℕ+} (hglobal : taoSection6GlobalTypical CA n full) (hcross : taoSection6PrefixCrossing (taoCor63StarQ CA n) full k) : taoSection6HeadGate CA n k (taoTupleWeight (full.take (k + 1))) (full.take (k + 1)) := by refine ⟨?_, ?_, ?_, ?_, rfl⟩ · have hkN : k < n := by simpa [taoSection6GlobalTypical_length hglobal] using hcross.1 exact taoSection6GlobalTypical_take_length hglobal (Nat.succ_le_of_lt hkN) · exact taoSection6GlobalTypical_take_typical hglobal · simpa [List.take_take, Nat.min_eq_left (Nat.le_succ k)] using hcross.2.1 · exact hcross.2.2 /-- Every sufficiently large local head gate lies in the crude weight box `l<2n`. -/ theorem taoSection6HeadGate_weight_lt_two_mul {CA : ℝ} {n k l : ℕ} {head : List ℕ+} (hCA : 17 ≤ CA) (hn : 1 ≤ n) (hlog : 4 ≤ Real.log (n : ℝ)) (hgate : taoSection6HeadGate CA n k l head) : l < 2 * n := by have hrange := taoSection6HeadGate_starLRange_of_log_ge_four hCA hlog hgate have hlog0 : 0 ≤ Real.log (n : ℝ) := by linarith have hcoeff : -CA ^ 2 + 2 * CA ≤ 0 := by nlinarith have hremainder : (-CA ^ 2 + 2 * CA) * Real.log (n : ℝ) ≤ 0 := mul_nonpos_of_nonpos_of_nonneg hcoeff hlog0 have hnreal : (0 : ℝ) < (n : ℝ) := by exact_mod_cast hn have hratio := mul_lt_mul_of_pos_left taoSection6_log_three_div_log_two_lt_eight_fifths hnreal have hlreal : (l : ℝ) < 2 * (n : ℝ) := by unfold taoCor63StarLRange taoCor63StarQ at hrange nlinarith [hrange.2, hratio, hremainder] exact_mod_cast hlreal /-- Eventual unique bounded `(k,l)` partition datum produced by the repaired global event. -/ theorem exists_taoSection6GlobalTypical_unique_bounded_headGate (CA : ℝ) (hCA : 17 ≤ CA) : ∃ N0 : ℕ, ∀ n : ℕ, N0 ≤ n → ∀ full : List ℕ+, taoSection6GlobalTypical CA n full → ∃! kl : ℕ × ℕ, kl.1 < n ∧ kl.2 < 2 * n ∧ taoSection6HeadGate CA n kl.1 kl.2 (full.take (kl.1 + 1)) := by obtain ⟨Ncross, hNcross⟩ := exists_taoSection6GlobalTypical_starQ_crossing CA hCA obtain ⟨Nlog, hNlog⟩ := taoCor63_exists_nat_log_gt 4 refine ⟨max Ncross (max Nlog 2), ?_⟩ intro n hn full hglobal have hncross : Ncross ≤ n := (le_max_left _ _).trans hn have hnlog : Nlog ≤ n := (le_max_left _ _).trans ((le_max_right Ncross (max Nlog 2)).trans hn) have hn2 : 2 ≤ n := (le_max_right Nlog 2).trans ((le_max_right Ncross (max Nlog 2)).trans hn) have hQ := hNcross n hncross full hglobal obtain ⟨k, hk, hkUnique⟩ := existsUnique_taoSection6PrefixCrossing hQ.1 hQ.2 let l := taoTupleWeight (full.take (k + 1)) have hgate : taoSection6HeadGate CA n k l (full.take (k + 1)) := by simpa [l] using taoSection6HeadGate_of_global_prefixCrossing hglobal hk have hl : l < 2 * n := taoSection6HeadGate_weight_lt_two_mul hCA (by omega) (le_of_lt (hNlog n hnlog)) hgate refine ⟨(k, l), ⟨by simpa [taoSection6GlobalTypical_length hglobal] using hk.1, hl, hgate⟩, ?_⟩ rintro ⟨k', l'⟩ hkl have hcross' : taoSection6PrefixCrossing (taoCor63StarQ CA n) full k' := by have hgate' := hkl.2.2 have hgateCross := taoSection6HeadGate_crossing hgate' refine ⟨by simpa [taoSection6GlobalTypical_length hglobal] using hkl.1, ?_, ?_⟩ · simpa [List.take_take, Nat.min_eq_left (Nat.le_succ k')] using hgateCross.1 · exact hgateCross.2 have hkEq : k' = k := hkUnique k' hcross' subst k' have hlEq : l' = l := by have hw := taoSection6HeadGate_weight hkl.2.2 simpa [l] using hw.symm subst l' rfl end end Tao end Erdos1135