import Erdos1135.Tao.Section5.PassScheduleFacts /-! # Section 5 Repaired Power-Interior Facts This leaf proves the deterministic geometry of the repaired source interior `[y^(1+delta), y^(alpha-delta)]`. It establishes rounding-safe odd nonemptiness and inclusion in the original branch source window. Logarithmic mass loss and trajectory localization remain separate probability consumers. -/ namespace Erdos1135 namespace Tao open Filter open scoped Topology noncomputable section theorem one_le_taoSection5BranchExponent (branch : TaoSection5SourceBranch) : 1 ≤ taoSection5BranchExponent branch := by cases branch <;> norm_num [taoSection5BranchExponent, taoAlpha] private theorem eventually_taoSection5PowerInteriorDelta_lt : ∀ᶠ B : ℕ in atTop, taoSection5PowerInteriorDelta B < (1 / 4000 : ℝ) := by have hlog : Tendsto (fun B : ℕ => Real.log (B : ℝ)) atTop atTop := Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop have hdelta : Tendsto (fun B : ℕ => (Real.log B) ^ (-(1 / 10 : ℝ))) atTop (𝓝 0) := by simpa only [Function.comp_def] using (tendsto_rpow_neg_atTop (by norm_num : 0 < (1 / 10 : ℝ))).comp hlog filter_upwards [hdelta.eventually_lt_const (by norm_num : (0 : ℝ) < 1 / 4000)] with B hB unfold taoSection5PowerInteriorDelta simpa only [show (-(1 / 10 : ℝ)) = (-1 / 10 : ℝ) by norm_num] using hB private theorem eventually_three_le_threshold_rpow_one_div_two_thousand : ∀ᶠ B : ℕ in atTop, (3 : ℝ) ≤ (B : ℝ) ^ (1 / 2000 : ℝ) := by have htendsto : Tendsto (fun B : ℕ => (B : ℝ) ^ (1 / 2000 : ℝ)) atTop atTop := by simpa only [Function.comp_def] using (tendsto_rpow_atTop (by norm_num : 0 < (1 / 2000 : ℝ))).comp tendsto_natCast_atTop_atTop exact htendsto.eventually_ge_atTop 3 /-- Eventual deterministic geometry of the repaired branch interiors. -/ structure TaoSection5PowerInteriorFacts (B : ℕ) : Prop where schedule : TaoSection5PassScheduleFacts B delta_pos : 0 < taoSection5PowerInteriorDelta B delta_lt : taoSection5PowerInteriorDelta B < (1 / 4000 : ℝ) source_le_lower : ∀ branch, taoSection5SourceY B branch ≤ taoSection5PowerInteriorLower B branch upper_le_source : ∀ branch, taoSection5PowerInteriorUpper B branch ≤ Real.rpow (taoSection5SourceY B branch) taoAlpha width : ∀ branch, taoSection5PowerInteriorLower B branch + 2 ≤ taoSection5PowerInteriorUpper B branch endpoint_room : ∀ branch, Nat.ceil (taoSection5PowerInteriorLower B branch) + 1 ≤ Nat.floor (taoSection5PowerInteriorUpper B branch) nonempty : ∀ branch, (taoSection5PowerInterior B branch).Nonempty mass_pos : ∀ branch, 0 < logFinsetMass (taoSection5PowerInterior B branch) subset_source : ∀ branch, taoSection5PowerInterior B branch ⊆ oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch) theorem eventually_taoSection5PowerInteriorFacts : ∀ᶠ B : ℕ in atTop, TaoSection5PowerInteriorFacts B := by filter_upwards [eventually_taoSection5PassScheduleFacts, eventually_taoSection5PowerInteriorDelta_lt, eventually_three_le_threshold_rpow_one_div_two_thousand] with B schedule hdeltaSmall hBgrowth have hBne : B ≠ 1 := by intro hB subst B norm_num at hBgrowth have hBgt : 1 < B := lt_of_le_of_ne schedule.one_le_B hBne.symm have hBcast : (1 : ℝ) ≤ B := by exact_mod_cast hBgt.le have hBpos : (0 : ℝ) < B := zero_lt_one.trans_le hBcast have hlogPos : 0 < Real.log B := Real.log_pos (by exact_mod_cast hBgt) have hdeltaPos : 0 < taoSection5PowerInteriorDelta B := by unfold taoSection5PowerInteriorDelta exact Real.rpow_pos_of_pos hlogPos _ have hgap : (1 / 2000 : ℝ) < taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B := by have halpha : taoAlpha - 1 = (1 / 1000 : ℝ) := by norm_num [taoAlpha] rw [halpha] nlinarith have hsourceOne (branch : TaoSection5SourceBranch) : 1 ≤ taoSection5SourceY B branch := by rw [taoSection5SourceY_eq_branch_rpow] exact Real.one_le_rpow hBcast (zero_le_one.trans (one_le_taoSection5BranchExponent branch)) have hsourcePos (branch : TaoSection5SourceBranch) : 0 < taoSection5SourceY B branch := zero_lt_one.trans_le (hsourceOne branch) have hsourceGe (branch : TaoSection5SourceBranch) : (B : ℝ) ≤ taoSection5SourceY B branch := by rw [taoSection5SourceY_eq_branch_rpow] exact Real.self_le_rpow_of_one_le hBcast (one_le_taoSection5BranchExponent branch) have hlowerOne (branch : TaoSection5SourceBranch) : 1 ≤ taoSection5PowerInteriorLower B branch := by unfold taoSection5PowerInteriorLower exact Real.one_le_rpow (hsourceOne branch) (by linarith) have hlowerNonneg (branch : TaoSection5SourceBranch) : 0 ≤ taoSection5PowerInteriorLower B branch := zero_le_one.trans (hlowerOne branch) have hsourceLower (branch : TaoSection5SourceBranch) : taoSection5SourceY B branch ≤ taoSection5PowerInteriorLower B branch := by unfold taoSection5PowerInteriorLower exact Real.self_le_rpow_of_one_le (hsourceOne branch) (by linarith) have hupperSource (branch : TaoSection5SourceBranch) : taoSection5PowerInteriorUpper B branch ≤ Real.rpow (taoSection5SourceY B branch) taoAlpha := by unfold taoSection5PowerInteriorUpper exact Real.rpow_le_rpow_of_exponent_le (hsourceOne branch) (by linarith) have hfactor (branch : TaoSection5SourceBranch) : (3 : ℝ) ≤ Real.rpow (taoSection5SourceY B branch) (taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B) := by have hBExponent : (B : ℝ) ^ (1 / 2000 : ℝ) ≤ (B : ℝ) ^ (taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B) := Real.rpow_le_rpow_of_exponent_le hBcast hgap.le have hbase : (B : ℝ) ^ (taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B) ≤ (taoSection5SourceY B branch) ^ (taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B) := Real.rpow_le_rpow (Nat.cast_nonneg B) (hsourceGe branch) ((by norm_num : (0 : ℝ) ≤ 1 / 2000).trans hgap.le) exact hBgrowth.trans (hBExponent.trans hbase) have hfactorization (branch : TaoSection5SourceBranch) : taoSection5PowerInteriorUpper B branch = taoSection5PowerInteriorLower B branch * Real.rpow (taoSection5SourceY B branch) (taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B) := by unfold taoSection5PowerInteriorUpper taoSection5PowerInteriorLower have hadd := Real.rpow_add (hsourcePos branch) (1 + taoSection5PowerInteriorDelta B) (taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B) calc Real.rpow (taoSection5SourceY B branch) (taoAlpha - taoSection5PowerInteriorDelta B) = Real.rpow (taoSection5SourceY B branch) ((1 + taoSection5PowerInteriorDelta B) + (taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B)) := by congr 2 ring _ = Real.rpow (taoSection5SourceY B branch) (1 + taoSection5PowerInteriorDelta B) * Real.rpow (taoSection5SourceY B branch) (taoAlpha - 1 - 2 * taoSection5PowerInteriorDelta B) := hadd have hwidth (branch : TaoSection5SourceBranch) : taoSection5PowerInteriorLower B branch + 2 ≤ taoSection5PowerInteriorUpper B branch := by have hmul := mul_le_mul_of_nonneg_left (hfactor branch) (hlowerNonneg branch) have hthree : taoSection5PowerInteriorLower B branch + 2 ≤ taoSection5PowerInteriorLower B branch * 3 := by nlinarith [hlowerOne branch] rw [hfactorization branch] exact hthree.trans hmul have hroom (branch : TaoSection5SourceBranch) : Nat.ceil (taoSection5PowerInteriorLower B branch) + 1 ≤ Nat.floor (taoSection5PowerInteriorUpper B branch) := by have hnonempty := oddLogWindow_ceil_floor_nonempty_of_add_two_le (hlowerNonneg branch) (hwidth branch) have hceil := Nat.ceil_lt_add_one (hlowerNonneg branch) apply Nat.le_floor norm_num at hceil ⊢ linarith [hwidth branch] have hnonempty (branch : TaoSection5SourceBranch) : (taoSection5PowerInterior B branch).Nonempty := by unfold taoSection5PowerInterior exact oddLogWindow_nonempty_of_succ_le (hroom branch) have hmass (branch : TaoSection5SourceBranch) : 0 < logFinsetMass (taoSection5PowerInterior B branch) := by unfold taoSection5PowerInterior exact logFinsetMass_oddLogWindow_pos_of_succ_le (hroom branch) have hsubset (branch : TaoSection5SourceBranch) : taoSection5PowerInterior B branch ⊆ oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch) := by unfold taoSection5PowerInterior taoSection5SourceLo taoSection5SourceHi taoNyLo taoNyHi apply oddLogWindow_mono · exact Nat.ceil_le_ceil (hsourceLower branch) · exact Nat.floor_mono (hupperSource branch) exact { schedule := schedule delta_pos := hdeltaPos delta_lt := hdeltaSmall source_le_lower := hsourceLower upper_le_source := hupperSource width := hwidth endpoint_room := hroom nonempty := hnonempty mass_pos := hmass subset_source := hsubset } end end Tao end Erdos1135