import Erdos1135.ND.Discrepancy.A5ReferenceSection3AmbientTerminal /-! # All-source A5 `T^*`-to-`L^*` no-hit estimate This leaf exposes the Section 3 terminal trace in the source-facing odd logarithmic block. It absorbs the finitely many targets below the ambient cutoff into one constant and handles sources whose upper endpoint is already below the target at iterate zero. -/ namespace Erdos1135 namespace ND noncomputable section /-- The natural no-hit probability in a Tao `N_y` window is the same PMF probability in the ND `oddBlock` convention. -/ private theorem syracuseNoHitWindowProb_taoNy_eq_pmfProb_logOddBlockPMF (B : Nat) (y : Real) (hmassTao : 0 < Tao.logFinsetMass (Tao.oddLogWindow (Tao.taoNyLo y) (Tao.taoNyHi y Tao.taoAlpha))) (hmassND : 0 < Tao.logFinsetMass (oddBlock y)) : Tao.syracuseNoHitWindowProb B (Tao.taoNyLo y) (Tao.taoNyHi y Tao.taoAlpha) hmassTao = Tao.pmfProb (logOddBlockPMF y hmassND) {N | ¬ Tao.syracuseHitsAtMost N.1 B} := by unfold Tao.syracuseNoHitWindowProb Tao.syracuseNoHitAtMost Tao.oddLogWindowPMF logOddBlockPMF oddBlock Tao.taoNyOddWindow rfl /-- Starting the ambient trace at `X = y^alpha` makes its terminal boundary exactly `y`. -/ private theorem taoSection3AmbientBoundary_rpow_taoAlpha_one {y : Real} (hy : 0 <= y) : Tao.taoSection3AmbientBoundary (y ^ Tao.taoAlpha) 1 = y := by simpa only [Tao.taoSection3AmbientBoundary, Tao.taoSection3AmbientRatio, pow_one] using Real.rpow_rpow_inv hy Tao.taoAlpha_pos.ne' /-- If the upper endpoint of the source block is at most `B`, every source hits the target at iterate zero. -/ private theorem syracuseHitsAtMost_of_mem_oddBlock_of_rpow_le {B : Nat} {y : Real} (hy : 0 <= y) (hupper : y ^ Tao.taoAlpha <= (B : Real)) {N : Nat} (hN : N ∈ oddBlock y) : Tao.syracuseHitsAtMost N B := by have hNwindow : N ∈ Tao.taoNyOddWindow y Tao.taoAlpha := by simpa only [oddBlock, alpha] using hN have hNleReal : (N : Real) <= y ^ Tao.taoAlpha := ((Tao.taoNyOddWindow_mem (alpha := Tao.taoAlpha) hy).mp hNwindow).2.1 have hNB : N <= B := by exact_mod_cast hNleReal.trans hupper exact ⟨0, by simpa only [Function.iterate_zero, id_eq] using hNB⟩ /-- The fixed-target Section 3 trace, with all ambient cutoffs hidden, holds uniformly for every natural target and every positive odd logarithmic source block. The constant is selected before `N0`, `y`, and the source-mass proof. -/ theorem exists_ndA5ReferenceAllSourceTStarL {c kappa d : Real} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∃ C : Real, 0 <= C ∧ ∀ (N0 : Nat), 2 <= N0 -> ∀ (y : Real), 2 <= y -> ∀ hmass : 0 < Tao.logFinsetMass (oddBlock y), Tao.pmfProb (logOddBlockPMF y hmass) {N | N0 < syracuseOrbitMin N.1} <= C * (Real.log (N0 : Real)) ^ (-d) := by rcases exists_ndA5ReferenceSection3AmbientTerminal hPhase hd hd20 hdk with ⟨Bstar, hBstar, hCtrace, hterminal⟩ let Ctrace := ndA5ReferenceSection3AmbientTraceConstant d let Csmall := (Real.log (Bstar : Real)) ^ d let C := max Ctrace Csmall have hBstarOne : (1 : Real) < (Bstar : Real) := by exact_mod_cast (show 1 < Bstar by omega) have hlogBstarPos : 0 < Real.log (Bstar : Real) := Real.log_pos hBstarOne have hCsmall : 0 <= Csmall := Real.rpow_nonneg hlogBstarPos.le d have hCtrace' : 0 <= Ctrace := by simpa only [Ctrace] using hCtrace have hC : 0 <= C := by exact hCtrace'.trans (le_max_left Ctrace Csmall) refine ⟨C, hC, ?_⟩ intro N0 hN0 y hy hmass have hN0One : (1 : Real) < (N0 : Real) := by exact_mod_cast (show 1 < N0 by omega) have hlogN0Pos : 0 < Real.log (N0 : Real) := Real.log_pos hN0One have hrateNonneg : 0 <= (Real.log (N0 : Real)) ^ (-d) := Real.rpow_nonneg hlogN0Pos.le (-d) by_cases hsmall : N0 < Bstar · have hN0leBstar : N0 <= Bstar := hsmall.le have hcastLe : (N0 : Real) <= (Bstar : Real) := by exact_mod_cast hN0leBstar have hlogLe : Real.log (N0 : Real) <= Real.log (Bstar : Real) := Real.log_le_log (Nat.cast_pos.mpr (show 0 < N0 by omega)) hcastLe have hpowLe : (Real.log (N0 : Real)) ^ d <= (Real.log (Bstar : Real)) ^ d := Real.rpow_le_rpow hlogN0Pos.le hlogLe hd.le have hcancel : (Real.log (N0 : Real)) ^ d * (Real.log (N0 : Real)) ^ (-d) = 1 := by rw [← Real.rpow_add hlogN0Pos] norm_num have hOneSmall : 1 <= Csmall * (Real.log (N0 : Real)) ^ (-d) := by calc 1 = (Real.log (N0 : Real)) ^ d * (Real.log (N0 : Real)) ^ (-d) := hcancel.symm _ <= (Real.log (Bstar : Real)) ^ d * (Real.log (N0 : Real)) ^ (-d) := mul_le_mul_of_nonneg_right hpowLe hrateNonneg _ = Csmall * (Real.log (N0 : Real)) ^ (-d) := by rfl have hsmallToC : Csmall * (Real.log (N0 : Real)) ^ (-d) <= C * (Real.log (N0 : Real)) ^ (-d) := mul_le_mul_of_nonneg_right (le_max_right Ctrace Csmall) hrateNonneg calc Tao.pmfProb (logOddBlockPMF y hmass) {N | N0 < syracuseOrbitMin N.1} <= Tao.pmfProb (logOddBlockPMF y hmass) Set.univ := Tao.pmfProb_mono _ (Set.subset_univ _) _ = 1 := Tao.pmfProb_univ _ _ <= Csmall * (Real.log (N0 : Real)) ^ (-d) := hOneSmall _ <= C * (Real.log (N0 : Real)) ^ (-d) := hsmallToC · have hBstarN0 : Bstar <= N0 := Nat.le_of_not_gt hsmall by_cases hmain : (N0 : Real) < y ^ Tao.taoAlpha · rcases hterminal hBstarN0 hmain with ⟨hmassAmbient, hterminalBound⟩ have hy0 : 0 <= y := le_trans (by norm_num) hy have hboundary : Tao.taoSection3AmbientBoundary (y ^ Tao.taoAlpha) 1 = y := taoSection3AmbientBoundary_rpow_taoAlpha_one hy0 have hmassTao : 0 < Tao.logFinsetMass (Tao.oddLogWindow (Tao.taoNyLo y) (Tao.taoNyHi y Tao.taoAlpha)) := by simpa only [Tao.taoSection3AmbientWindow, Tao.taoNyOddWindow, hboundary] using hmassAmbient have htrace : Tao.pmfProb (logOddBlockPMF y hmass) {N | ¬ Tao.syracuseHitsAtMost N.1 N0} <= Ctrace * (Real.log (N0 : Real)) ^ (-d) := by rw [← syracuseNoHitWindowProb_taoNy_eq_pmfProb_logOddBlockPMF N0 y hmassTao hmass] simpa only [hboundary, Ctrace] using hterminalBound have htraceOrbit : Tao.pmfProb (logOddBlockPMF y hmass) {N | N0 < syracuseOrbitMin N.1} <= Ctrace * (Real.log (N0 : Real)) ^ (-d) := by simpa only [not_syracuseHitsAtMost_iff_lt_orbitMin] using htrace exact htraceOrbit.trans (mul_le_mul_of_nonneg_right (le_max_left Ctrace Csmall) hrateNonneg) · have hy0 : 0 <= y := le_trans (by norm_num) hy have hupper : y ^ Tao.taoAlpha <= (N0 : Real) := le_of_not_gt hmain have hempty : {N : {n // n ∈ oddBlock y} | N0 < syracuseOrbitMin N.1} = ∅ := by ext N simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false] intro hbad have hhit : Tao.syracuseHitsAtMost N.1 N0 := syracuseHitsAtMost_of_mem_oddBlock_of_rpow_le hy0 hupper N.property exact ((not_syracuseHitsAtMost_iff_lt_orbitMin).2 hbad) hhit rw [hempty, Tao.pmfProb_empty] exact mul_nonneg hC hrateNonneg end end ND end Erdos1135