import Erdos1135.ND.Discrepancy.A5ReferenceSection3GeometricBudget /-! # Same-witness A5 Section 3 ambient terminal trace This leaf combines one fixed facade witness, its geometric budget, the uniform forward-window mass cutoff, and the weak ambient least crossing. It exports only the terminal harmonic no-hit estimate consumed by the later all-source theorem. -/ namespace Erdos1135 namespace ND open Filter open scoped Topology BigOperators noncomputable section /-- The explicit fixed-`d` loss from the first active threshold `q_0 = b_(j+3)` to the fixed target `B`. -/ noncomputable def ndA5ReferenceSection3AmbientTraceConstant (d : Real) : Real := (960084 * (Tao.taoSection3AmbientRatio ^ 3) ^ (-d)) / (1 - Tao.taoAlpha ^ (-d)) /-- One sufficiently large natural target controls the complete active ambient trace uniformly in its top `X`. The hidden facade cutoff, scalar step cutoff, mass cutoff, crossing index, and trace length are all selected before or hidden from the public `B,X` tail. -/ theorem exists_ndA5ReferenceSection3AmbientTerminal {c kappa d : Real} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∃ Bstar : Nat, 2 <= Bstar ∧ 0 <= ndA5ReferenceSection3AmbientTraceConstant d ∧ ∀ {B : Nat}, Bstar <= B -> ∀ {X : Real}, (B : Real) < X -> ∃ hmass : 0 < Tao.logFinsetMass (Tao.taoSection3AmbientWindow X 1), Tao.syracuseNoHitWindowProb B (Tao.taoNyLo (Tao.taoSection3AmbientBoundary X 1)) (Tao.taoNyHi (Tao.taoSection3AmbientBoundary X 1) Tao.taoAlpha) hmass <= ndA5ReferenceSection3AmbientTraceConstant d * (Real.log (B : Real)) ^ (-d) := by rcases exists_ndA5ReferenceSection3ForwardGeometricPacket hPhase hd hd20 hdk with ⟨B0, Qstep, _hB0, hQstep, _hnonneg, hbounds, hgeom⟩ rcases Tao.exists_taoSection3ForwardFamilyCutoff with ⟨Ymass, hYmass, hfamily⟩ have hratioLt : Tao.taoAlpha ^ (-d) < 1 := Real.rpow_lt_one_of_one_lt_of_neg Tao.taoAlpha_one_lt (neg_lt_zero.mpr hd) have hdenomPos : 0 < 1 - Tao.taoAlpha ^ (-d) := sub_pos.mpr hratioLt have hCtrace : 0 <= ndA5ReferenceSection3AmbientTraceConstant d := by unfold ndA5ReferenceSection3AmbientTraceConstant exact div_nonneg (mul_nonneg (by norm_num) (Real.rpow_nonneg (pow_nonneg Tao.taoSection3AmbientRatio_pos.le 3) _)) hdenomPos.le have hpow2 : Tendsto (fun B : Nat => (B : Real) ^ (Tao.taoSection3AmbientRatio ^ 2)) atTop atTop := by simpa only [Function.comp_def] using (tendsto_rpow_atTop (pow_pos Tao.taoSection3AmbientRatio_pos 2)).comp tendsto_natCast_atTop_atTop have hpow3 : Tendsto (fun B : Nat => (B : Real) ^ (Tao.taoSection3AmbientRatio ^ 3)) atTop atTop := by simpa only [Function.comp_def] using (tendsto_rpow_atTop (pow_pos Tao.taoSection3AmbientRatio_pos 3)).comp tendsto_natCast_atTop_atTop have hlarge : ∀ᶠ B : Nat in atTop, 2 <= B ∧ Ymass <= (B : Real) ^ (Tao.taoSection3AmbientRatio ^ 2) ∧ Qstep <= (B : Real) ^ (Tao.taoSection3AmbientRatio ^ 3) := by filter_upwards [eventually_ge_atTop (2 : Nat), hpow2.eventually_ge_atTop Ymass, hpow3.eventually_ge_atTop Qstep] with B hB hmass hstep exact ⟨hB, hmass, hstep⟩ rcases eventually_atTop.1 hlarge with ⟨Bstar, hBstar⟩ refine ⟨Bstar, (hBstar Bstar le_rfl).1, hCtrace, ?_⟩ intro B hBBstar X hBX rcases hBstar B hBBstar with ⟨hB, hmassB, hstepB⟩ rcases Tao.exists_taoSection3AmbientTraceCut hB hBX with ⟨j, _hBj, _hpredLower, hpredUpper, hseedLower, _hseedUpper, hthresholdLower, _hthresholdUpper⟩ have hBnonneg : (0 : Real) <= (B : Real) := Nat.cast_nonneg B have hXnonneg : (0 : Real) <= X := hBnonneg.trans hBX.le have hident := Tao.taoSection3AmbientForwardTrace_identifications hXnonneg j have hyMass : Ymass <= Tao.taoSection3AmbientBoundary X (j + 2) := hmassB.trans hseedLower.le have hyOne : 1 <= Tao.taoSection3AmbientBoundary X (j + 2) := hYmass.trans hyMass have hseed : Tao.taoSection3ForwardScale (Tao.taoSection3AmbientBoundary X (j + 2)) 1 <= (B : Real) := by rw [hident.1] exact hpredUpper have hpartial := Tao.taoSection3Forward_allPartial_of_realRateBounds (Jmax := j + 1) hbounds (show 1 <= B by omega) hyOne (fun m _hm => (hfamily hyMass m).2) hseed let hmassForward : 0 < Tao.logFinsetMass (Tao.taoSection3ForwardWindow (Tao.taoSection3AmbientBoundary X (j + 2)) (j + 1)) := (hfamily hyMass (j + 1)).2 have hpartialTerminal := hpartial (j + 1) le_rfl have hnoHitForward := Tao.taoSection3ForwardNoHitProb_le_sum_of_partial hmassForward hpartialTerminal have hq0Large : Qstep < Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0 := by calc Qstep <= (B : Real) ^ (Tao.taoSection3AmbientRatio ^ 3) := hstepB _ < Tao.taoSection3AmbientBoundary X (j + 3) := hthresholdLower _ = Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0 := hident.2.1.symm have hsum := hgeom hyOne hq0Large.le (j + 1) have hBpos : 0 < (B : Real) := by positivity have hBone : (1 : Real) < (B : Real) := by exact_mod_cast (show 1 < B by omega) have hlogBpos : 0 < Real.log (B : Real) := Real.log_pos hBone have hboundaryLogLower : Tao.taoSection3AmbientRatio ^ 3 * Real.log (B : Real) < Real.log (Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0) := by calc Tao.taoSection3AmbientRatio ^ 3 * Real.log (B : Real) = Real.log ((B : Real) ^ (Tao.taoSection3AmbientRatio ^ 3)) := (Real.log_rpow hBpos (Tao.taoSection3AmbientRatio ^ 3)).symm _ < Real.log (Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0) := Real.log_lt_log (Real.rpow_pos_of_pos hBpos (Tao.taoSection3AmbientRatio ^ 3)) (by calc (B : Real) ^ (Tao.taoSection3AmbientRatio ^ 3) < Tao.taoSection3AmbientBoundary X (j + 3) := hthresholdLower _ = Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0 := hident.2.1.symm) have hlogRateCompare : (Real.log (Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0)) ^ (-d) <= (Tao.taoSection3AmbientRatio ^ 3) ^ (-d) * (Real.log (B : Real)) ^ (-d) := by calc (Real.log (Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0)) ^ (-d) <= (Tao.taoSection3AmbientRatio ^ 3 * Real.log (B : Real)) ^ (-d) := Real.rpow_le_rpow_of_nonpos (mul_pos (pow_pos Tao.taoSection3AmbientRatio_pos 3) hlogBpos) hboundaryLogLower.le (neg_nonpos.mpr hd.le) _ = (Tao.taoSection3AmbientRatio ^ 3) ^ (-d) * (Real.log (B : Real)) ^ (-d) := Real.mul_rpow (pow_nonneg Tao.taoSection3AmbientRatio_pos.le 3) hlogBpos.le have hfactorNonneg : 0 <= (960084 : Real) / (1 - Tao.taoAlpha ^ (-d)) := by exact div_nonneg (by norm_num) hdenomPos.le have hsumRate : (∑ m ∈ Finset.range (j + 1), Tao.taoSection3ForwardStepError (ndA5ReferenceSection3NoHitError B0) (ndA5ReferenceSection3PassTVError B0 d) (Tao.taoSection3AmbientBoundary X (j + 2)) m) <= ndA5ReferenceSection3AmbientTraceConstant d * (Real.log (B : Real)) ^ (-d) := by refine hsum.trans ?_ calc (960084 * (Real.log (Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0)) ^ (-d)) / (1 - Tao.taoAlpha ^ (-d)) = ((960084 : Real) / (1 - Tao.taoAlpha ^ (-d))) * (Real.log (Tao.taoSection3ForwardThreshold (Tao.taoSection3AmbientBoundary X (j + 2)) 0)) ^ (-d) := by ring _ <= ((960084 : Real) / (1 - Tao.taoAlpha ^ (-d))) * ((Tao.taoSection3AmbientRatio ^ 3) ^ (-d) * (Real.log (B : Real)) ^ (-d)) := mul_le_mul_of_nonneg_left hlogRateCompare hfactorNonneg _ = ndA5ReferenceSection3AmbientTraceConstant d * (Real.log (B : Real)) ^ (-d) := by unfold ndA5ReferenceSection3AmbientTraceConstant ring have hmass : 0 < Tao.logFinsetMass (Tao.taoSection3AmbientWindow X 1) := by rw [← hident.2.2.2] exact hmassForward refine ⟨hmass, ?_⟩ have hnoHitTerminal : Tao.syracuseNoHitWindowProb B (Tao.taoNyLo (Tao.taoSection3AmbientBoundary X 1)) (Tao.taoNyHi (Tao.taoSection3AmbientBoundary X 1) Tao.taoAlpha) hmass <= ∑ m ∈ Finset.range (j + 1), Tao.taoSection3ForwardStepError (ndA5ReferenceSection3NoHitError B0) (ndA5ReferenceSection3PassTVError B0 d) (Tao.taoSection3AmbientBoundary X (j + 2)) m := by simpa only [hident.2.2.1] using hnoHitForward exact hnoHitTerminal.trans hsumRate end end ND end Erdos1135