import Erdos1135.ND.Band.A5ReferenceExteriorTupleTail /-! # A5 retained-carrier terminal cap This leaf bounds the terminal Geom(2) coordinate while an actual original endpoint witness is still available. Only after this bound is obtained may a later consumer enlarge the exterior carrier to an ambient exact-length event. -/ namespace Erdos1135 namespace ND noncomputable section /-- Logarithmic budget for the terminal valuation coordinate of an endpoint in the canonical lost window. -/ noncomputable def ndA5EndpointTerminalLogBudget (B : ℕ) : ℝ := Real.log B + Tao.taoSection5CanonicalLostSlack B + 2 * (Tao.taoSection5M0 B : ℝ) * Real.log 2 + Real.log (8 / 3 : ℝ) /-- Scalar cap that converts the endpoint terminal budget into a multiple of the ambient logarithm. Its eventual truth is a later scalar-rate fact. -/ def NDA5EndpointTerminalCapBudget (B : ℕ) : Prop := ndA5EndpointTerminalLogBudget B ≤ 2 * Real.log 2 * Real.log B /-- Every tuple in a predicate-filtered carrier retains enough endpoint data to bound its terminal coordinate before any ambient enlargement. -/ theorem terminalValue_mul_logTwo_lt_ndA5EndpointTerminalLogBudget {B q : ℕ} {E : Set ℕ} {X : ZMod (3 ^ q)} {P : ℕ → Prop} [DecidablePred P] {bs : List ℕ+} (facts : Tao.TaoSection5PassLostWindowFacts B) (hbs : bs ∈ ndA5EndpointValuationFilterCarrier B E q X P) : (Tao.geom2PNatListTerminalValue bs : ℝ) * Real.log 2 < ndA5EndpointTerminalLogBudget B := by rcases exists_endpoint_of_mem_ndA5EndpointValuationFilterCarrier hbs with ⟨M, hM, _hP⟩ let m := Tao.taoSection5M0 B have hmOne : 1 ≤ m := by simpa only [m] using facts.one_le_m0 have hm : m - 1 + 1 = m := Nat.sub_add_cancel hmOne have hodd := Tao.taoSection5Endpoint_odd M have hfirst : Tao.syracuseFirstHitAtMost B M.1 m := by simpa only [m] using (Tao.mem_taoSection5EPrime_iff.mp M.2).2.2.2.1 have hfirst' : Tao.syracuseFirstHitAtMost B M.1 (m - 1 + 1) := by simpa only [hm] using hfirst have hroom : 2 * 3 ^ (m - 1) ≤ B := by exact_mod_cast facts.preterminal_offset_room have hMposNat : 0 < M.1 := Odd.pos hodd have hMpos : (0 : ℝ) < M.1 := by exact_mod_cast hMposNat have hBposNat : 0 < B := (by positivity : 0 < 2 * 3 ^ (m - 1)).trans_le hroom have hBpos : (0 : ℝ) < B := by exact_mod_cast hBposNat have hval := (Tao.mem_taoSection5EndpointValuationFiber_iff.mp hM).1 have hfullNat : 2 ^ Tao.taoTupleWeight bs < 8 * 3 ^ (m - 1) * M.1 := by have hfull := Tao.syracuseFirstHitAtMost_terminal_lower_cleared (r := m - 1) hodd hroom hfirst' have hfullVal : 2 ^ Tao.taoTupleWeight (Tao.taoSection5EndpointValuation M) < 8 * 3 ^ (m - 1) * M.1 := by simpa only [hm, Tao.taoSection5EndpointValuation] using hfull simpa only [hval] using hfullVal have hfull : (2 : ℝ) ^ Tao.taoTupleWeight bs < 8 * (3 : ℝ) ^ (m - 1) * (M.1 : ℝ) := by exact_mod_cast hfullNat have hlogFull := Real.log_lt_log (pow_pos (by norm_num : (0 : ℝ) < 2) _) hfull rw [Real.log_pow, Real.log_mul (mul_ne_zero (by norm_num : (8 : ℝ) ≠ 0) (pow_ne_zero _ (by norm_num : (3 : ℝ) ≠ 0))) hMpos.ne', Real.log_mul (by norm_num : (8 : ℝ) ≠ 0) (pow_ne_zero _ (by norm_num : (3 : ℝ) ≠ 0)), Real.log_pow] at hlogFull have hMupperNat : M.1 ≤ Nat.floor (Tao.taoSection5CanonicalLostUpper B) := (Tao.mem_taoSection5EPrime_iff.mp M.2).2.1 have hlostUpperNonneg : 0 ≤ Tao.taoSection5CanonicalLostUpper B := by unfold Tao.taoSection5CanonicalLostUpper Tao.taoSection5CanonicalLostScale positivity have hMupper : (M.1 : ℝ) ≤ Tao.taoSection5CanonicalLostUpper B := (Nat.le_floor_iff hlostUpperNonneg).mp hMupperNat have hlogM := Real.log_le_log hMpos hMupper unfold Tao.taoSection5CanonicalLostUpper Tao.taoSection5CanonicalLostScale at hlogM rw [Real.log_mul (Real.exp_ne_zero _) (mul_ne_zero (pow_ne_zero _ (by norm_num : (4 / 3 : ℝ) ≠ 0)) hBpos.ne'), Real.log_exp, Real.log_mul (pow_ne_zero _ (by norm_num : (4 / 3 : ℝ) ≠ 0)) hBpos.ne', Real.log_pow] at hlogM have hmCast : (((m - 1 : ℕ) : ℝ) + 1) = (m : ℝ) := by exact_mod_cast hm have hmCast' : (((Tao.taoSection5M0 B - 1 : ℕ) : ℝ) + 1) = (Tao.taoSection5M0 B : ℝ) := by simpa only [m] using hmCast have hlogFourThirds : Real.log (4 / 3 : ℝ) = 2 * Real.log 2 - Real.log 3 := by rw [Real.log_div (by norm_num : (4 : ℝ) ≠ 0) (by norm_num : (3 : ℝ) ≠ 0), show (4 : ℝ) = 2 ^ 2 by norm_num, Real.log_pow] ring have hlogEight : Real.log (8 : ℝ) = 3 * Real.log 2 := by rw [show (8 : ℝ) = 2 ^ 3 by norm_num, Real.log_pow] norm_num have hlogEightThirds : Real.log (8 / 3 : ℝ) = 3 * Real.log 2 - Real.log 3 := by rw [Real.log_div (by norm_num : (8 : ℝ) ≠ 0) (by norm_num : (3 : ℝ) ≠ 0), hlogEight] rw [hlogEight] at hlogFull rw [hlogFourThirds, ← hmCast'] at hlogM have hweightLog : (Tao.taoTupleWeight bs : ℝ) * Real.log 2 < ndA5EndpointTerminalLogBudget B := by unfold ndA5EndpointTerminalLogBudget rw [hlogEightThirds, ← hmCast'] dsimp only [m] at hlogFull nlinarith [hlogFull, hlogM] have hterminalLe : Tao.geom2PNatListTerminalValue bs ≤ Tao.taoTupleWeight bs := by have hsplit := Tao.taoTupleWeight_eq_dropLast_add_terminalValue bs omega have hterminalLeReal : (Tao.geom2PNatListTerminalValue bs : ℝ) ≤ (Tao.taoTupleWeight bs : ℝ) := by exact_mod_cast hterminalLe exact (mul_le_mul_of_nonneg_right hterminalLeReal (Real.log_nonneg (by norm_num))).trans_lt hweightLog /-- Pure algebraic conversion from a terminal logarithmic cap to the native base/mass comparison. -/ theorem geom2PNatListNativeBase_le_log_add_one_mul_mass {B : ℕ} {bs : List ℕ+} {budget : ℝ} (hterminal : (Tao.geom2PNatListTerminalValue bs : ℝ) * Real.log 2 < budget) (hcap : budget ≤ 2 * Real.log 2 * Real.log B) : Tao.geom2PNatListNativeBase bs ≤ (Real.log B + 1) * Tao.geom2PNatListMass bs := by have hlogTwo : 0 < Real.log (2 : ℝ) := Real.log_pos (by norm_num) have hterminalLt : (Tao.geom2PNatListTerminalValue bs : ℝ) < 2 * Real.log B := by have hmul : (Tao.geom2PNatListTerminalValue bs : ℝ) * Real.log 2 < (2 * Real.log B) * Real.log 2 := by calc (Tao.geom2PNatListTerminalValue bs : ℝ) * Real.log 2 < budget := hterminal _ ≤ 2 * Real.log 2 * Real.log B := hcap _ = (2 * Real.log B) * Real.log 2 := by ring exact lt_of_mul_lt_mul_right hmul hlogTwo.le have hmass : 0 ≤ Tao.geom2PNatListMass bs := by rw [Tao.geom2PNatListMass_eq_pow] positivity rw [Tao.geom2PNatListNativeBase_eq_half_mass] calc (1 / 2 : ℝ) * (Tao.geom2PNatListMass bs * (Tao.geom2PNatListTerminalValue bs : ℝ) + 2 * Tao.geom2PNatListMass bs) = ((Tao.geom2PNatListTerminalValue bs : ℝ) / 2 + 1) * Tao.geom2PNatListMass bs := by ring _ ≤ (Real.log B + 1) * Tao.geom2PNatListMass bs := by apply mul_le_mul_of_nonneg_right _ hmass linarith /-- Predicate-carrier native-base cap, applied before ambient enlargement. -/ theorem geom2PNatListNativeBase_le_log_add_one_mul_mass_of_mem_filterCarrier {B q : ℕ} {E : Set ℕ} {X : ZMod (3 ^ q)} {P : ℕ → Prop} [DecidablePred P] {bs : List ℕ+} (facts : Tao.TaoSection5PassLostWindowFacts B) (hcap : NDA5EndpointTerminalCapBudget B) (hbs : bs ∈ ndA5EndpointValuationFilterCarrier B E q X P) : Tao.geom2PNatListNativeBase bs ≤ (Real.log B + 1) * Tao.geom2PNatListMass bs := by exact geom2PNatListNativeBase_le_log_add_one_mul_mass (terminalValue_mul_logTwo_lt_ndA5EndpointTerminalLogBudget facts hbs) hcap /-- Exterior specialization on the retained original-endpoint carrier. -/ theorem ndA5ReferenceExteriorCarrier_nativeBase_le_log_add_one_mul_mass {B q : ℕ} {C : ℝ} {E : Set ℕ} {X : ZMod (3 ^ q)} {bs : List ℕ+} (facts : Tao.TaoSection5PassLostWindowFacts B) (hcap : NDA5EndpointTerminalCapBudget B) (hbs : bs ∈ ndA5ReferenceExteriorEndpointValuationCarrier B C E q X) : Tao.geom2PNatListNativeBase bs ≤ (Real.log B + 1) * Tao.geom2PNatListMass bs := by classical exact geom2PNatListNativeBase_le_log_add_one_mul_mass_of_mem_filterCarrier (P := ndA5ReferenceEndpointExteriorAt B C) facts hcap (by simpa only [ndA5ReferenceExteriorEndpointValuationCarrier] using hbs) end end ND end Erdos1135