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f4b3ad1 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 | import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticPatternErrorRate
import OAI.NumberTheory.Ostmann.Arithmetic.PrimePatternFrequencyNormSum
namespace OAI
/-! # The full numerical norm bound with the original equality patterns -/
namespace Ostmann
open Filter
open scoped Classical BigOperators
/-- The actual pattern sum and paired frequency sum have an absolute bulk
constant. The cutoff includes the constructed bottom reserve `2 sqrt(m)`.
Only the genuine comparison errors use the full number of histories. -/
theorem arithmetic_two_prime_pattern_norm_rate (ψ : SchwartzMap ℝ ℂ) (n k : ℕ)
(hk : 0 < k) (hn : n ≤ k) (B Cfreq Cprior ε : ℝ)
(hCfreq : 0 ≤ Cfreq) (hCprior : 1 ≤ Cprior) (hε : 0 < ε)
(hdepth : 8 * (Cprior + 1) ≤ (k : ℝ) ^ 3) :
∀ᶠ L : ℝ in atTop, let m := spectatorBulkCount k L
∃ D : ℝ, 0 ≤ D ∧
(∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * Cfreq * m) →
(q.divisors.card : ℝ) ≤ D) ∧
∀ (S : Finset ℤ) (N V : ℕ) (Δ err : ℝ),
(S.card : ℝ) ≤ Real.exp (Cfreq * m) →
(∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N) →
(N : ℝ) ≤ Real.exp (Cfreq * m) →
(V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) →
0 ≤ err → err ≤ Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) +
3 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L)) →
let _ := sampleSetoidFintype (Bool × MovingSampleIndex n)
let K := ((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) *
B ^ (2 ^ n - 1)) ^ 2 *
((2 : ℝ) ^ (2 ^ n * m) * 4 * 3 ^ (2 ^ n * m)) * 2 ^ (2 ^ n * m)
∀ R : Setoid (Bool × MovingSampleIndex n) → FrequencyTree (S × S) n → ℂ,
(∀ s t, ‖R s t‖ ≤
((4 : ℝ) ^ Fintype.card (Quotient s) *
(Real.exp (Cprior * L)) ^ (4 * n * 2 ^ n - Fintype.card (Quotient s))) *
(err + K * (frequencyLeafWeight (pairedFrequencyLeaf S V) n t *
((frequencySplitList S n t).map (pairFrequencySupportBound D)).prod))) →
‖∑ s, ∑ t, R s t‖ ≤
Real.exp ((2 ^ n : ℕ) * Δ + (Real.log 12 + 1) * (2 ^ n : ℕ) * m + ε * m) +
4 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by
have hCplus : 1 ≤ Cprior + 1 := by linarith
filter_upwards [arithmetic_prime_pattern_budget_rate ψ n k hk hn B Cfreq (Cprior + 1) ε
hCfreq hCplus hε hdepth,
movingPattern_four_errors_rate n k hk (Cprior + 1) Cfreq hCplus hCfreq,
eventually_ge_atTop (2 : ℝ)] with L hbudget herrors hL
dsimp only
obtain ⟨D, hD, hdiv, hbudget⟩ := hbudget
refine ⟨D, hD, hdiv, ?_⟩
intro S N V Δ err hcard hS hN hV herr herrbound R hR
have hdivN : ∀ q : ℕ, q ≠ 0 → q ≤ N ^ 2 → (q.divisors.card : ℝ) ≤ D := by
intro q hq hqN
apply hdiv q hq
calc
(q : ℝ) ≤ (N : ℝ) ^ 2 := by exact_mod_cast hqN
_ ≤ (Real.exp (Cfreq * spectatorBulkCount k L)) ^ 2 :=
pow_le_pow_left₀ (Nat.cast_nonneg N) hN 2
_ = Real.exp (2 * Cfreq * spectatorBulkCount k L) := by
rw [← Real.exp_nat_mul]
congr 1
ring
have hexp : (4 : ℝ) ≤ Real.exp ((Cprior + 1) * L) := by
have hCL := mul_le_mul hCplus hL (by norm_num : (0 : ℝ) ≤ 2) (by linarith)
have hCL' := mul_le_mul_of_nonneg_right hCprior (by linarith : 0 ≤ L)
linarith [Real.add_one_le_exp ((Cprior + 1) * L)]
have hmono : max 4 (Real.exp (Cprior * L)) ≤ max 2 (Real.exp ((Cprior + 1) * L)) := by
apply max_le
· exact hexp.trans (le_max_right _ _)
· exact (Real.exp_le_exp.mpr (by linarith)).trans (le_max_right _ _)
have hpriorcost :
(2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * (max 4 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n) ≤
(2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * (max 2 (Real.exp ((Cprior + 1) * L))) ^ (4 * n * 2 ^ n) :=
mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) hmono _) (by positivity)
have hsum := prime_pattern_frequency_norm_sum_le S n N V D (Real.exp (Cprior * L))
(((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) *
B ^ (2 ^ n - 1)) ^ 2 *
((2 : ℝ) ^ (2 ^ n * spectatorBulkCount k L) * 4 * 3 ^ (2 ^ n * spectatorBulkCount k L)) *
2 ^ (2 ^ n * spectatorBulkCount k L)) err hD (Real.exp_nonneg _) (by positivity)
herr hS hdivN R hR
have hcost := (mul_le_mul_of_nonneg_right hpriorcost (by positivity)).trans (hbudget N V Δ hN hV)
have he := (mul_le_mul_of_nonneg_left herrbound
(show 0 ≤ ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) *
(max 4 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) *
(Fintype.card (FrequencyTree (S × S) n) : ℝ) by positivity)).trans
((mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hpriorcost
(Nat.cast_nonneg _)) (by positivity)).trans (herrors S hcard))
apply hsum.trans
nlinarith only [hcost, he]
/-- The actual pattern sum and paired frequency sum have an absolute bulk
constant. The cutoff includes the constructed bottom reserve `2 sqrt(m)`.
The fifth error is the literal giant comparison; only comparison errors use the full number of histories. -/
theorem arithmetic_two_prime_original_pattern_norm_rate (ψ : SchwartzMap ℝ ℂ) (n k : ℕ)
(hk : 0 < k) (hn : n ≤ k) (B Cfreq Cprior ε : ℝ)
(hCfreq : 0 ≤ Cfreq) (hCprior : 1 ≤ Cprior) (hε : 0 < ε)
(hdepth : 8 * (Cprior + 1) ≤ (k : ℝ) ^ 3) :
∀ᶠ L : ℝ in atTop, let m := spectatorBulkCount k L
∃ D : ℝ, 0 ≤ D ∧
(∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * Cfreq * m) →
(q.divisors.card : ℝ) ≤ D) ∧
∀ (S : Finset ℤ) (N V : ℕ) (Δ err : ℝ),
(S.card : ℝ) ≤ Real.exp (Cfreq * m) →
(∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N) →
(N : ℝ) ≤ Real.exp (Cfreq * m) →
(V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) →
0 ≤ err → err ≤ Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) +
4 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L)) →
let _ := sampleSetoidFintype (Bool × MovingSampleIndex n)
let K := ((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) *
B ^ (2 ^ n - 1)) ^ 2 *
((2 : ℝ) ^ (2 ^ n * m) * 4 * 3 ^ (2 ^ n * m)) * 2 ^ (2 ^ n * m)
∀ R : Setoid (Bool × MovingSampleIndex n) → FrequencyTree (S × S) n → ℂ,
(∀ s t, ‖R s t‖ ≤
((4 : ℝ) ^ Fintype.card (Quotient s) *
(Real.exp (Cprior * L)) ^ (4 * n * 2 ^ n - Fintype.card (Quotient s))) *
(err + K * (frequencyLeafWeight (pairedFrequencyLeaf S V) n t *
((frequencySplitList S n t).map (pairFrequencySupportBound D)).prod))) →
‖∑ s, ∑ t, R s t‖ ≤
Real.exp ((2 ^ n : ℕ) * Δ + (Real.log 12 + 1) * (2 ^ n : ℕ) * m + ε * m) +
5 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by
have hCplus : 1 ≤ Cprior + 1 := by linarith
filter_upwards [arithmetic_prime_pattern_budget_rate ψ n k hk hn B Cfreq (Cprior + 1) ε
hCfreq hCplus hε hdepth,
movingPattern_five_errors_rate n k hk (Cprior + 1) Cfreq hCplus hCfreq,
eventually_ge_atTop (2 : ℝ)] with L hbudget herrors hL
dsimp only
obtain ⟨D, hD, hdiv, hbudget⟩ := hbudget
refine ⟨D, hD, hdiv, ?_⟩
intro S N V Δ err hcard hS hN hV herr herrbound R hR
have hdivN : ∀ q : ℕ, q ≠ 0 → q ≤ N ^ 2 → (q.divisors.card : ℝ) ≤ D := by
intro q hq hqN
apply hdiv q hq
calc
(q : ℝ) ≤ (N : ℝ) ^ 2 := by exact_mod_cast hqN
_ ≤ (Real.exp (Cfreq * spectatorBulkCount k L)) ^ 2 :=
pow_le_pow_left₀ (Nat.cast_nonneg N) hN 2
_ = Real.exp (2 * Cfreq * spectatorBulkCount k L) := by
rw [← Real.exp_nat_mul]
congr 1
ring
have hexp : (4 : ℝ) ≤ Real.exp ((Cprior + 1) * L) := by
have hCL := mul_le_mul hCplus hL (by norm_num : (0 : ℝ) ≤ 2) (by linarith)
have hCL' := mul_le_mul_of_nonneg_right hCprior (by linarith : 0 ≤ L)
linarith [Real.add_one_le_exp ((Cprior + 1) * L)]
have hmono : max 4 (Real.exp (Cprior * L)) ≤ max 2 (Real.exp ((Cprior + 1) * L)) := by
apply max_le
· exact hexp.trans (le_max_right _ _)
· exact (Real.exp_le_exp.mpr (by linarith)).trans (le_max_right _ _)
have hpriorcost :
(2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * (max 4 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n) ≤
(2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * (max 2 (Real.exp ((Cprior + 1) * L))) ^ (4 * n * 2 ^ n) :=
mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) hmono _) (by positivity)
have hsum := prime_pattern_frequency_norm_sum_le S n N V D (Real.exp (Cprior * L))
(((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) *
B ^ (2 ^ n - 1)) ^ 2 *
((2 : ℝ) ^ (2 ^ n * spectatorBulkCount k L) * 4 * 3 ^ (2 ^ n * spectatorBulkCount k L)) *
2 ^ (2 ^ n * spectatorBulkCount k L)) err hD (Real.exp_nonneg _) (by positivity)
herr hS hdivN R hR
have hcost := (mul_le_mul_of_nonneg_right hpriorcost (by positivity)).trans (hbudget N V Δ hN hV)
have he := (mul_le_mul_of_nonneg_left herrbound
(show 0 ≤ ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) *
(max 4 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) *
(Fintype.card (FrequencyTree (S × S) n) : ℝ) by positivity)).trans
((mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hpriorcost
(Nat.cast_nonneg _)) (by positivity)).trans (herrors S hcard))
apply hsum.trans
nlinarith only [hcost, he]
end Ostmann
end OAI
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