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785c727 | 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 | import OAI.Geometry.EinsteinFour.Moments
import OAI.Geometry.EinsteinFour.MomentObstruction
namespace OAI
/-! The scalar assembly of the balanced-Weyl-moments argument. The upper
volume estimate and the weighted variance estimate remain separate geometric
obligations; neither is declared to be a published input here. -/
namespace PositiveEinsteinFour
open MeasureTheory Filter MomentObstruction
noncomputable section
private theorem J_nonneg_of_variance {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {z : Ω → ℝ}
(hzi : Integrable z μ) (hz2i : Integrable (fun ω => z ω ^ 2) μ)
(hHi : Integrable (fun ω => H (z ω) ^ 2) μ)
(hvar : (∫ ω, H (z ω) ^ 2 ∂μ) - (∫ ω, H (z ω) ∂μ) ^ 2 ≤
9 / 16 * ((∫ ω, z ω ^ 2 ∂μ) - ∫ ω, z ω ∂μ)) :
0 ≤ 16 * (∫ ω, H (z ω) ∂μ) ^ 2 - ∫ ω, Phi (z ω) ∂μ := by
have hid : (fun ω => Phi (z ω)) =
(fun ω => 16 * H (z ω) ^ 2 + 9 * z ω - 9 * z ω ^ 2) := by
funext ω
unfold Phi
ring
have hi1 := integral_sub ((hHi.const_mul 16).add (hzi.const_mul 9)) (hz2i.const_mul 9)
have hi2 := integral_add (hHi.const_mul 16) (hzi.const_mul 9)
simp only [Pi.add_apply] at hi1
rw [hid, hi1, hi2]
simp only [integral_const_mul]
linarith
/-- In the ordered case, the exact integer and moment constraints force equal
moments once the two-constrained-moments obstruction is applied. -/
theorem ordered_balanced_moments {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ] {v b : Ω → ℝ}
(n m : ℕ) (volume : ℝ)
(hbm : AEStronglyMeasurable b μ)
(hv : ∀ᵐ ω ∂μ, v ω ∈ Set.Icc (0 : ℝ) 2)
(hb : ∀ᵐ ω ∂μ, b ω ∈ Set.Icc (0 : ℝ) 2)
(hsum : ∀ᵐ ω ∂μ, v ω + b ω ≤ 2)
(hbi : Integrable b μ)
(hv2i : Integrable (fun ω => v ω ^ 2) μ)
(hb2i : Integrable (fun ω => b ω ^ 2) μ)
(hHbi : Integrable (fun ω => H (b ω) ^ 2) μ)
(hvgap : 1 ≤ ∫ ω, v ω ^ 2 ∂μ) (hbgap : 1 ≤ ∫ ω, b ω ^ 2 ∂μ)
(horder : (∫ ω, b ω ^ 2 ∂μ) ≤ ∫ ω, v ω ^ 2 ∂μ)
(hmoments : (∫ ω, v ω ^ 2 ∂μ) + (∫ ω, b ω ^ 2 ∂μ) < 4)
(hvolume : 0 < volume)
(hupper : volume ≤ 2 * (1 - 3 / 40 *
((∫ ω, v ω ^ 2 ∂μ) + ∫ ω, b ω ^ 2 ∂μ)))
(heuler : 2 + (n : ℝ) + (m : ℝ) = volume *
(1 + (∫ ω, v ω ^ 2 ∂μ) + ∫ ω, b ω ^ 2 ∂μ))
(hsignature : (n : ℝ) - (m : ℝ) = 2 / 3 * volume *
((∫ ω, v ω ^ 2 ∂μ) - ∫ ω, b ω ^ 2 ∂μ))
(hvariance : (∫ ω, H (b ω) ^ 2 ∂μ) - (∫ ω, H (b ω) ∂μ) ^ 2 ≤
9 / 16 * ((∫ ω, b ω ^ 2 ∂μ) - ∫ ω, b ω ∂μ)) :
(∫ ω, v ω ^ 2 ∂μ) = (∫ ω, b ω ^ 2 ∂μ) := by
rcases ordered_harmonic_counts n m _ _ volume hvgap hbgap horder hmoments
hvolume hupper heuler hsignature with hcounts | ⟨hn, hm⟩
· have hnreal : (n : ℝ) = (m : ℝ) := congrArg Nat.cast hcounts
have hprod : volume * ((∫ ω, v ω ^ 2 ∂μ) - ∫ ω, b ω ^ 2 ∂μ) = 0 := by
rw [hnreal] at hsignature
linarith
have := (mul_eq_zero.mp hprod).resolve_left (ne_of_gt hvolume)
linarith
· subst n
subst m
norm_num at heuler hsignature
have hrelation := exceptional_moment_relation _ _ volume heuler hsignature
have hi : Integrable (fun ω => 4 - 4 * b ω + b ω ^ 2) μ :=
((integrable_const 4).sub (hbi.const_mul 4)).add hb2i
have hbound : (∫ ω, v ω ^ 2 ∂μ) ≤ 4 - 4 * (∫ ω, b ω ∂μ) + ∫ ω, b ω ^ 2 ∂μ := by
have h := integral_mono_ae hv2i hi (by
filter_upwards [hv, hb, hsum] with ω hvω hbω hsω
have hp : 0 ≤ (2 - b ω - v ω) * (2 - b ω + v ω) := by
apply mul_nonneg <;> linarith [hvω.1, hbω.2]
nlinarith)
have hi1 := integral_add ((integrable_const (4 : ℝ)).sub (hbi.const_mul 4)) hb2i
have hi2 := integral_sub (integrable_const (4 : ℝ)) (hbi.const_mul 4)
simp only [Pi.sub_apply] at hi1
rw [hi1, hi2] at h
simpa only [integral_const_mul, integral_const, probReal_univ, one_smul] using h
have hconstraint := exceptional_moment_constraint _ _ _ hrelation hbound
have hconstraint' : (∫ ω, 28 * b ω + 6 * b ω ^ 2 ∂μ) ≤ 25 := by
rw [integral_add (hbi.const_mul 28) (hb2i.const_mul 6)]
simpa only [integral_const_mul] using hconstraint
have hnegative := two_constrained_moments hbm hb hbgap hconstraint'
have hnonnegative := J_nonneg_of_variance hbi hb2i hHbi hvariance
linarith
/-- Symmetric scalar form of the balanced-moment conclusion. The geometric
volume and weighted-variance estimates are explicit premises, pending their
independent formalization. -/
theorem balanced_moments {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ] {v b : Ω → ℝ}
(n m : ℕ) (volume : ℝ)
(hvm : AEStronglyMeasurable v μ) (hbm : AEStronglyMeasurable b μ)
(hv : ∀ᵐ ω ∂μ, v ω ∈ Set.Icc (0 : ℝ) 2)
(hb : ∀ᵐ ω ∂μ, b ω ∈ Set.Icc (0 : ℝ) 2)
(hsum : ∀ᵐ ω ∂μ, v ω + b ω ≤ 2)
(hvi : Integrable v μ) (hbi : Integrable b μ)
(hv2i : Integrable (fun ω => v ω ^ 2) μ)
(hb2i : Integrable (fun ω => b ω ^ 2) μ)
(hHvi : Integrable (fun ω => H (v ω) ^ 2) μ)
(hHbi : Integrable (fun ω => H (b ω) ^ 2) μ)
(hvgap : 1 ≤ ∫ ω, v ω ^ 2 ∂μ) (hbgap : 1 ≤ ∫ ω, b ω ^ 2 ∂μ)
(hmoments : (∫ ω, v ω ^ 2 ∂μ) + (∫ ω, b ω ^ 2 ∂μ) < 4)
(hvolume : 0 < volume)
(hupper : volume ≤ 2 * (1 - 3 / 40 *
((∫ ω, v ω ^ 2 ∂μ) + ∫ ω, b ω ^ 2 ∂μ)))
(heuler : 2 + (n : ℝ) + (m : ℝ) = volume *
(1 + (∫ ω, v ω ^ 2 ∂μ) + ∫ ω, b ω ^ 2 ∂μ))
(hsignature : (n : ℝ) - (m : ℝ) = 2 / 3 * volume *
((∫ ω, v ω ^ 2 ∂μ) - ∫ ω, b ω ^ 2 ∂μ))
(hvvariance : (∫ ω, H (v ω) ^ 2 ∂μ) - (∫ ω, H (v ω) ∂μ) ^ 2 ≤
9 / 16 * ((∫ ω, v ω ^ 2 ∂μ) - ∫ ω, v ω ∂μ))
(hbvariance : (∫ ω, H (b ω) ^ 2 ∂μ) - (∫ ω, H (b ω) ∂μ) ^ 2 ≤
9 / 16 * ((∫ ω, b ω ^ 2 ∂μ) - ∫ ω, b ω ∂μ)) :
(∫ ω, v ω ^ 2 ∂μ) = (∫ ω, b ω ^ 2 ∂μ) ∧ n = m := by
have hequal : (∫ ω, v ω ^ 2 ∂μ) = (∫ ω, b ω ^ 2 ∂μ) := by
rcases le_total (∫ ω, b ω ^ 2 ∂μ) (∫ ω, v ω ^ 2 ∂μ) with horder | horder
· exact ordered_balanced_moments n m volume hbm hv hb hsum hbi hv2i hb2i hHbi
hvgap hbgap horder hmoments hvolume hupper heuler hsignature hbvariance
· symm
apply ordered_balanced_moments m n volume hvm hb hv
(hsum.mono fun ω hω => by linarith) hvi hb2i hv2i hHvi hbgap hvgap horder
(by linarith) hvolume (by convert hupper using 1; ring)
(by nlinarith [heuler]) (by nlinarith [hsignature]) hvvariance
refine ⟨hequal, ?_⟩
rw [hequal] at hsignature
have hreal : (n : ℝ) = (m : ℝ) := by linarith
exact_mod_cast hreal
end
end PositiveEinsteinFour
end OAI
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