| 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 |
|
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| end OAI |
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|
