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