import Erdos1135.ND.Band.A5SourceMixture import Erdos1135.ND.Discrepancy.A5ReferenceGenuineBandRate /-! # Genuine A5 whole-source passage rate at the common center This leaf averages the checked factor-seven band estimates through the exact whole-source mixtures. The possible inclusive top endpoint costs one further copy of the reference rate. The results concern genuine passage events; the later common-target totalizer is intentionally separate. -/ namespace Erdos1135 namespace ND open Filter open scoped BigOperators Topology noncomputable section private theorem abs_top_sub_weight_mul_le_forty_mul {t w Z : ℝ} (ht0 : 0 ≤ t) (htw : t ≤ w) (hw0 : 0 ≤ w) (hZ : Z ∈ Set.Icc (0 : ℝ) 40) : |t - w * Z| ≤ 40 * w := by rw [abs_le] have hwZ0 : 0 ≤ w * Z := mul_nonneg hw0 hZ.1 have hwZ40 : w * Z ≤ w * 40 := mul_le_mul_of_nonneg_left hZ.2 hw0 constructor <;> nlinarith private theorem abs_weighted_sum_add_top_sub_center_le_eight {ι : Type*} (S : Finset ι) (w f : ι → ℝ) (t p Z R : ℝ) (hw : ∀ i ∈ S, 0 ≤ w i) (hpoint : ∀ i ∈ S, |f i - Z| ≤ 7 * R) (hnorm : (∑ i ∈ S, w i) + p = 1) (hp0 : 0 ≤ p) (htop : |t - p * Z| ≤ R) (hR0 : 0 ≤ R) : |(∑ i ∈ S, w i * f i) + t - Z| ≤ 8 * R := by have hweighted := abs_sum_mul_sub_center_mul_sum_le S w f (fun _ ↦ 7 * R) Z hw hpoint have hsum_le_one : (∑ i ∈ S, w i) ≤ 1 := by linarith have hweighted' : |(∑ i ∈ S, w i * f i) - Z * (∑ i ∈ S, w i)| ≤ 7 * R := by calc |(∑ i ∈ S, w i * f i) - Z * (∑ i ∈ S, w i)| ≤ ∑ i ∈ S, w i * (7 * R) := hweighted _ = (∑ i ∈ S, w i) * (7 * R) := by rw [Finset.sum_mul] _ ≤ 1 * (7 * R) := mul_le_mul_of_nonneg_right hsum_le_one (mul_nonneg (by norm_num) hR0) _ = 7 * R := by ring calc |(∑ i ∈ S, w i * f i) + t - Z| = |((∑ i ∈ S, w i * f i) - Z * (∑ i ∈ S, w i)) + (t - p * Z)| := by congr 1 linear_combination Z * hnorm _ ≤ |(∑ i ∈ S, w i * f i) - Z * (∑ i ∈ S, w i)| + |t - p * Z| := abs_add_le _ _ _ ≤ 7 * R + R := add_le_add hweighted' htop _ = 8 * R := by ring /-- Every valid whole harmonic and flat A5 source has genuine passage mass within eight reference-rate copies of the common target. -/ theorem eventually_ndA5GenuineSourcePassDecay_referenceLevel_le_eight {c kappa d : ℝ} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∀ᶠ B : ℕ in atTop, ∀ (branch : Tao.TaoSection5SourceBranch) (E : Set ℕ) (hB : 1 ≤ B) (hcount : 0 < ndA5BandCount B branch), (|((ndA5HarmonicSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E| ≤ 8 * ndA5ReferenceInteriorScalarRate B d) ∧ (|((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E| ≤ 8 * ndA5ReferenceInteriorScalarRate B d) := by filter_upwards [eventually_ndA5GenuineBandPassDecay_referenceLevel_le_seven hPhase hd hd20 hdk, eventually_ndA5ReferenceCommonProfile_mem_Icc_forty, eventually_one_hundred_sixty_div_natCast_le_ndA5ReferenceInteriorScalarRate d] with B hband hprofile htopRate intro branch E hB hcount let A : Set Tao.TaoOddNat := Tao.taoSection5PassEvent B E let Z : ℝ := ndA5ReferenceCommonProfile B E let R : ℝ := ndA5ReferenceInteriorScalarRate B d have hR0 : 0 ≤ R := by dsimp only [R, ndA5ReferenceInteriorScalarRate] exact Real.rpow_nonneg (Real.log_nonneg (by exact_mod_cast hB)) _ have hpointH : ∀ j ∈ Finset.range (ndA5BandCount B branch), |((ndA5HarmonicBandPMF B branch j hB hcount).toOuterMeasure A).toReal - Z| ≤ 7 * R := by intro j hj simpa only [A, Z, R] using (hband branch j E hB hcount (Finset.mem_range.mp hj)).1 have hpointF : ∀ j ∈ Finset.range (ndA5BandCount B branch), |((ndA5FlatBandPMF B branch j hB hcount).toOuterMeasure A).toReal - Z| ≤ 7 * R := by intro j hj simpa only [A, Z, R] using (hband branch j E hB hcount (Finset.mem_range.mp hj)).2 have htopH40 : |ndA5HarmonicTopRestrictedMass B branch A - ndA5HarmonicTopWeight B branch * Z| ≤ 40 * ndA5HarmonicTopWeight B branch := abs_top_sub_weight_mul_le_forty_mul (ndA5HarmonicTopRestrictedMass_nonneg hB hcount A) (ndA5HarmonicTopRestrictedMass_le_weight hB hcount A) (ndA5HarmonicTopWeight_nonneg B branch) (hprofile E) have htopH : |ndA5HarmonicTopRestrictedMass B branch A - ndA5HarmonicTopWeight B branch * Z| ≤ R := by calc _ ≤ 40 * ndA5HarmonicTopWeight B branch := htopH40 _ ≤ 40 * (4 / (B : ℝ)) := mul_le_mul_of_nonneg_left (ndA5HarmonicTopWeight_le_four_div hB branch hcount) (by norm_num) _ = 160 / (B : ℝ) := by ring _ = 160 * Real.rpow (B : ℝ) (-1) := by simpa only [div_eq_mul_inv] using congrArg (fun x : ℝ ↦ 160 * x) (Real.rpow_neg_one (B : ℝ)).symm _ ≤ R := by simpa only [R] using htopRate have htopF40 : |ndA5FlatTopRestrictedMass B branch A - ndA5FlatTopWeight B branch * Z| ≤ 40 * ndA5FlatTopWeight B branch := abs_top_sub_weight_mul_le_forty_mul (ndA5FlatTopRestrictedMass_nonneg hB hcount A) (ndA5FlatTopRestrictedMass_le_weight hB hcount A) (ndA5FlatTopWeight_nonneg B branch) (hprofile E) have htopF : |ndA5FlatTopRestrictedMass B branch A - ndA5FlatTopWeight B branch * Z| ≤ R := by calc _ ≤ 40 * ndA5FlatTopWeight B branch := htopF40 _ ≤ 40 * (4 / (B : ℝ)) := mul_le_mul_of_nonneg_left (ndA5FlatTopWeight_le_four_div hB branch hcount) (by norm_num) _ = 160 / (B : ℝ) := by ring _ = 160 * Real.rpow (B : ℝ) (-1) := by simpa only [div_eq_mul_inv] using congrArg (fun x : ℝ ↦ 160 * x) (Real.rpow_neg_one (B : ℝ)).symm _ ≤ R := by simpa only [R] using htopRate constructor · rw [ndA5HarmonicSourcePMF_eventMass_eq_band_mixture_add_top hB hcount A] exact abs_weighted_sum_add_top_sub_center_le_eight (Finset.range (ndA5BandCount B branch)) (ndA5HarmonicBandWeight B branch) (fun j ↦ ((ndA5HarmonicBandPMF B branch j hB hcount).toOuterMeasure A).toReal) (ndA5HarmonicTopRestrictedMass B branch A) (ndA5HarmonicTopWeight B branch) Z R (fun j _ ↦ ndA5HarmonicBandWeight_nonneg B branch j) hpointH (sum_ndA5HarmonicBandWeight_add_topWeight_eq_one hB branch hcount) (ndA5HarmonicTopWeight_nonneg B branch) htopH hR0 · rw [ndA5FlatSourcePMF_eventMass_eq_band_mixture_add_top hB hcount A] exact abs_weighted_sum_add_top_sub_center_le_eight (Finset.range (ndA5BandCount B branch)) (ndA5FlatBandWeight B branch) (fun j ↦ ((ndA5FlatBandPMF B branch j hB hcount).toOuterMeasure A).toReal) (ndA5FlatTopRestrictedMass B branch A) (ndA5FlatTopWeight B branch) Z R (fun j _ ↦ ndA5FlatBandWeight_nonneg B branch j) hpointF (sum_ndA5FlatBandWeight_add_topWeight_eq_one hB branch hcount) (ndA5FlatTopWeight_nonneg B branch) htopF hR0 /-- On one source branch, the genuine harmonic and flat passage masses differ by at most sixteen reference-rate copies. -/ theorem eventually_abs_ndA5HarmonicSourcePassMass_sub_flat_le_sixteen {c kappa d : ℝ} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∀ᶠ B : ℕ in atTop, ∀ (branch : Tao.TaoSection5SourceBranch) (E : Set ℕ) (hB : 1 ≤ B) (hcount : 0 < ndA5BandCount B branch), |((ndA5HarmonicSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal| ≤ 16 * ndA5ReferenceInteriorScalarRate B d := by filter_upwards [eventually_ndA5GenuineSourcePassDecay_referenceLevel_le_eight hPhase hd hd20 hdk] with B hsource intro branch E hB hcount have hpair := hsource branch E hB hcount calc |((ndA5HarmonicSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal| = |(((ndA5HarmonicSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E) - (((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E)| := by ring _ ≤ |((ndA5HarmonicSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E| + |((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E| := by simpa only [sub_eq_add_neg, abs_neg] using (abs_add_le (((ndA5HarmonicSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E) (-(((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E))) _ ≤ 8 * ndA5ReferenceInteriorScalarRate B d + 8 * ndA5ReferenceInteriorScalarRate B d := add_le_add hpair.1 hpair.2 _ = 16 * ndA5ReferenceInteriorScalarRate B d := by ring /-- Across two source branches, the genuine harmonic passage masses differ by at most sixteen reference-rate copies. -/ theorem eventually_abs_ndA5HarmonicSourcePassMass_sub_harmonic_le_sixteen {c kappa d : ℝ} (hPhase : PhaseGap c kappa) (hd : 0 < d) (hd20 : d < 1 / 20) (hdk : d < 1 / (2 * kappa)) : ∀ᶠ B : ℕ in atTop, ∀ (branch₁ branch₂ : Tao.TaoSection5SourceBranch) (E : Set ℕ) (hB : 1 ≤ B) (hcount₁ : 0 < ndA5BandCount B branch₁) (hcount₂ : 0 < ndA5BandCount B branch₂), |((ndA5HarmonicSourcePMF B branch₁ hB hcount₁).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ((ndA5HarmonicSourcePMF B branch₂ hB hcount₂).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal| ≤ 16 * ndA5ReferenceInteriorScalarRate B d := by filter_upwards [eventually_ndA5GenuineSourcePassDecay_referenceLevel_le_eight hPhase hd hd20 hdk] with B hsource intro branch₁ branch₂ E hB hcount₁ hcount₂ have h₁ := (hsource branch₁ E hB hcount₁).1 have h₂ := (hsource branch₂ E hB hcount₂).1 calc |((ndA5HarmonicSourcePMF B branch₁ hB hcount₁).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ((ndA5HarmonicSourcePMF B branch₂ hB hcount₂).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal| = |(((ndA5HarmonicSourcePMF B branch₁ hB hcount₁).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E) - (((ndA5HarmonicSourcePMF B branch₂ hB hcount₂).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E)| := by ring _ ≤ |((ndA5HarmonicSourcePMF B branch₁ hB hcount₁).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E| + |((ndA5HarmonicSourcePMF B branch₂ hB hcount₂).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E| := by simpa only [sub_eq_add_neg, abs_neg] using (abs_add_le (((ndA5HarmonicSourcePMF B branch₁ hB hcount₁).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E) (-(((ndA5HarmonicSourcePMF B branch₂ hB hcount₂).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ndA5ReferenceCommonProfile B E))) _ ≤ 8 * ndA5ReferenceInteriorScalarRate B d + 8 * ndA5ReferenceInteriorScalarRate B d := add_le_add h₁ h₂ _ = 16 * ndA5ReferenceInteriorScalarRate B d := by ring end end ND end Erdos1135