import Erdos1135.ND.Discrepancy.A5ReferenceGenuineSourceRate import Erdos1135.ND.Probability.PassTotalizer import Erdos1135.Tao.Probability.FullL1 /-! # A5 common-target totalized source rates This leaf applies the neutral complement totalizer and one finite Hahn step to the checked genuine whole-source comparisons. Both source branches use the same passage target `B` and therefore the same finite output carrier. -/ namespace Erdos1135 namespace ND open Filter open scoped Topology noncomputable section private def ndNatCastPassCarrierEquiv (B : ℕ) : {M : ℕ // M ≤ B} ≃ {M : ℕ // M ≤ Nat.floor (B : ℝ)} where toFun M := ⟨M.1, by simpa using M.2⟩ invFun M := ⟨M.1, by simpa using M.2⟩ left_inv M := by apply Subtype.ext; rfl right_inv M := by apply Subtype.ext; rfl private theorem val_syracusePassLocationAtMostOrOne_eq_of_eq {B B' N : ℕ} (hBB' : B = B') (hB : 1 ≤ B) (hB' : 1 ≤ B') : (Tao.syracusePassLocationAtMostOrOne B N hB).1 = (Tao.syracusePassLocationAtMostOrOne B' N hB').1 := by subst B' rfl private theorem taoTV_map_equiv {α β : Type*} [Fintype α] [Fintype β] (p q : PMF α) (e : α ≃ β) : Tao.taoTV (p.map e) (q.map e) = Tao.taoTV p q := by apply le_antisymm · exact Tao.taoTV_map_le p q e · have h := Tao.taoTV_map_le (p.map e) (q.map e) e.symm have he : e.symm ∘ e = id := by funext x exact e.symm_apply_apply x rw [PMF.map_comp, PMF.map_comp, he, PMF.map_id, PMF.map_id] at h exact h /-- The totalized flat and harmonic passage laws on either source branch are within thirty-two reference-rate copies in project full L1. -/ theorem eventually_ndA5FlatHarmonicTotalizedPassTV_le_thirty_two {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) (hB : 1 ≤ B) (hcount : 0 < ndA5BandCount B branch), Tao.taoTV (ndTotalizedPassLaw B hB (ndA5FlatSourcePMF B branch hB hcount)) (ndTotalizedPassLaw B hB (ndA5HarmonicSourcePMF B branch hB hcount)) ≤ 32 * ndA5ReferenceInteriorScalarRate B d := by filter_upwards [eventually_abs_ndA5HarmonicSourcePassMass_sub_flat_le_sixteen hPhase hd hd20 hdk] with B hsource intro branch hB hcount have hgen : ∀ E : Set ℕ, |((ndA5FlatSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal - ((ndA5HarmonicSourcePMF B branch hB hcount).toOuterMeasure (Tao.taoSection5PassEvent B E)).toReal| ≤ 16 * ndA5ReferenceInteriorScalarRate B d := by intro E simpa only [abs_sub_comm] using hsource branch E hB hcount have hTV := taoTV_ndTotalizedPassLaw_le_two_mul_of_genuine hB (ndA5FlatSourcePMF B branch hB hcount) (ndA5HarmonicSourcePMF B branch hB hcount) hgen calc Tao.taoTV (ndTotalizedPassLaw B hB (ndA5FlatSourcePMF B branch hB hcount)) (ndTotalizedPassLaw B hB (ndA5HarmonicSourcePMF B branch hB hcount)) ≤ 2 * (16 * ndA5ReferenceInteriorScalarRate B d) := hTV _ = 32 * ndA5ReferenceInteriorScalarRate B d := by ring /-- The two harmonic source branches have totalized passage laws within thirty-two reference-rate copies, the LOG-BLOCK sibling endpoint. -/ theorem eventually_ndA5HarmonicBranchesTotalizedPassTV_le_thirty_two {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) (hB : 1 ≤ B) (hcount₁ : 0 < ndA5BandCount B branch₁) (hcount₂ : 0 < ndA5BandCount B branch₂), Tao.taoTV (ndTotalizedPassLaw B hB (ndA5HarmonicSourcePMF B branch₁ hB hcount₁)) (ndTotalizedPassLaw B hB (ndA5HarmonicSourcePMF B branch₂ hB hcount₂)) ≤ 32 * ndA5ReferenceInteriorScalarRate B d := by filter_upwards [eventually_abs_ndA5HarmonicSourcePassMass_sub_harmonic_le_sixteen hPhase hd hd20 hdk] with B hsource intro branch₁ branch₂ hB hcount₁ hcount₂ have hgen : ∀ E : Set ℕ, |((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 intro E exact hsource branch₁ branch₂ E hB hcount₁ hcount₂ have hTV := taoTV_ndTotalizedPassLaw_le_two_mul_of_genuine hB (ndA5HarmonicSourcePMF B branch₁ hB hcount₁) (ndA5HarmonicSourcePMF B branch₂ hB hcount₂) hgen calc Tao.taoTV (ndTotalizedPassLaw B hB (ndA5HarmonicSourcePMF B branch₁ hB hcount₁)) (ndTotalizedPassLaw B hB (ndA5HarmonicSourcePMF B branch₂ hB hcount₂)) ≤ 2 * (16 * ndA5ReferenceInteriorScalarRate B d) := hTV _ = 32 * ndA5ReferenceInteriorScalarRate B d := by ring /-- At a natural passage target, the ND public full-L1 convention is exactly the common-target totalized A5 flat/harmonic distance. -/ theorem passFullL1_natCast_taoSection5SourceY_eq_ndA5TotalizedPassTV {B : ℕ} (hB : 1 ≤ B) (branch : Tao.TaoSection5SourceBranch) (hcount : 0 < ndA5BandCount B branch) : passFullL1 (B : ℝ) (Tao.taoSection5SourceY B branch) (show (1 : ℝ) ≤ B by exact_mod_cast hB) (ndA5OddBlock_sourceY_nonempty hB branch hcount) (logFinsetMass_ndA5OddBlock_sourceY_pos hB branch hcount) = Tao.taoTV (ndTotalizedPassLaw B hB (ndA5FlatSourcePMF B branch hB hcount)) (ndTotalizedPassLaw B hB (ndA5HarmonicSourcePMF B branch hB hcount)) := by let e := ndNatCastPassCarrierEquiv B let p := ndTotalizedPassLaw B hB (ndA5FlatSourcePMF B branch hB hcount) let q := ndTotalizedPassLaw B hB (ndA5HarmonicSourcePMF B branch hB hcount) have hflat : uniformPassLaw (B : ℝ) (Tao.taoSection5SourceY B branch) (show (1 : ℝ) ≤ B by exact_mod_cast hB) (ndA5OddBlock_sourceY_nonempty hB branch hcount) = p.map e := by dsimp only [p, e] unfold uniformPassLaw ndA5FlatSourcePMF ndTotalizedPassLaw rw [PMF.map_comp, PMF.map_comp] congr 1 funext N apply Subtype.ext change (Tao.syracusePassLocationAtMostOrOne (Nat.floor (B : ℝ)) N.1 _).1 = (Tao.syracusePassLocationAtMostOrOne B N.1 hB).1 exact val_syracusePassLocationAtMostOrOne_eq_of_eq (Nat.floor_natCast B) _ _ have hharmonic : logPassLaw (B : ℝ) (Tao.taoSection5SourceY B branch) (show (1 : ℝ) ≤ B by exact_mod_cast hB) (logFinsetMass_ndA5OddBlock_sourceY_pos hB branch hcount) = q.map e := by dsimp only [q, e] unfold logPassLaw ndA5HarmonicSourcePMF ndTotalizedPassLaw rw [PMF.map_comp, PMF.map_comp] congr 1 funext N apply Subtype.ext change (Tao.syracusePassLocationAtMostOrOne (Nat.floor (B : ℝ)) N.1 _).1 = (Tao.syracusePassLocationAtMostOrOne B N.1 hB).1 exact val_syracusePassLocationAtMostOrOne_eq_of_eq (Nat.floor_natCast B) _ _ unfold passFullL1 rw [hflat, hharmonic] exact taoTV_map_equiv p q e /-- Consumer-shaped natural-target specialization of the ND public `passFullL1` convention. -/ theorem eventually_passFullL1_natCast_taoSection5SourceY_le_thirty_two {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) (hB : 1 ≤ B) (hcount : 0 < ndA5BandCount B branch), passFullL1 (B : ℝ) (Tao.taoSection5SourceY B branch) (show (1 : ℝ) ≤ B by exact_mod_cast hB) (ndA5OddBlock_sourceY_nonempty hB branch hcount) (logFinsetMass_ndA5OddBlock_sourceY_pos hB branch hcount) ≤ 32 * ndA5ReferenceInteriorScalarRate B d := by filter_upwards [eventually_ndA5FlatHarmonicTotalizedPassTV_le_thirty_two hPhase hd hd20 hdk] with B hTV intro branch hB hcount rw [passFullL1_natCast_taoSection5SourceY_eq_ndA5TotalizedPassTV hB branch hcount] exact hTV branch hB hcount end end ND end Erdos1135