| import OAI.AlgebraicGeometry.PlaneCurves.BasisNoncancellation |
| import OAI.AlgebraicGeometry.PlaneCurves.MarkedDisplacements |
| import OAI.AlgebraicGeometry.PlaneCurves.UniversalSections |
|
|
| namespace OAI |
|
|
| /-! |
| # Collision for geometric section families and necessary sections |
| -/ |
|
|
| section |
|
|
| /-! |
| ## Source collision on the actual Wτ |
| -/ |
|
|
| noncomputable section |
| namespace Nagata.W20 |
| open scoped BigOperators Topology |
| open Nagata.Workers.W12 Nagata.Workers.W10 Nagata.FiniteExponents |
| open Nagata.CoefficientSpaces Nagata.W22 |
|
|
| / |
| and actual jet map. Finite basis and all analytic coefficient premises have been |
| discharged. Only the universal necessary-W geometric input remains. -/ |
| theorem source_collision_of_every_configuration |
| {τ : ℝ} (hτ : 0 < τ) (hτone : τ < 1) |
| (marks : Fin 9 → ℝ) (a delta x0 : ℝ) (d : ℤ) (m q : ℕ) |
| (hs : (∑ i, marks i) = 9 * x0 - delta) |
| (hd : 3 * (m : ℤ) < d) |
| (hfirst : ∀ j : ℤ, 0 ≤ j → j ≤ (m : ℤ) → |
| firstLower d (m : ℤ) j a delta ∉ Set.range (Int.cast : ℤ → ℝ)) |
| (hsecond : ∀ j : ℤ, (m : ℤ) < j → j ≤ d / 3 → 0 < d - 3 * j → |
| secondLower d j a delta ∉ Set.range (Int.cast : ℤ → ℝ)) |
| {U : Set ℂ} (hU : IsOpen U) (h0 : (0 : ℂ) ∈ U) |
| (hEvery : ∀ p : Fin q → ℂ, Function.Injective p → (∀ i, p i ∈ U) → |
| ∃ F : genuineSourceSections hτ hτone marks a d m, F ≠ 0 ∧ |
| ∀ i, Nagata.Workers.W28.HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun x : ℂ × ℂ => localScalar F (tauPower τ x0) x.1 x.2) (p i, 0) m) : |
| ∃ F₀ : genuineSourceSections hτ hτone marks a d m, |
| F₀ ≠ 0 ∧ genuineSourceJetMap hτ hτone marks a x0 d m q F₀ = 0 := by |
| exact actual_kernel_of_every_configuration |
| (fun z hz => (rayMarkedSection_analyticOnNhd hτ hτone marks z hz).differentiableAt) |
| (tauPower_ne_zero hτ x0) |
| (genuineSourceBasis hτ hτone marks a delta x0 d m hs hd hfirst hsecond) |
| hU h0 hEvery |
|
|
| / |
| fixed covering frame. Row first coordinate is b, second is ell. -/ |
| theorem genuineSourceJetMap_eq_mixed_derivative |
| {τ : ℝ} (hτ : 0 < τ) (hτone : τ < 1) |
| (marks : Fin 9 → ℝ) (a x0 : ℝ) (d : ℤ) (m q : ℕ) |
| (F : genuineSourceSections hτ hτone marks a d m) |
| (row : Nagata.Workers.W30.JetIndex q m) : |
| genuineSourceJetMap hτ hτone marks a x0 d m q F row = |
| iteratedDeriv row.2.val |
| (fun x => iteratedDeriv row.1.val |
| (fun y => localScalar F (tauPower τ x0) x y) 0) 0 := by |
| exact actualJetMap_apply_eq_mixed_derivative _ _ F row |
|
|
| end Nagata.W20 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.W20 |
| open scoped BigOperators Topology |
| open Nagata.Workers.W12 Nagata.Workers.W10 Nagata.CoefficientSpaces |
| open Nagata.Workers.W17 Nagata.Workers.W28 Nagata.ProjectiveGeometry |
|
|
| / |
| by the source basis and collision map, with all scalar identifications discharged. -/ |
| theorem source_necessary_W_of_cubic_geometry {q d m : ℕ} {τ : ℝ} |
| (hτ : 0 < τ) (hτone : τ < 1) |
| (hUniversal : Nagata.W13.UniversalSupport (9 + q) d (fun _ => m)) |
| (marks : Fin 9 → ℝ) (a x0 : ℝ) |
| (hdisjoint : Pairwise (fun i j => ∀ k : ℤ, |
| ((τ ^ marks i : ℝ) : ℂ) ≠ ((τ ^ marks j : ℝ) : ℂ) * (τ : ℂ) ^ k)) |
| (ξ : Fin q → ℂ) |
| (hpP : ∀ i, (Nagata.W22.rayMarkedSection hτ hτone marks).val |
| (tauPower τ x0 * Complex.exp (ξ i)) ≠ 0) |
| (C : MvPolynomial (Fin 3) ℂ) (hC : C.IsHomogeneous 3) |
| (X : Fin 3 → Nagata.W08.automorphicSections (τ : ℂ) 3 |
| (tauPower τ (((∑ i, marks i) + a) / 3))) |
| (hrelations : Nagata.Workers.W25.polynomialRelationIdeal (fun i => (X i).val) = |
| Ideal.span ({C} : Set _)) |
| (chart : Fin (9 + q) → Fin 3) |
| (hc : ∀ b, (X (chart b)).val |
| (markedNormalBase (fun i => ((τ ^ marks i : ℝ) : ℂ)) |
| (logarithmicPoints (tauPower τ x0) ξ) b) ≠ 0) |
| (hbase : Function.Injective (fun b => chartPoint₂ (chart b) |
| (planeCoordinate (normalizedCurvePoint (fun i => (X i).val) (chart b) |
| (markedNormalBase (fun i => ((τ ^ marks i : ℝ) : ℂ)) |
| (logarithmicPoints (tauPower τ x0) ξ) b))))) |
| (hpartial : ∀ b, planePolynomialEval |
| (MvPolynomial.pderiv 1 (Nagata.W27.directChartHom (chart b) C)) |
| (normalizedCurvePoint (fun i => (X i).val) (chart b) |
| (markedNormalBase (fun i => ((τ ^ marks i : ℝ) : ℂ)) |
| (logarithmicPoints (tauPower τ x0) ξ) b)) ≠ 0) : |
| ∃ F : genuineSourceSections hτ hτone marks a (d : ℤ) m, F ≠ 0 ∧ |
| ∀ i, HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun x : ℂ × ℂ => localScalar F (tauPower τ x0) x.1 x.2) (ξ i, 0) m := by |
| have ht : ‖(τ : ℂ)‖ < 1 := by |
| simpa only [Complex.norm_real, Real.norm_eq_abs, abs_of_pos hτ] using hτone |
| have ht0 : (τ : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hτ.ne' |
| have hm : ∀ i, ((τ ^ marks i : ℝ) : ℂ) ≠ 0 := |
| fun i => Complex.ofReal_ne_zero.mpr (Real.rpow_pos_of_pos hτ (marks i)).ne' |
| have heq := nineThetaSection_ray_eq hτ hτone ht ht0 marks hm |
| have hp : ∀ i, (nineThetaSection ht ht0 (fun i => ((τ ^ marks i : ℝ) : ℂ)) hm).val |
| ((logarithmicPoints (tauPower τ x0) ξ) i).1 ≠ 0 := by |
| simpa only [heq, logarithmicPoints] using hpP |
| have hW := universalSupport_necessary_W_local hUniversal ht ht0 |
| (tauPower_ne_zero hτ (((∑ i, marks i) + a) / 3)) |
| (fun i => ((τ ^ marks i : ℝ) : ℂ)) hm hdisjoint |
| (tauPower τ x0) (tauPower_ne_zero hτ x0) ξ hp C hC X hrelations chart hc hbase hpartial |
| change ∃ F : ActualSection (τ : ℂ) |
| (tauPower τ (((∑ i, marks i) + a) / 3)) (∏ i, -((τ ^ marks i : ℝ) : ℂ)) |
| (d : ℤ) m (Nagata.W22.rayMarkedSection hτ hτone marks).val, F ≠ 0 ∧ |
| ∀ i, HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun x : ℂ × ℂ => localScalar F (tauPower τ x0) x.1 x.2) (ξ i, 0) m at hW |
| exact Eq.mp (congrArg (fun γ : ℂ => |
| ∃ F : ActualSection (τ : ℂ) (tauPower τ (((∑ i, marks i) + a) / 3)) γ |
| (d : ℤ) m (Nagata.W22.rayMarkedSection hτ hτone marks).val, F ≠ 0 ∧ |
| ∀ i, HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun x : ℂ × ℂ => localScalar F (tauPower τ x0) x.1 x.2) (ξ i, 0) m) |
| (rayMarkedMultiplier hτ marks)) hW |
|
|
| end Nagata.W20 |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Exact fixed-τ source collision from universal support and concrete cubic |
| chart geometry. The common marked-point disk and Pi nonvanishing are supplied |
| by the proved fixed-period configuration construction, not by caller assumptions. -/ |
| noncomputable section |
| namespace Nagata.W20 |
| open scoped BigOperators Topology |
| open Nagata.Workers.W12 Nagata.Workers.W10 Nagata.CoefficientSpaces |
| open Nagata.Workers.W17 Nagata.Workers.W28 Nagata.ProjectiveGeometry |
| open Nagata.FiniteExponents |
|
|
| / |
| on the common exponential chart. -/ |
| theorem source_markedPi_cover_eq {τ : ℝ} (hτ : 0 < τ) (hτone : τ < 1) |
| (marks : Fin 9 → ℝ) (x0 : ℝ) (x : ℂ) : |
| (Nagata.W22.rayMarkedSection hτ hτone marks).val |
| (tauPower τ x0 * Complex.exp x) = |
| Nagata.W06.TorusConfiguration.markedPi τ marks |
| (Nagata.Workers.W05.chartPoint τ x0 x : ℂ) := by |
| have hz : tauPower τ x0 * Complex.exp x ≠ 0 := |
| mul_ne_zero (tauPower_ne_zero hτ x0) (Complex.exp_ne_zero x) |
| unfold Nagata.W22.rayMarkedSection |
| rw [Nagata.W08.markedThetaSection_apply _ _ _ _ hz] |
| unfold Nagata.W06.TorusConfiguration.markedPi |
| simp only [Nagata.Workers.W05.chartPoint_eq_real_rpow hτ, |
| Nagata.Workers.W05.rayPoint_eq_real_rpow hτ, tauPower] |
|
|
| / |
| theorem source_marked_orbits_disjoint {τ : ℝ} (hτ : 0 < τ) (hτone : τ < 1) |
| (marks : Fin 9 → ℝ) (hinj : Function.Injective marks) |
| (hb : ∀ i, 0 < marks i ∧ marks i < 1 / 2) : |
| Pairwise (fun i j => ∀ k : ℤ, |
| ((τ ^ marks i : ℝ) : ℂ) ≠ ((τ ^ marks j : ℝ) : ℂ) * (τ : ℂ) ^ k) := by |
| intro i j hij k he |
| apply hij |
| apply hinj |
| have heq : Nagata.W21.torusPointMk (Nagata.Workers.W05.positivePeriod τ hτ) |
| (Nagata.Workers.W05.rayPoint τ (marks j)) = |
| Nagata.W21.torusPointMk (Nagata.Workers.W05.positivePeriod τ hτ) |
| (Nagata.Workers.W05.rayPoint τ (marks i)) := by |
| apply Quotient.sound |
| apply (Nagata.W21.torusOrbitSetoid_iff_coe _ _ _).mpr |
| refine ⟨k, ?_⟩ |
| simpa only [Nagata.Workers.W05.rayPoint_eq_real_rpow hτ, |
| Nagata.Workers.W05.positivePeriod, Units.val_mk0, mul_comm] using he |
| have hj : marks j ∈ Set.Ioo (0 : ℝ) 1 := ⟨(hb j).1, by linarith [(hb j).2]⟩ |
| have hi : marks i ∈ Set.Ioo (0 : ℝ) 1 := ⟨(hb i).1, by linarith [(hb i).2]⟩ |
| exact (Nagata.W06.TorusConfiguration.rayTorusPoint_injective_on_unit_interval |
| hτ hτone hj hi heq).symm |
|
|
| / |
| cubic chart geometry. No finite basis, analytic section, Pi-nonvanishing, local |
| multiplicity, or moving-sequence hypothesis remains. -/ |
| theorem source_collision_of_universalSupport_and_cubic_geometry |
| {τ : ℝ} (hτ : 0 < τ) (hτone : τ < 1) |
| (marks : Fin 9 → ℝ) (hmarks : Function.Injective marks) |
| (hmbounds : ∀ i, 0 < marks i ∧ marks i < 1 / 2) |
| (a delta : ℝ) (d m q : ℕ) |
| (hs : (∑ i, marks i) = 9 * (1 / 2 : ℝ) - delta) |
| (hd : 3 * (m : ℤ) < (d : ℤ)) |
| (hfirst : ∀ j : ℤ, 0 ≤ j → j ≤ (m : ℤ) → |
| firstLower (d : ℤ) (m : ℤ) j a delta ∉ Set.range (Int.cast : ℤ → ℝ)) |
| (hsecond : ∀ j : ℤ, (m : ℤ) < j → j ≤ (d : ℤ) / 3 → 0 < (d : ℤ) - 3 * j → |
| secondLower (d : ℤ) j a delta ∉ Set.range (Int.cast : ℤ → ℝ)) |
| (hUniversal : Nagata.W13.UniversalSupport (9 + q) d (fun _ => m)) |
| (C : MvPolynomial (Fin 3) ℂ) (hC : C.IsHomogeneous 3) |
| (X : Fin 3 → Nagata.W08.automorphicSections (τ : ℂ) 3 |
| (tauPower τ (((∑ i, marks i) + a) / 3))) |
| (hrelations : Nagata.Workers.W25.polynomialRelationIdeal (fun i => (X i).val) = |
| Ideal.span ({C} : Set _)) |
| (hcharts : ∀ ξ : Fin q → ℂ, |
| Function.Injective (Nagata.W06.TorusConfiguration.torusTuple τ hτ (1 / 2) marks ξ) → |
| ∃ chart : Fin (9 + q) → Fin 3, |
| (∀ b, (X (chart b)).val |
| (markedNormalBase (fun i => ((τ ^ marks i : ℝ) : ℂ)) |
| (logarithmicPoints (tauPower τ (1 / 2)) ξ) b) ≠ 0) ∧ |
| Function.Injective (fun b => chartPoint₂ (chart b) |
| (planeCoordinate (normalizedCurvePoint (fun i => (X i).val) (chart b) |
| (markedNormalBase (fun i => ((τ ^ marks i : ℝ) : ℂ)) |
| (logarithmicPoints (tauPower τ (1 / 2)) ξ) b)))) ∧ |
| (∀ b, planePolynomialEval |
| (MvPolynomial.pderiv 1 (Nagata.W27.directChartHom (chart b) C)) |
| (normalizedCurvePoint (fun i => (X i).val) (chart b) |
| (markedNormalBase (fun i => ((τ ^ marks i : ℝ) : ℂ)) |
| (logarithmicPoints (tauPower τ (1 / 2)) ξ) b)) ≠ 0)) : |
| ∃ F₀ : genuineSourceSections hτ hτone marks a (d : ℤ) m, |
| F₀ ≠ 0 ∧ genuineSourceJetMap hτ hτone marks a (1 / 2) (d : ℤ) m q F₀ = 0 := by |
| obtain ⟨R, hR, hcfg⟩ := Nagata.W06.TorusConfiguration.exists_disk_for_fixed_period |
| hτ hτone marks hmarks hmbounds |
| apply source_collision_of_every_configuration hτ hτone marks a delta (1 / 2) |
| (d : ℤ) m q hs hd hfirst hsecond Metric.isOpen_ball (Metric.mem_ball_self hR) |
| intro ξ hξ hξU |
| have hnorm : ∀ i, ‖ξ i‖ < R := by |
| intro i |
| simpa only [Metric.mem_ball, dist_zero_right] using hξU i |
| obtain ⟨hinj, hPi⟩ := hcfg q ξ hξ hnorm |
| obtain ⟨chart, hc, hbase, hpartial⟩ := hcharts ξ hinj |
| apply source_necessary_W_of_cubic_geometry hτ hτone hUniversal marks a (1 / 2) |
| (source_marked_orbits_disjoint hτ hτone marks hmarks hmbounds) ξ ?_ |
| C hC X hrelations chart hc hbase hpartial |
| intro i |
| rw [source_markedPi_cover_eq] |
| exact hPi i |
|
|
| end Nagata.W20 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
