import OAI.Geometry.HarmonicGrowth.Cycling import OAI.Geometry.HarmonicGrowth.Intrinsic namespace OAI noncomputable section open Filter open scoped BigOperators Topology ContDiff namespace HarmonicCounterexample open Matrix Berger Cartesian variable {n : ℕ} lemma laplace_congr (g : SmoothMetric n) {f h : Space n → ℝ} {x : Space n} (he : f =ᶠ[𝓝 x] h) : laplaceBeltrami g f x = laplaceBeltrami g h x := by have hd (i j : Fin n) : coordDeriv i (coordDeriv j f) x = coordDeriv i (coordDeriv j h) x := coordDeriv_congr (he.eventually_nhds.mono fun _ hy => coordDeriv_congr hy j) i simp only [laplaceBeltrami,coordDeriv_congr he,hd] lemma laplace_eq_euclidean_of_flat (g : SmoothMetric n) {x : Space n} (hg : ∀ᶠ y in 𝓝 x, g.coeff y = 1) (f : Space n → ℝ) : laplaceBeltrami g f x = euclideanLaplacian f x := by have hd (k i j : Fin n) : coordDeriv k (fun y => g.coeff y i j) x = 0 := by rw [coordDeriv_congr (hg.mono fun _ hy => congrFun (congrFun hy i) j) k] exact coordDeriv_const _ _ _ have hΓ (k i j : Fin n) : christoffel g x k i j = 0 := by simp [christoffel,hd] simp [laplaceBeltrami,hΓ,hg.self_of_nhds,Matrix.one_apply,euclideanLaplacian,hessian, Matrix.trace,Matrix.diag] end HarmonicCounterexample end noncomputable section open Filter open scoped BigOperators Topology ContDiff namespace HarmonicCounterexample.Cartesian open Matrix Berger variable {n : ℕ} {ι : Type*} [Fintype ι] lemma synthesized_harmonic_at_ne_zero (g : SmoothMetric n) (J : ComplexStructure (Fin n)) {l : ℝ} {P : ι → Space n → ℝ} {Y Y1 Y2 : ℝ → ι → ℝ} (hY : ∀ t i, HasDerivAt (fun s => Y s i) (Y1 t i) t) (hY1 : ∀ t i, HasDerivAt (fun s => Y1 s i) (Y2 t i) t) (hP : ∀ i, ContDiff ℝ ∞ (P i)) (hE : ∀ i x, euler (P i) x = l*P i x) (hH : ∀ i x, euclideanLaplacian (P i) x = 0) {x : Space n} (hx : x ≠ 0) {A q : ℝ} (hA : A ≠ 0) (hq : q ≠ 0) (u v : ℝ) (hg : (g.coeff x)⁻¹ = polarTensor J x A⁻¹ (A*q)⁻¹) (hΓ : ∀ k i j, christoffel g x k i j = normalizedConnectionCoeff J x A q u v k i j) (H : Matrix ι ι ℝ) (hW : ∀ i, hopfDerivative J (hopfDerivative J (P i)) x = ∑ j, H i j*P j x) (hODE : ∀ i, Y2 (logRadius x) i+((n:ℝ)-2+((n:ℝ)-1)*u+v)*Y1 (logRadius x) i = A⁻¹*l*(l+(n:ℝ)-2)*Y (logRadius x) i- ((A*q)⁻¹-A⁻¹)*∑ j, H j i*Y (logRadius x) j) : laplaceBeltrami g (synthesized l P Y) x = 0 := by have hdiff (i : ι) {y : Space n} (hy : y ≠ 0) : DifferentiableAt ℝ (fun y => regularCoefficient l Y i (logRadius y)*P i y) y := (logProfile_differentiableAt (fun t => regularCoefficient_deriv (hY t) i) hy).mul ((hP i).differentiable (by simp) y) have hnd (i : ι) : ∀ᶠ y in 𝓝 x, DifferentiableAt ℝ (fun y => regularCoefficient l Y i (logRadius y)*P i y) y := (eventually_ne_nhds hx).mono fun _ hy => hdiff i hy have hdd (i : ι) (j : Fin n) : DifferentiableAt ℝ (coordDeriv j (fun y => regularCoefficient l Y i (logRadius y)*P i y)) x := by exact coordDeriv_product_differentiableAt ((eventually_ne_nhds hx).mono fun _ hy => logProfile_differentiableAt (fun t => regularCoefficient_deriv (hY t) i) hy) (Filter.Eventually.of_forall ((hP i).differentiable (by simp))) (logProfile_coord_differentiableAt (fun t => regularCoefficient_deriv (hY t) i) (fun t => regularCoefficient_second (hY t) (hY1 t) i) hx) (fun j => (coordDeriv_smooth (hP i) j).differentiable (by simp) x) j apply (mul_eq_zero.mp ?_).resolve_left (squareRadius_pos hx).ne' change (x ⬝ᵥ x)*laplaceBeltrami g (synthesized l P Y) x = 0 unfold synthesized rw [laplace_sum g hnd hdd,Finset.mul_sum] simp only [laplace_regular_term g J hY hY1 (hP _) (hE _) (hH _) hx hA hq u v hg hΓ, hODE,hW] let d := ((A*q)⁻¹-A⁻¹) have he (j : ι) : (A⁻¹*l*(l+(n:ℝ)-2)*Y (logRadius x) j-d*(∑ i,H i j*Y (logRadius x) i)- A⁻¹*l*(l+(n:ℝ)-2)*Y (logRadius x) j)*P j x+ d*Y (logRadius x) j*(∑ i,H j i*P i x) = d*((∑ i,Y (logRadius x) j*H j i*P i x)- ∑ i,Y (logRadius x) i*H i j*P j x) := by have h1 : (∑ i,H i j*Y (logRadius x) i)*P j x = ∑ i,Y (logRadius x) i*H i j*P j x := by rw [Finset.sum_mul] apply Finset.sum_congr rfl intro i _ ring have h2 : Y (logRadius x) j*(∑ i,H j i*P i x) = ∑ i,Y (logRadius x) j*H j i*P i x := by rw [Finset.mul_sum] apply Finset.sum_congr rfl intro i _ ring calc _ = d*(Y (logRadius x) j*(∑ i,H j i*P i x)- (∑ i,H i j*Y (logRadius x) i)*P j x) := by ring _ = _ := by rw [h1,h2] simp only [← show d = ((A*q)⁻¹-A⁻¹) from rfl,he,← Finset.mul_sum, Finset.sum_sub_distrib] have hs : (∑ j,∑ i,Y (logRadius x) j*H j i*P i x) = ∑ j,∑ i,Y (logRadius x) i*H i j*P j x := Finset.sum_comm rw [hs,sub_self,mul_zero,mul_zero] end HarmonicCounterexample.Cartesian end noncomputable section open Filter open scoped BigOperators Topology ContDiff namespace HarmonicCounterexample.Angular open MvPolynomial Matrix Berger Cartesian variable {n : ℕ} abbrev evaluate (P : PolynomialSpace n) (x : Space n) : ℝ := MvPolynomial.eval x P lemma evaluate_smooth (P : PolynomialSpace n) : ContDiff ℝ ∞ (evaluate P) := by unfold evaluate induction P using MvPolynomial.induction_on with | C c => simpa [Angular.evaluate] using (contDiff_const : ContDiff ℝ ∞ (fun _ : Space n => c)) | add P Q hp hq => simpa only [Angular.evaluate, eval_add] using hp.add hq | mul_X P i hp => simpa only [Angular.evaluate, eval_mul, eval_X] using hp.mul (contDiff_apply ℝ ℝ i) lemma coordDeriv_evaluate (P : PolynomialSpace n) (i : Fin n) : coordDeriv i (evaluate P) = evaluate (pderiv i P) := by ext x unfold evaluate induction P using MvPolynomial.induction_on with | C c => simp [coordDeriv_const] | add P Q hp hq => simp only [eval_add] change coordDeriv i (fun y => eval y P+eval y Q) x = _ rw [coordDeriv_add ((evaluate_smooth P).differentiable (by simp) x) ((evaluate_smooth Q).differentiable (by simp) x),hp,hq] simp | mul_X P j hp => simp only [eval_mul,eval_X] change coordDeriv i (fun y => eval y P*y j) x = _ rw [coordDeriv_mul ((evaluate_smooth P).differentiable (by simp) x) (hasFDerivAt_apply j x).differentiableAt,hp,coordDeriv_coordinate] simp only [Angular.evaluate,pderiv_mul,pderiv_X,eval_add,eval_mul,eval_X] by_cases h : i = j · subst j; simp [add_comm,mul_comm] · simp [Ne.symm h,mul_comm] lemma euler_evaluate {P : PolynomialSpace n} {l : ℕ} (hP : P.IsHomogeneous l) (x : Space n) : euler (evaluate P) x = (l : ℝ)*evaluate P x := by have h := congrArg (eval x) hP.sum_X_mul_pderiv simpa only [map_sum,eval_mul,eval_X,map_nsmul,nsmul_eq_mul,map_natCast, euler,Berger.gradient,dotProduct,coordDeriv_evaluate,Angular.evaluate] using h lemma euclideanLaplacian_evaluate (P : PolynomialSpace n) (x : Space n) : euclideanLaplacian (evaluate P) x = evaluate (polynomialLaplacian P) x := by simp [euclideanLaplacian,hessian,Matrix.trace,Matrix.diag,coordDeriv_evaluate,Angular.evaluate, polynomialLaplacian] lemma harmonicPolynomials_euler {l : ℕ} (P : harmonicPolynomials n l) (x : Space n) : euler (evaluate (P : PolynomialSpace n)) x = (l:ℝ)*evaluate (P : PolynomialSpace n) x := euler_evaluate P.property.1 x lemma harmonicPolynomials_harmonic {l : ℕ} (P : harmonicPolynomials n l) (x : Space n) : euclideanLaplacian (evaluate (P : PolynomialSpace n)) x = 0 := by rw [euclideanLaplacian_evaluate,show polynomialLaplacian (P : PolynomialSpace n) = 0 from P.property.2] simp [Angular.evaluate] end HarmonicCounterexample.Angular end noncomputable section open Filter open scoped BigOperators Topology ContDiff namespace HarmonicCounterexample.Angular open MvPolynomial Matrix Berger Cartesian variable {n : ℕ} lemma evaluate_injective : Function.Injective (evaluate (n := n)) := by intro P Q h apply MvPolynomial.funext intro x exact congrFun h x lemma evaluate_homogeneous {P : PolynomialSpace n} {l : ℕ} (hP : P.IsHomogeneous l) (c : ℝ) (x : Space n) : evaluate P (c • x) = c^l*evaluate P x := by unfold evaluate conv_lhs => rw [P.as_sum] conv_rhs => arg 2; rw [P.as_sum] simp only [map_sum,Finset.mul_sum] apply Finset.sum_congr rfl intro d hd have hdeg : Finsupp.degree d = l := by rw [Finsupp.degree_eq_weight_one] exact hP (MvPolynomial.mem_support_iff.1 hd) simp only [eval_monomial,Finsupp.prod,Pi.smul_apply,smul_eq_mul,mul_pow, Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum,← Finsupp.degree_apply,hdeg] ring lemma evaluate_zero_of_pos_degree {P : PolynomialSpace n} {l : ℕ} (hP : P.IsHomogeneous l) (hl : 0 < l) : evaluate P 0 = 0 := by simpa [zero_pow (Nat.ne_of_gt hl)] using evaluate_homogeneous hP 0 (0 : Space n) lemma evaluate_sum {ι : Type*} (s : Finset ι) (P : ι → PolynomialSpace n) (x : Space n) : evaluate (∑ i ∈ s,P i) x = ∑ i ∈ s,evaluate (P i) x := by simp [Angular.evaluate] lemma evaluate_smul (c : ℝ) (P : PolynomialSpace n) (x : Space n) : evaluate (c • P) x = c*evaluate P x := by simp [Angular.evaluate] lemma coordinate_radius_smul (c : ℝ) (x : Space n) : Cartesian.radius (c • x) = |c| *Cartesian.radius x := by simp only [Cartesian.radius,Cartesian.squareRadius,dotProduct,Pi.smul_apply,smul_eq_mul,mul_pow, show ∀ a : ℝ, a*a = a^2 from fun a => (pow_two a).symm,← Finset.mul_sum,Real.sqrt_mul (sq_nonneg c),Real.sqrt_sq_eq_abs] end HarmonicCounterexample.Angular end noncomputable section open Filter open scoped BigOperators Topology ContDiff namespace HarmonicCounterexample.Angular open MvPolynomial Matrix Berger Cartesian variable {n : ℕ} lemma hopf_evaluate (J : ComplexStructure (Fin n)) (P : PolynomialSpace n) (x : Space n) : hopfDerivative J (evaluate P) x = evaluate (polynomialRotation J.matrix P) x := by simp only [hopfDerivative,Berger.gradient,dotProduct,Matrix.mulVec,coordDeriv_evaluate, polynomialRotation_apply,evaluate_sum,evaluate_smul,Angular.evaluate,eval_mul,eval_X, Finset.sum_mul] apply Finset.sum_congr rfl intro i _ apply Finset.sum_congr rfl intro j _ ring end HarmonicCounterexample.Angular end noncomputable section open Filter open scoped BigOperators Topology ContDiff namespace HarmonicCounterexample.Angular open MvPolynomial Matrix Berger Cartesian variable {n : ℕ} lemma evaluate_zero_of_core {P : PolynomialSpace n} (hP : ∀ x, squareRadius x < 1 → evaluate P x = 0) : P = 0 := by let d : ℝ := 1/((n:ℝ)+1) have hn : (0:ℝ) ≤ n := Nat.cast_nonneg n have hd : 0 < d := by positivity have hnd : (n:ℝ)*d^2 < 1 := by dsimp [d] rw [div_pow,one_pow,← mul_div_assoc,mul_one,div_lt_one (sq_pos_of_pos (by positivity))] nlinarith [sq_nonneg (n:ℝ)] apply MvPolynomial.funext_set (fun _ => Set.Ioo (-d) d) (fun _ => Set.Ioo_infinite (by linarith)) intro x hx rw [map_zero] apply hP have hi (i : Fin n) : x i*x i ≤ d^2 := by have hh := hx i (Set.mem_univ i) nlinarith [hh.1,hh.2] have hs : squareRadius x ≤ (n:ℝ)*d^2 := by unfold squareRadius dotProduct calc _ ≤ ∑ i : Fin n,d^2 := Finset.sum_le_sum fun i _ => hi i _ = _ := by simp linarith /-- Equality on the flat core determines the actual polynomial, not merely its restriction to a formal angular vector space. -/ lemma evaluate_eq_of_core {P Q : PolynomialSpace n} (h : ∀ x, squareRadius x < 1 → evaluate P x = evaluate Q x) : P = Q := by apply sub_eq_zero.mp apply evaluate_zero_of_core intro x hx simp only [Angular.evaluate,eval_sub,h x hx,sub_self] lemma rotation_evaluate (J : ComplexStructure (Fin n)) (l : ℕ) (P : harmonicPolynomials n l) (x : Space n) : hopfDerivative J (evaluate (P : PolynomialSpace n)) x = evaluate ((rotation J l P) : PolynomialSpace n) x := hopf_evaluate J P x end HarmonicCounterexample.Angular end noncomputable section open Filter open scoped BigOperators Topology ContDiff namespace HarmonicCounterexample.Cartesian open Matrix Berger Angular MvPolynomial variable {n : ℕ} {ι : Type*} [Fintype ι] lemma synthesized_core_eq {l : ℝ} {P : ι → PolynomialSpace n} {Y : ℝ → ι → ℝ} {c : ι → ℝ} (hc : ∀ t ≤ 0, Y t = Real.exp (l*t) • c) {x : Space n} (hx : squareRadius x < 1) : synthesized l (fun i => evaluate (P i)) Y x = evaluate (∑ i,c i • P i) x := by have ht : logRadius x ≤ 0 := by unfold logRadius exact mul_nonpos_of_nonneg_of_nonpos (by norm_num) (Real.log_nonpos (squareRadius_nonneg x) hx.le) simp only [synthesized,regularCoefficient_core hc ht,evaluate_sum,evaluate_smul] lemma synthesized_harmonic_at_zero (g : SmoothMetric n) (hg : ∀ᶠ x in 𝓝 0, g.coeff x = 1) {l : ℕ} (P : ι → harmonicPolynomials n l) {Y : ℝ → ι → ℝ} {c : ι → ℝ} (hc : ∀ t ≤ 0, Y t = Real.exp ((l:ℝ)*t) • c) : laplaceBeltrami g (synthesized l (fun i => evaluate (P i : PolynomialSpace n)) Y) 0 = 0 := by have he : synthesized l (fun i => evaluate (P i : PolynomialSpace n)) Y =ᶠ[𝓝 0] evaluate (∑ i,c i • (P i : PolynomialSpace n)) := by have hr : ∀ᶠ x : Space n in 𝓝 0,squareRadius x < 1 := squareRadius_smooth.continuous.continuousAt.eventually (eventually_lt_nhds (by simp)) exact hr.mono fun _ hx => synthesized_core_eq hc hx rw [laplace_congr g he,laplace_eq_euclidean_of_flat g hg,euclideanLaplacian_evaluate] have hh : polynomialLaplacian (∑ i,c i • (P i : PolynomialSpace n)) = 0 := by simp only [map_sum,map_smul,show ∀ i, polynomialLaplacian (P i : PolynomialSpace n) = 0 from fun i => (P i).property.2,smul_zero,Finset.sum_const_zero] simp only [hh,Angular.evaluate,map_zero] /-- Global vanishing of synthesized functions forces vanishing of the initial polynomial. No limiting separation argument, ODE inverse, or unique-continuation claim is hidden here: the regular flow equals its polynomial on the open core. -/ lemma synthesized_injective_from_core {l : ℝ} {P : ι → PolynomialSpace n} {Y : ℝ → ι → ℝ} {c : ι → ℝ} (hc : ∀ t ≤ 0, Y t = Real.exp (l*t) • c) (hz : synthesized l (fun i => evaluate (P i)) Y = 0) : ∑ i,c i • P i = 0 := by apply evaluate_zero_of_core intro x hx rw [← synthesized_core_eq hc hx,hz] rfl end HarmonicCounterexample.Cartesian end noncomputable section namespace HarmonicCounterexample.Angular open MvPolynomial Matrix Berger open scoped ContDiff Topology InnerProductSpace variable {n : ℕ} /-- The round eigenvalue in degree l on S^(n-1). -/ def roundEigenvalue (n l : ℕ) : ℝ := (l:ℝ)*((l:ℝ)+(n:ℝ)-2) lemma roundEigenvalue_nonneg {n : ℕ} (hn : 2 ≤ n) (l : ℕ) : 0 ≤ roundEigenvalue n l := by have hn' : (2:ℝ) ≤ n := by exact_mod_cast hn unfold roundEigenvalue exact mul_nonneg (Nat.cast_nonneg l) (by linarith [Nat.cast_nonneg (α := ℝ) l]) /-- The actual negative Berger Laplacian on the genuine harmonic-polynomial space. Here c is the horizontal metric coefficient and q the squared Hopf stretch (the manuscript q). -/ def angularOperator (J : ComplexStructure (Fin n)) (l : ℕ) (c q : ℝ) : harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l := c⁻¹ • (roundEigenvalue n l • (1 : harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l) + (1-(q)⁻¹) • (rotationCLM J l*rotationCLM J l)) lemma angularOperator_round (J : ComplexStructure (Fin n)) (l : ℕ) (c : ℝ) : angularOperator J l c 1 = (c⁻¹*roundEigenvalue n l) • 1 := by ext P simp only [angularOperator,inv_one,sub_self,_root_.smul_apply,_root_.add_apply, one_apply_eq_self,_root_.zero_smul,_root_.add_zero,_root_.smul_smul] lemma angularOperator_inner (J : ComplexStructure (Fin n)) (l : ℕ) (c q : ℝ) (P : harmonicPolynomials n l) : ⟪P,angularOperator J l c q P⟫_ℝ = c⁻¹*(roundEigenvalue n l*‖P‖^2+ (1-(q)⁻¹)*⟪P,(rotationCLM J l*rotationCLM J l) P⟫_ℝ) := by simp only [angularOperator,_root_.smul_apply,_root_.add_apply, one_apply_eq_self,_root_.inner_smul_right,_root_.inner_add_right,real_inner_self_eq_norm_sq] lemma angularOperator_symmetric (J : ComplexStructure (Fin n)) (l : ℕ) (c q : ℝ) : (angularOperator J l c q).IsSymmetric := by intro P Q change ⟪angularOperator J l c q P,Q⟫_ℝ = ⟪P,angularOperator J l c q Q⟫_ℝ simp only [angularOperator,_root_.smul_apply,_root_.add_apply, one_apply_eq_self,_root_.inner_smul_left,_root_.inner_smul_right,_root_.inner_add_left,_root_.inner_add_right, conj_trivial] have hs : ⟪(rotationCLM J l*rotationCLM J l) P,Q⟫_ℝ = ⟪P,(rotationCLM J l*rotationCLM J l) Q⟫_ℝ := rotationCLM_square_symmetric J l P Q rw [hs] lemma angularOperator_lower (J : ComplexStructure (Fin n)) (l : ℕ) {c q : ℝ} (hc : 0 < c) (hq : 1 ≤ q) (P : harmonicPolynomials n l) : c⁻¹*(roundEigenvalue n l-(1-(q)⁻¹)*(l:ℝ)^2)*‖P‖^2 ≤ ⟪P,angularOperator J l c q P⟫_ℝ := by have hd : 0 ≤ 1-(q)⁻¹ := sub_nonneg.2 (inv_le_one_of_one_le₀ hq) have h := mul_le_mul_of_nonneg_left (rotationCLM_square_lower J l P) hd rw [angularOperator_inner] have hi : 0 ≤ c⁻¹ := inv_nonneg.2 hc.le have hh := mul_le_mul_of_nonneg_left h hi nlinarith lemma angularOperator_nonnegative_of_one_le {n : ℕ} (hn : 2 ≤ n) (J : ComplexStructure (Fin n)) (l : ℕ) {c q : ℝ} (hc : 0 < c) (hq : 1 ≤ q) (P : harmonicPolynomials n l) : 0 ≤ ⟪P,angularOperator J l c q P⟫_ℝ := by apply le_trans _ (angularOperator_lower J l hc hq P) have hn' : (2:ℝ) ≤ n := by exact_mod_cast hn have hq' : 0 ≤ (q)⁻¹ := inv_nonneg.2 (le_trans zero_le_one hq) have he : roundEigenvalue n l-(1-(q)⁻¹)*(l:ℝ)^2 = (l:ℝ)*((n:ℝ)-2)+(q)⁻¹*(l:ℝ)^2 := by unfold roundEigenvalue; ring rw [he] exact mul_nonneg (mul_nonneg (inv_nonneg.2 hc.le) (add_nonneg (mul_nonneg (Nat.cast_nonneg l) (by linarith)) (mul_nonneg hq' (sq_nonneg _)))) (sq_nonneg _) lemma angularOperator_positive_lower {n : ℕ} (hn : 2 ≤ n) (J : ComplexStructure (Fin n)) (l : ℕ) {c q : ℝ} (hc : 0 < c) (hq : 0 < q) (P : harmonicPolynomials n l) : c⁻¹*min 1 q⁻¹*(l:ℝ)^2*‖P‖^2 ≤ ⟪P,angularOperator J l c q P⟫_ℝ := by have hn' : (2:ℝ) ≤ n := by exact_mod_cast hn have hi : 0 ≤ c⁻¹ := inv_nonneg.2 hc.le by_cases h : 1 ≤ q · apply le_trans _ (angularOperator_lower J l hc h P) have he : roundEigenvalue n l-(1-q⁻¹)*(l:ℝ)^2 = (l:ℝ)*((n:ℝ)-2)+q⁻¹*(l:ℝ)^2 := by unfold roundEigenvalue; ring rw [min_eq_right (inv_le_one_of_one_le₀ h),he] have h0 : 0 ≤ (l:ℝ)*((n:ℝ)-2) := mul_nonneg (Nat.cast_nonneg l) (by linarith) have := mul_nonneg (mul_nonneg hi h0) (sq_nonneg ‖P‖) nlinarith · have hqi : 1 ≤ q⁻¹ := (one_le_inv₀ hq).2 (le_of_not_ge h) have hrot := mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.2 hqi) (rotationCLM_square_nonpos J l P) rw [min_eq_left hqi,angularOperator_inner] have he : (l:ℝ)^2 ≤ roundEigenvalue n l := by unfold roundEigenvalue nlinarith [mul_nonneg (Nat.cast_nonneg (α := ℝ) l) (sub_nonneg.2 hn')] have hmul := mul_le_mul_of_nonneg_right he (sq_nonneg ‖P‖) have hh := mul_le_mul_of_nonneg_left hmul hi have hpos := mul_nonneg hi hrot nlinarith lemma angularOperator_nonnegative {n : ℕ} (hn : 2 ≤ n) (J : ComplexStructure (Fin n)) (l : ℕ) {c q : ℝ} (hc : 0 < c) (hq : 0 < q) (P : harmonicPolynomials n l) : 0 ≤ ⟪P,angularOperator J l c q P⟫_ℝ := by apply le_trans _ (angularOperator_positive_lower hn J l hc hq P) positivity lemma angularOperator_norm_le (J : ComplexStructure (Fin n)) (l : ℕ) (c q : ℝ) : ‖angularOperator J l c q‖ ≤ |c⁻¹| *(|roundEigenvalue n l|+|1-(q)⁻¹| *(l:ℝ)^2) := by rw [angularOperator,norm_smul,Real.norm_eq_abs] apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) calc _ ≤ ‖roundEigenvalue n l • (1 : harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l)‖+ ‖(1-(q)⁻¹) • (rotationCLM J l*rotationCLM J l)‖ := @norm_add_le (harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l) _ _ _ _ ≤ |roundEigenvalue n l|+|1-(q)⁻¹| *(l:ℝ)^2 := by simp only [norm_smul,Real.norm_eq_abs] apply add_le_add · exact (mul_le_mul_of_nonneg_left (ContinuousLinearMap.norm_id_le) (abs_nonneg _)).trans_eq (mul_one _) · apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) calc _ ≤ ‖rotationCLM J l‖*‖rotationCLM J l‖ := ContinuousLinearMap.opNorm_comp_le (rotationCLM J l) (rotationCLM J l) _ ≤ (l:ℝ)*(l:ℝ) := mul_self_le_mul_self (@norm_nonneg (harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l) _ (rotationCLM J l)) (rotationCLM_opNorm_le J l) _ = _ := (sq _).symm lemma angularOperator_smooth (J : ComplexStructure (Fin n)) (l : ℕ) {c q : ℝ → ℝ} (hc : ContDiff ℝ ∞ c) (hq : ContDiff ℝ ∞ q) (hc0 : ∀ t,c t ≠ 0) (hq0 : ∀ t,q t ≠ 0) : ContDiff ℝ ∞ (fun t => angularOperator J l (c t) (q t)) := by let : Module ℝ (harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l) := ContinuousLinearMap.module exact (hc.inv hc0).smul (contDiff_const.add ((contDiff_const.sub (hq.inv hq0)).smul (contDiff_const : ContDiff ℝ ∞ (fun _ : ℝ => rotationCLM J l*rotationCLM J l)))) /-- Switching the genuine angular rotation on an open round gap is smooth, although the label itself need not be continuous there. -/ lemma angularOperator_glued_smooth (J : ℝ → ComplexStructure (Fin n)) (l : ℕ) {c q : ℝ → ℝ} (hc : ContDiff ℝ ∞ c) (hq : ContDiff ℝ ∞ q) (hc0 : ∀ t,c t ≠ 0) (hq0 : ∀ t,q t ≠ 0) (hJ : ∀ t,(∀ᶠ s in nhds t,J s = J t) ∨ (∀ᶠ s in nhds t,q s = 1)) : ContDiff ℝ ∞ (fun t => angularOperator (J t) l (c t) (q t)) := by let : Module ℝ (harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l) := ContinuousLinearMap.module rw [contDiff_iff_contDiffAt] intro t rcases hJ t with hfixed | hround · apply ((angularOperator_smooth (J t) l hc hq hc0 hq0).contDiffAt).congr_of_eventuallyEq exact hfixed.mono fun s hs => by dsimp only; rw [hs] · have hs : ContDiff ℝ ∞ (fun s => ((c s)⁻¹*roundEigenvalue n l) • (1 : harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l)) := ((hc.inv hc0).mul (contDiff_const : ContDiff ℝ ∞ (fun _ : ℝ => roundEigenvalue n l))).smul (contDiff_const : ContDiff ℝ ∞ (fun _ : ℝ => (1 : harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l))) apply hs.contDiffAt.congr_of_eventuallyEq exact hround.mono fun s hqs => by dsimp only; rw [hqs,angularOperator_round] end HarmonicCounterexample.Angular end noncomputable section namespace HarmonicCounterexample.Angular open Module MvPolynomial Matrix Berger Cartesian open scoped ContDiff variable {n l : ℕ} {ι : Type*} [Fintype ι] /-- Coordinates in a fixed basis of actual harmonic polynomials. -/ def coordinates (b : Basis ι ℝ (harmonicPolynomials n l)) : harmonicPolynomials n l ≃L[ℝ] (ι → ℝ) := b.equivFun.toContinuousLinearEquiv lemma coordinates_sum (b : Basis ι ℝ (harmonicPolynomials n l)) (P : harmonicPolynomials n l) : ∑ i, coordinates b P i • b i = P := b.sum_equivFun P lemma basis_evaluate (b : Basis ι ℝ (harmonicPolynomials n l)) (P : harmonicPolynomials n l) (x : Space n) : (∑ i, coordinates b P i*evaluate (b i : PolynomialSpace n) x) = evaluate (P : PolynomialSpace n) x := by have h := congrArg (fun Q : harmonicPolynomials n l => evaluate (Q : PolynomialSpace n) x) (coordinates_sum b P) simpa only [Submodule.coe_sum, Submodule.coe_smul,evaluate_sum,evaluate_smul] using h lemma basis_operator_coordinates (b : Basis ι ℝ (harmonicPolynomials n l)) (T : harmonicPolynomials n l →L[ℝ] harmonicPolynomials n l) (P : harmonicPolynomials n l) (i : ι) : coordinates b (T P) i = ∑ j,coordinates b (T (b j)) i*coordinates b P j := by have h := congrArg (fun Q => coordinates b (T Q) i) (coordinates_sum b P) rw [← h] simp only [map_sum,map_smul,Finset.sum_apply,Pi.smul_apply,smul_eq_mul] apply Finset.sum_congr rfl intro j _ ring lemma square_rotation_basis_evaluate (b : Basis ι ℝ (harmonicPolynomials n l)) (J : ComplexStructure (Fin n)) (i : ι) (x : Space n) : hopfDerivative J (hopfDerivative J (evaluate (b i : PolynomialSpace n))) x = ∑ j,coordinates b ((rotationCLM J l*rotationCLM J l) (b i)) j* evaluate (b j : PolynomialSpace n) x := by rw [basis_evaluate] have he (P : harmonicPolynomials n l) : hopfDerivative J (evaluate (P : PolynomialSpace n)) = evaluate (rotationCLM J l P : PolynomialSpace n) := funext (rotation_evaluate J l P) rw [he,he] rfl lemma angularOperator_basis_coordinates (b : Basis ι ℝ (harmonicPolynomials n l)) (J : ComplexStructure (Fin n)) (c q : ℝ) (P : harmonicPolynomials n l) (i : ι) : coordinates b (angularOperator J l c q P) i = c⁻¹*roundEigenvalue n l*coordinates b P i + c⁻¹*(1-q⁻¹)*(∑ j,coordinates b ((rotationCLM J l*rotationCLM J l) (b j)) i* coordinates b P j) := by change coordinates b (c⁻¹ • (roundEigenvalue n l • P + (1-q⁻¹) • ((rotationCLM J l*rotationCLM J l) P))) i = _ simp only [map_smul,map_add,Pi.smul_apply,Pi.add_apply,smul_eq_mul] rw [basis_operator_coordinates b (rotationCLM J l*rotationCLM J l) P i] ring end HarmonicCounterexample.Angular end noncomputable section namespace HarmonicCounterexample.Construction open Filter Set Schedule AngularStream LinearODE Berger Cartesian Pulses open FiniteControl.SmoothWord open scoped Topology ContDiff InnerProductSpace variable (L:ℕ) [NeZero L] /-- A single constructed infinite history, including the thresholds and exact whole-period equalities, for any prescribed determinant-one simultaneous target. -/ structure Stream (target:∀l:Fin L,Matrix.SpecialLinearGroup (Index l.val) ℝ) where word : PhysicalWord CS j0 : ℕ s : ℕ→Fin word.n→ℝ admissible : ∀j,s j∈word.admissible zero : ∀j,jd0 l.val) j0 j l+ (theta l.val) • (geometry L).legV (fun l=>d0 l.val) j0 j l)*(target l:Matrix _ _ ℝ)) lemma exists_stream (target:∀l:Fin L,Matrix.SpecialLinearGroup (Index l.val) ℝ) : Nonempty (Stream L target) := by obtain ⟨w,N,ht⟩ := global_actual_transmission (system L) (geometry L) (fun l=>d0 l.val) (fun l=>by convert Dround_diagonal l.val using 1; simp only [system,geometry]) (generator_trace L) (generator_full L) target let : NeZero w.n := ⟨w.n_ne⟩ obtain ⟨N',hN'⟩ := scalar_all_curvatures w mu_pos mu_le profile_limit let j0 := max N N' obtain ⟨s,hs,hzero,hpacket,htrans⟩ := ht j0 (le_max_left _ _) obtain ⟨d,hd,hdlocal⟩ := exists_unit_round_gap_selector w.n obtain ⟨hqb,hcurv⟩ := hN' j0 (le_max_right _ _) s hs refine ⟨⟨w,j0,s,hs,hzero,hpacket,d,(fun t i hi=>(hd t i hi).1),hdlocal,?_,hcurv,?_⟩⟩ · intro t exact ⟨(hqb t).1,(hqb t).2.trans (by norm_num [qstar])⟩ · intro j hj l obtain ⟨D,hD,hDp,z,hz,he⟩ := htrans j hj l refine ⟨z,hz,?_⟩ rw [hD] at he exact he variable {L} {target:∀l:Fin L,Matrix.SpecialLinearGroup (Index l.val) ℝ} namespace Stream omit [NeZero L] variable (F:Stream L target) instance : NeZero F.word.n := ⟨F.word.n_ne⟩ def q : ℝ→ℝ := scalar F.s qstar F.j0 def c : ℝ→ℝ := horizontalScale mu F.q def J (t:ℝ) : ComplexStructure (Fin 16) := (globalSlot F.d F.j0 t).elim Jround (fun i=>(F.word.slot i).reindex (finSumFinEquiv (m:=8) (n:=8)).symm) lemma q_pos (t:ℝ) : 0by simp only [J,hu]) · exact Or.inr h lemma q_center (t:ℝ) (ht:t≤0) : F.q t=1 := scalar_initial F.s (ht.trans (time_pos F.j0).le) lemma c_center (t:ℝ) (ht:t≤0) : F.c t=1 := horizontalScale_initial F.s F.j0 mu ht def metric : SmoothMetric 16 := bergerMetric F.c F.q F.J F.c_smooth F.q_smooth F.c_pos F.q_pos F.J_local F.c_center F.q_center lemma geometry_main : Complete F.metric ∧ EuclideanNearOrigin F.metric ∧ PositiveSubunitAVR F.metric ∧ ∀x,distance F.metric 0 x=radius x := profile_metric_global mu_pos.le F.q_smooth F.qbounds F.J F.J_local F.c_center F.q_center profile_limit lemma ricci : RicciNonnegative F.metric := by apply bergerMetric_ricciNonnegative F.c F.q (horizontalSlope mu F.q) (logSlope F.q) F.J F.c_smooth F.q_smooth F.c_pos F.q_pos F.J_local F.c_center F.q_center · exact horizontalScale_hasDeriv (F.q_smooth.differentiable (by simp)) F.q_pos mu · exact logSlope_hasDeriv (F.q_smooth.differentiable (by simp)) F.q_pos · exact (horizontalSlope_smooth F.q_smooth F.q_pos mu).differentiable (by simp) · exact (logSlope_smooth F.q_smooth F.q_pos).differentiable (by simp) · intro t;exact (F.curvature t).1 · intro t;exact (F.curvature t).2.1 · intro t;exact (F.curvature t).2.2 lemma drift_eq (t:ℝ) : drift t=14+15*horizontalSlope mu F.q t+logSlope F.q t := by unfold drift horizontalSlope;ring lemma coefficient_eq (l:Fin L) (t:ℝ) : (geometry L).coefficient F.word.slot F.s F.j0 l t= Angular.angularOperator (F.J t) (l.val+2) (F.c t) (F.q t) := by rw [HistoryGeometry.coefficient,angular_globalSlot _ _ _ _ F.d F.selects] have hD : ((globalSlot F.d F.j0 t).elim ((geometry L).D₀ l) (fun i=>(system L).D l (F.word.slot i)))= Angular.rotationCLM (F.J t) (l.val+2)*Angular.rotationCLM (F.J t) (l.val+2) := by cases h : globalSlot F.d F.j0 t <;> simp only [J,h,Option.elim,system,geometry,Dround,D,Jround] rw [hD] have hB : (system L).B l=Angular.roundEigenvalue 16 (l.val+2) := by simp only [system,B,degree,Angular.roundEigenvalue,Nat.cast_add,Nat.cast_ofNat];ring change alpha t • PulseTaylor.angular (1/15) ((system L).B l • 1) (Angular.rotationCLM (F.J t) (l.val+2)*Angular.rotationCLM (F.J t) (l.val+2)) (F.q t)=_ rw [hB] have hc : (F.c t)⁻¹=alpha t*(F.q t)^(1/15:ℝ) := by rw [c,horizontalScale,_root_.mul_inv_rev,show (-1:ℝ)/15=-(1/15) by ring,Real.rpow_neg (F.q_pos t).le] simp only [inv_inv,alpha];ring have hr : (F.q t)^(1/15:ℝ)-(F.q t)^((1/15:ℝ)-1)= (F.q t)^(1/15:ℝ)*(1-(F.q t)⁻¹) := by rw [Real.rpow_sub_one (F.q_pos t).ne'];ring rw [PulseTaylor.angular,Angular.angularOperator,hc,hr] module end Stream end HarmonicCounterexample.Construction end end OAI