Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.NumberTheory.CubicGram.CommonFactors

namespace OAI

/-! Finite cubic sieve operators, bilinear norms and coprime completion. -/

section

noncomputable section
open scoped BigOperators ContDiff
open Set Filter MeasureTheory Topology
attribute [local instance] Classical.propDecidable
namespace CubicFirstMoment

def primarySmoothedSieveMass (S : Finset Eisenstein) (u : Eisenstein → ℂ)
    (W : ℝ → ℂ) (M : ℝ) : ℂ :=
  ∑' x : Eisenstein, if primary x then
    W (norm x / M) * (‖∑ n ∈ S, u n * cubicSymbol n x‖^2 : ℝ) else 0

lemma primarySmoothedSieveMass_gram (S : Finset Eisenstein)
    (hS : ∀ n ∈ S, primary n) (u : Eisenstein → ℂ)
    (W : ℝ → ℂ) (hW : HasCompactSupport W) (hW' : ContDiff ℝ ∞ W)
    {M : ℝ} (hM : 0 < M) :
    primarySmoothedSieveMass S u W M =
      ∑ a ∈ S, ∑ b ∈ S, u a * star (u b) * primaryCharacterGram a b W M := by
  let F (a b x : Eisenstein) : ℂ := u a * star (u b) *
    (if primary x then W (norm x / M) * cubicSymbol a x * star (cubicSymbol b x) else 0)
  have hF (a : Eisenstein) (ha : a ∈ S) (b : Eisenstein) (hb : b ∈ S) :
      Summable (F a b) := by
    apply Summable.mul_left
    simpa only [and_true] using
      summable_gram_restricted (hS a ha) (hS b hb) W hW hW' hM (fun _ => True)
  have hpoint (x : Eisenstein) :
      (if primary x then W (norm x / M) *
        (‖∑ n ∈ S, u n * cubicSymbol n x‖^2 : ℝ) else 0) =
      ∑ a ∈ S, ∑ b ∈ S, F a b x := by
    by_cases hx : primary x
    · simp only [ite_eq_left hx, ← Complex.normSq_eq_norm_sq, ← Complex.mul_conj,
        map_sum, map_mul, F]
      simp_rw [Finset.sum_mul, Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro a ha
      apply Finset.sum_congr rfl
      intro b hb
      simp only [starRingEnd_apply]
      ring
    · simp [hx, F]
  unfold primarySmoothedSieveMass
  simp_rw [hpoint]
  rw [Summable.tsum_finsetSum (fun a ha => summable_sum (fun b hb => hF a ha b hb))]
  apply Finset.sum_congr rfl
  intro a ha
  rw [Summable.tsum_finsetSum (hF a ha)]
  apply Finset.sum_congr rfl
  intro b hb
  exact tsum_mul_left

end CubicFirstMoment

open UniqueFactorizationMonoid
namespace CubicFirstMoment

lemma primaryPrime_mem_factors {n p : Eisenstein} (hn : primary n)
    (hp : primaryPrime p) (hd : p ∣ n) : p ∈ primaryPrimeFactors n := by
  obtain ⟨q,hq,hpq⟩ := exists_mem_normalizedFactors_of_dvd
    (primary_ne_zero hn) hp.2.irreducible hd
  apply Finset.mem_image.mpr
  refine ⟨q, Multiset.mem_toFinset.mpr hq, ?_⟩
  apply primary_associated_eq
    (primaryNormalize_primary (unit_residue_of_dvd_primary hn (dvd_of_mem_normalizedFactors hq))) hp.1
  exact (primaryNormalize_associated q).symm.trans hpq.symm

lemma primaryPrime_mem_factors_iff {n p : Eisenstein} (hn : primary n) :
    p ∈ primaryPrimeFactors n ↔ primaryPrime p ∧ p ∣ n :=
  ⟨primaryPrimeFactor_spec hn, fun h => primaryPrime_mem_factors hn h.1 h.2⟩

lemma primeSet_subset_factors_of_prod_dvd {n : Eisenstein} (hn : primary n)
    {s : Finset Eisenstein} (hs : ∀ p ∈ s, primaryPrime p)
    (hd : (∏ p ∈ s, p) ∣ n) : s ⊆ primaryPrimeFactors n := by
  intro p hp
  exact primaryPrime_mem_factors hn (hs p hp)
    ((Finset.dvd_prod_of_mem (fun p : Eisenstein => p) hp).trans hd)

lemma primary_moebius_universal_indicator {U : Finset Eisenstein}
    (hU : ∀ p ∈ U, primaryPrime p) {a b : Eisenstein}
    (ha : primary a) (hsa : Squarefree a) (haU : primaryPrimeFactors a ⊆ U) :
    (if IsCoprime a b then (1 : ℂ) else 0) =
      ∑ s ∈ U.powerset, (idealMoebius (∏ p ∈ s, p) : ℂ) *
        (if (∏ p ∈ s, p) ∣ a ∧ (∏ p ∈ s, p) ∣ b then (1 : ℂ) else 0) := by
  rw [primary_moebius_coprimality ha hsa]
  calc
    _ = ∑ s ∈ (primaryPrimeFactors a).powerset,
        (idealMoebius (∏ p ∈ s, p) : ℂ) *
          (if (∏ p ∈ s, p) ∣ a ∧ (∏ p ∈ s, p) ∣ b then (1 : ℂ) else 0) := by
      apply Finset.sum_congr rfl
      intro s hs
      have hd : (∏ p ∈ s, p) ∣ a :=
        (primary_subsets_prod_dvd ha (Finset.mem_powerset.mp hs) a).mpr
          (fun p hp => (primaryPrimeFactor_spec ha (Finset.mem_powerset.mp hs hp)).2)
      simp only [hd, true_and]
    _ = _ := by
      apply Finset.sum_subset (Finset.powerset_mono.mpr haU)
      intro s hs hn
      have hnd : ¬(∏ p ∈ s, p) ∣ a := by
        intro hd
        apply hn
        exact Finset.mem_powerset.mpr (primeSet_subset_factors_of_prod_dvd ha
          (fun p hp => hU p (Finset.mem_powerset.mp hs hp)) hd)
      simp [hnd]

theorem coprime_bilinear_moebius (S T U : Finset Eisenstein)
    (hS : ∀ a ∈ S, primary a ∧ Squarefree a)
    (hU : ∀ p ∈ U, primaryPrime p)
    (hSU : ∀ a ∈ S, primaryPrimeFactors a ⊆ U) (v w : Eisenstein → ℂ) :
    (∑ a ∈ S, ∑ b ∈ T, if IsCoprime a b then v a * star (w b) else 0) =
      ∑ s ∈ U.powerset, (idealMoebius (∏ p ∈ s, p) : ℂ) *
        (∑ a ∈ S, if (∏ p ∈ s, p) ∣ a then v a else 0) *
          star (∑ b ∈ T, if (∏ p ∈ s, p) ∣ b then w b else 0) := by
  have hpoint (a : Eisenstein) (ha : a ∈ S) (b : Eisenstein) :
      (if IsCoprime a b then v a * star (w b) else 0) =
        ∑ s ∈ U.powerset, (idealMoebius (∏ p ∈ s, p) : ℂ) *
          ((if (∏ p ∈ s, p) ∣ a then v a else 0) *
            star (if (∏ p ∈ s, p) ∣ b then w b else 0)) := by
    have hi := congrArg (fun z : ℂ => z * (v a * star (w b)))
      (primary_moebius_universal_indicator hU (hS a ha).1 (hS a ha).2 (hSU a ha)
        (b := b))
    rw [Finset.sum_mul] at hi
    convert hi using 1
    · split_ifs <;> simp
    · apply Finset.sum_congr rfl
      intro s hs
      split_ifs <;> simp_all
  simp_rw [Finset.sum_congr rfl (fun a ha => Finset.sum_congr rfl
    (fun b _ => hpoint a ha b))]
  
  simp_rw [Finset.sum_comm (s := T) (t := U.powerset)]
  rw [Finset.sum_comm (s := S) (t := U.powerset)]
  apply Finset.sum_congr rfl
  intro s hs
  simp only [star_sum]
  simp only [Finset.sum_mul, Finset.mul_sum]
  rw [Finset.sum_comm (s := T) (t := S)]
  apply Finset.sum_congr rfl
  intro a ha
  apply Finset.sum_congr rfl
  intro b hb
  ring

end CubicFirstMoment

namespace CubicFirstMoment

def finiteCubicOperator (S H : Finset Eisenstein) :
    EuclideanSpace ℂ S →L[ℂ] EuclideanSpace ℂ H :=
  (Matrix.toEuclideanLin (fun (h : H) (a : S) => cubicSymbol a.val h.val)).toContinuousLinearMap

def finiteCubicBound (S H : Finset Eisenstein) : ℝ := ‖finiteCubicOperator S H‖^2

lemma finiteCubicBound_nonneg (S H : Finset Eisenstein) : 0 ≤ finiteCubicBound S H :=
  sq_nonneg _

lemma finiteCubicOperator_apply (S H : Finset Eisenstein) (u : Eisenstein → ℂ)
    (h : H) :
    finiteCubicOperator S H (WithLp.toLp 2 (fun a : S => u a.val)) h =
      ∑ a ∈ S, u a * cubicSymbol a h.val := by
  change (∑ a : S, cubicSymbol a.val h.val * u a.val) = _
  rw [← Finset.sum_subtype S (fun _ => Iff.rfl) (fun a => cubicSymbol a h.val * u a)]
  apply Finset.sum_congr rfl
  intro a ha
  ring

lemma finiteCubicBound_controls_mass (S H : Finset Eisenstein) (u : Eisenstein → ℂ) :
    (∑ h ∈ H, ‖∑ a ∈ S, u a * cubicSymbol a h‖^2) ≤
      finiteCubicBound S H * ∑ a ∈ S, ‖u a‖^2 := by
  let v : EuclideanSpace ℂ S := WithLp.toLp 2 (fun a : S => u a.val)
  have hb := (finiteCubicOperator S H).le_opNorm v
  have hs := pow_le_pow_left₀ (_root_.norm_nonneg _) hb 2
  rw [mul_pow, EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq] at hs
  change (∑ h : H, ‖finiteCubicOperator S H v h‖^2) ≤
      finiteCubicBound S H * ∑ a : S, ‖u a.val‖^2 at hs
  dsimp only [v] at hs
  simp_rw [finiteCubicOperator_apply] at hs
  simpa only [← Finset.sum_subtype H (fun _ => Iff.rfl)
    (fun h => ‖∑ a ∈ S, u a * cubicSymbol a h‖^2),
    ← Finset.sum_subtype S (fun _ => Iff.rfl) (fun a => ‖u a‖^2)] using hs

end CubicFirstMoment

namespace CubicFirstMoment

private def euclideanConj {ι : Type*} [Fintype ι] (v : EuclideanSpace ℂ ι) :
    EuclideanSpace ℂ ι := WithLp.toLp 2 (fun i => star (v i))

private lemma euclideanConj_norm {ι : Type*} [Fintype ι] (v : EuclideanSpace ℂ ι) :
    ‖euclideanConj v‖ = ‖v‖ := by
  apply (sq_eq_sq₀ (_root_.norm_nonneg _) (_root_.norm_nonneg _)).mp
  simp [EuclideanSpace.norm_sq_eq, euclideanConj]

private def matrixHilbertOperator {ι κ : Type*} [Fintype ι] [Fintype κ]
    (A : Matrix ι κ ℂ) : EuclideanSpace ℂ κ →L[ℂ] EuclideanSpace ℂ ι :=
  A.toEuclideanLin.toContinuousLinearMap

private lemma matrixHilbertOperator_conj {ι κ : Type*} [Fintype ι] [Fintype κ]
    (A : Matrix ι κ ℂ) (v : EuclideanSpace ℂ κ) :
    matrixHilbertOperator (A.map star) v =
      euclideanConj (matrixHilbertOperator A (euclideanConj v)) := by
  apply PiLp.ext
  intro i
  change (∑ j, star (A i j) * v j) = star (∑ j, A i j * star (v j))
  simp [star_sum, star_mul, mul_comm]

private lemma matrixHilbertOperator_conj_norm_le {ι κ : Type*} [Fintype ι] [Fintype κ]
    (A : Matrix ι κ ℂ) :
    ‖matrixHilbertOperator (A.map star)‖ ≤ ‖matrixHilbertOperator A‖ := by
  apply (ContinuousLinearMap.opNorm_le_iff (f := matrixHilbertOperator (A.map star))
    (_root_.norm_nonneg (matrixHilbertOperator A))).mpr
  intro v
  rw [matrixHilbertOperator_conj, euclideanConj_norm]
  simpa only [euclideanConj_norm] using
    (matrixHilbertOperator A).le_opNorm (euclideanConj v)

private lemma matrixHilbertOperator_conj_norm {ι κ : Type*} [Fintype ι] [Fintype κ]
    (A : Matrix ι κ ℂ) :
    ‖matrixHilbertOperator (A.map star)‖ = ‖matrixHilbertOperator A‖ := by
  apply le_antisymm (matrixHilbertOperator_conj_norm_le A)
  have he : (A.map star).map star = A := by ext i j; simp
  have h := matrixHilbertOperator_conj_norm_le (A.map star)
  rwa [he] at h

private lemma matrixHilbertOperator_transpose_norm {ι κ : Type*}
    [Fintype ι] [Fintype κ] (A : Matrix ι κ ℂ) :
    ‖matrixHilbertOperator A.transpose‖ = ‖matrixHilbertOperator A‖ := by
  have hc : A.transpose = (A.conjTranspose).map star := by
    ext i j
    simp
  rw [hc, matrixHilbertOperator_conj_norm]
  change ‖LinearMap.toContinuousLinearMap A.conjTranspose.toEuclideanLin‖ = _
  rw [Matrix.toEuclideanLin_conjTranspose_eq_adjoint, LinearMap.adjoint_toContinuousLinearMap]
  exact ContinuousLinearMap.adjoint.norm_map _

theorem finiteCubicBound_symm (S H : Finset Eisenstein)
    (hS : ∀ a ∈ S, primary a) (hH : ∀ h ∈ H, primary h) :
    finiteCubicBound S H = finiteCubicBound H S := by
  let A : Matrix H S ℂ := fun h a => cubicSymbol a.val h.val
  have hmat : (fun (a : S) (h : H) => cubicSymbol h.val a.val) = A.transpose := by
    funext a h
    exact cubic_reciprocity (hH h.val h.property) (hS a.val a.property)
  unfold finiteCubicBound finiteCubicOperator
  rw [hmat]
  exact congrArg (fun x : ℝ => x^2) (matrixHilbertOperator_transpose_norm A).symm

end CubicFirstMoment

namespace CubicFirstMoment

theorem finiteCubicBound_bilinear (S H : Finset Eisenstein)
    (v w phase : Eisenstein → ℂ) (hphase : ∀ h ∈ H, ‖phase h‖ ≤ 1) :
    ‖∑ h ∈ H, phase h * (∑ a ∈ S, v a * cubicSymbol a h) *
      star (∑ a ∈ S, w a * cubicSymbol a h)‖ ≤
      finiteCubicBound S H *
        ((∑ a ∈ S, ‖v a‖^2) + (∑ a ∈ S, ‖w a‖^2)) / 2 := by
  let V (h : Eisenstein) := ∑ a ∈ S, v a * cubicSymbol a h
  let W (h : Eisenstein) := ∑ a ∈ S, w a * cubicSymbol a h
  calc
    _ ≤ ∑ h ∈ H, ‖phase h * V h * star (W h)‖ := norm_sum_le _ _
    _ ≤ ∑ h ∈ H, (‖V h‖^2+‖W h‖^2)/2 := by
      apply Finset.sum_le_sum
      intro h hh
      rw [norm_mul, norm_mul, norm_star]
      calc
        _ ≤ 1 * ‖V h‖ * ‖W h‖ :=
          mul_le_mul_of_nonneg_right
            (mul_le_mul_of_nonneg_right (hphase h hh) (_root_.norm_nonneg _))
            (_root_.norm_nonneg _)
        _ ≤ _ := by nlinarith [sq_nonneg (‖V h‖-‖W h‖)]
    _ = ((∑ h ∈ H, ‖V h‖^2) + (∑ h ∈ H, ‖W h‖^2))/2 := by
      rw [← Finset.sum_div, Finset.sum_add_distrib]
    _ ≤ _ := by
      have hv := finiteCubicBound_controls_mass S H v
      have hw := finiteCubicBound_controls_mass S H w
      change (∑ h ∈ H, ‖V h‖^2) ≤ _ at hv
      change (∑ h ∈ H, ‖W h‖^2) ≤ _ at hw
      nlinarith

theorem finiteCubicBound_conjugate_bilinear (S H : Finset Eisenstein)
    (v w phase : Eisenstein → ℂ) (hphase : ∀ h ∈ H, ‖phase h‖ ≤ 1) :
    ‖∑ h ∈ H, phase h * (∑ a ∈ S, v a * star (cubicSymbol a h)) *
      star (∑ a ∈ S, w a * star (cubicSymbol a h))‖ ≤
      finiteCubicBound S H *
        ((∑ a ∈ S, ‖v a‖^2) + (∑ a ∈ S, ‖w a‖^2)) / 2 := by
  have h := finiteCubicBound_bilinear S H (fun a => star (w a))
    (fun a => star (v a)) phase hphase
  simp only [norm_star] at h
  have he : (∑ h ∈ H, phase h * (∑ a ∈ S, v a * star (cubicSymbol a h)) *
      star (∑ a ∈ S, w a * star (cubicSymbol a h))) =
      ∑ h ∈ H, phase h * (∑ a ∈ S, star (w a) * cubicSymbol a h) *
        star (∑ a ∈ S, star (v a) * cubicSymbol a h) := by
    apply Finset.sum_congr rfl
    intro h hh
    simp [star_sum, star_mul, mul_comm, mul_left_comm, mul_assoc]
  rw [he]
  simpa only [add_comm] using h

end CubicFirstMoment

namespace CubicFirstMoment

lemma coefficientVector_norm_sq (S : Finset Eisenstein) (u : Eisenstein → ℂ) :
    ‖(WithLp.toLp 2 (fun a : S => u a.val) : EuclideanSpace ℂ S)‖^2 =
      ∑ a ∈ S, ‖u a‖^2 := by
  rw [EuclideanSpace.norm_sq_eq]
  exact (Finset.sum_subtype S (fun _ => Iff.rfl) (fun a => ‖u a‖^2)).symm

lemma finiteCubicOperator_norm_sq (S H : Finset Eisenstein) (u : Eisenstein → ℂ) :
    ‖finiteCubicOperator S H (WithLp.toLp 2 (fun a : S => u a.val))‖^2 =
      ∑ h ∈ H, ‖∑ a ∈ S, u a * cubicSymbol a h‖^2 := by
  rw [EuclideanSpace.norm_sq_eq]
  change (∑ h : H, ‖finiteCubicOperator S H (WithLp.toLp 2 (fun a : S => u a.val)) h‖^2) = _
  simp_rw [finiteCubicOperator_apply]
  exact (Finset.sum_subtype H (fun _ => Iff.rfl)
    (fun h => ‖∑ a ∈ S, u a * cubicSymbol a h‖^2)).symm

lemma finiteCubicBound_le_iff (S H : Finset Eisenstein) {C : ℝ} (hC : 0 ≤ C) :
    finiteCubicBound S H ≤ C ↔ ∀ u : Eisenstein → ℂ,
      (∑ h ∈ H, ‖∑ a ∈ S, u a * cubicSymbol a h‖^2) ≤
        C * ∑ a ∈ S, ‖u a‖^2 := by
  constructor
  · intro h u
    exact (finiteCubicBound_controls_mass S H u).trans
      (mul_le_mul_of_nonneg_right h (Finset.sum_nonneg (fun _ _ => sq_nonneg _)))
  · intro h
    have hn : ‖finiteCubicOperator S H‖ ≤ Real.sqrt C := by
      apply (ContinuousLinearMap.opNorm_le_iff (f := finiteCubicOperator S H)
        (Real.sqrt_nonneg C)).mpr
      intro v
      let u (a : Eisenstein) : ℂ := if ha : a ∈ S then v ⟨a, ha⟩ else 0
      have hv : WithLp.toLp 2 (fun a : S => u a.val) = v := by
        apply PiLp.ext
        intro a
        simp [u]
      have hh : ‖finiteCubicOperator S H v‖^2 ≤ C*‖v‖^2 := by
        rw [← hv, finiteCubicOperator_norm_sq, coefficientVector_norm_sq]
        exact h u
      apply (sq_le_sq₀ (_root_.norm_nonneg _) (mul_nonneg (Real.sqrt_nonneg _) (_root_.norm_nonneg _))).mp
      simpa only [mul_pow, Real.sq_sqrt hC] using hh
    exact (pow_le_pow_left₀ (_root_.norm_nonneg _) hn 2).trans_eq (Real.sq_sqrt hC)

end CubicFirstMoment

namespace CubicFirstMoment

lemma finiteCubicBound_mono {S T H J : Finset Eisenstein} (hST : S ⊆ T) (hHJ : H ⊆ J) :
    finiteCubicBound S H ≤ finiteCubicBound T J := by
  apply (finiteCubicBound_le_iff S H (finiteCubicBound_nonneg T J)).mpr
  intro u
  let v (a : Eisenstein) := if a ∈ S then u a else 0
  have hi (h : Eisenstein) : (∑ a ∈ T, v a * cubicSymbol a h) =
      ∑ a ∈ S, u a * cubicSymbol a h := by
    calc
      _ = ∑ a ∈ S, v a * cubicSymbol a h :=
        (Finset.sum_subset hST (fun a _ ha => by simp [v, ha])).symm
      _ = _ := Finset.sum_congr rfl (fun a ha => by simp [v, ha])
  have hv : (∑ a ∈ T, ‖v a‖^2) = ∑ a ∈ S, ‖u a‖^2 := by
    calc
      _ = ∑ a ∈ S, ‖v a‖^2 :=
        (Finset.sum_subset hST (fun a _ ha => by simp [v, ha])).symm
      _ = _ := Finset.sum_congr rfl (fun a ha => by simp [v, ha])
  have hb := finiteCubicBound_controls_mass T J v
  simp_rw [hi, hv] at hb
  exact (Finset.sum_le_sum_of_subset_of_nonneg hHJ (fun _ _ _ => sq_nonneg _)).trans hb

lemma finiteCubicBound_mul_support_le (Q H : Finset Eisenstein)
    {m : Eisenstein} (hm : primary m) (hQ : ∀ b ∈ Q, primary b) :
    finiteCubicBound (Q.image (fun b => m*b)) H ≤ finiteCubicBound Q H := by
  apply (finiteCubicBound_le_iff _ _ (finiteCubicBound_nonneg Q H)).mpr
  intro u
  have hinj : ∀ a ∈ Q, ∀ b ∈ Q, m*a=m*b → a=b := by
    intro a ha b hb he
    exact mul_left_cancel₀ (primary_ne_zero hm) he
  have he (h : Eisenstein) :
      (∑ a ∈ Q.image (fun b => m*b), u a * cubicSymbol a h) =
        cubicSymbol m h * ∑ b ∈ Q, u (m*b) * cubicSymbol b h := by
    rw [Finset.sum_image hinj, Finset.mul_sum]
    apply Finset.sum_congr rfl
    intro b hb
    rw [cubicSymbol_mul_lower (primary_ne_zero hm) (primary_ne_zero (hQ b hb))]
    ring
  have hn (h : Eisenstein) :
      ‖∑ a ∈ Q.image (fun b => m*b), u a * cubicSymbol a h‖^2 ≤
        ‖∑ b ∈ Q, u (m*b) * cubicSymbol b h‖^2 := by
    rw [he, norm_mul]
    apply pow_le_pow_left₀ (by positivity)
    simpa only [one_mul] using mul_le_mul_of_nonneg_right
      (norm_cubicSymbol_le_one hm h) (_root_.norm_nonneg _)
  calc
    _ ≤ ∑ h ∈ H, ‖∑ b ∈ Q, u (m*b) * cubicSymbol b h‖^2 :=
      Finset.sum_le_sum (fun h _ => hn h)
    _ ≤ finiteCubicBound Q H * ∑ b ∈ Q, ‖u (m*b)‖^2 :=
      finiteCubicBound_controls_mass Q H (fun b => u (m*b))
    _ = _ := by rw [Finset.sum_image hinj]

end CubicFirstMoment

namespace CubicFirstMoment

lemma norm_idealMoebius_le_one (m : Eisenstein) : ‖(idealMoebius m : ℂ)‖ ≤ 1 := by
  have hs : (idealMoebius m : ℝ)^2 ≤ 1 := by
    rw [idealMoebius_sq]
    split_ifs <;> norm_num
  have hn : ‖(idealMoebius m : ℂ)‖ = |(idealMoebius m : ℝ)| :=
    Complex.norm_intCast _
  rw [hn]
  nlinarith [sq_abs (idealMoebius m : ℝ), abs_nonneg (idealMoebius m : ℝ)]

theorem finiteCubicBound_coprime_bilinear (S H U : Finset Eisenstein)
    (hS : ∀ a ∈ S, primary a ∧ Squarefree a)
    (hU : ∀ p ∈ U, primaryPrime p)
    (hSU : ∀ a ∈ S, primaryPrimeFactors a ⊆ U)
    (v w phase : Eisenstein → ℂ) (hphase : ∀ h ∈ H, ‖phase h‖ ≤ 1) :
    ‖∑ h ∈ H, phase h * ∑ a ∈ S, ∑ b ∈ S,
      if IsCoprime a b then
        (v a * star (cubicSymbol a h)) * star (w b * star (cubicSymbol b h)) else 0‖ ≤
      ∑ s ∈ U.powerset,
        finiteCubicBound (S.filter (fun a => (∏ p ∈ s, p) ∣ a)) H *
          ((∑ a ∈ S.filter (fun a => (∏ p ∈ s, p) ∣ a), ‖v a‖^2) +
           (∑ a ∈ S.filter (fun a => (∏ p ∈ s, p) ∣ a), ‖w a‖^2)) / 2 := by
  let V (s : Finset Eisenstein) (h : Eisenstein) : ℂ :=
    ∑ a ∈ S.filter (fun a => (∏ p ∈ s, p) ∣ a), v a * star (cubicSymbol a h)
  let W (s : Finset Eisenstein) (h : Eisenstein) : ℂ :=
    ∑ a ∈ S.filter (fun a => (∏ p ∈ s, p) ∣ a), w a * star (cubicSymbol a h)
  have he (h : Eisenstein) := coprime_bilinear_moebius S S U hS hU hSU
    (fun a => v a * star (cubicSymbol a h)) (fun a => w a * star (cubicSymbol a h))
  have he' (h : Eisenstein) :
      (∑ a ∈ S, ∑ b ∈ S, if IsCoprime a b then
        (v a * star (cubicSymbol a h)) * star (w b * star (cubicSymbol b h)) else 0) =
      ∑ s ∈ U.powerset, (idealMoebius (∏ p ∈ s, p) : ℂ) * V s h * star (W s h) := by
    simpa only [V, W, Finset.sum_filter] using he h
  simp_rw [he', Finset.mul_sum]
  rw [Finset.sum_comm]
  have hfactor (s : Finset Eisenstein) :
      (∑ h ∈ H, phase h * ((idealMoebius (∏ p ∈ s, p) : ℂ) * V s h * star (W s h))) =
      (idealMoebius (∏ p ∈ s, p) : ℂ) * ∑ h ∈ H, phase h * V s h * star (W s h) := by
    rw [Finset.mul_sum]
    apply Finset.sum_congr rfl
    intro h hh
    ring
  simp_rw [hfactor]
  apply (norm_sum_le _ _).trans
  apply Finset.sum_le_sum
  intro s hs
  rw [norm_mul]
  calc
    _ ≤ 1 * ‖∑ h ∈ H, phase h * V s h * star (W s h)‖ :=
      mul_le_mul_of_nonneg_right (norm_idealMoebius_le_one _) (_root_.norm_nonneg _)
    _ ≤ _ := by
      rw [one_mul]
      exact finiteCubicBound_conjugate_bilinear _ H v w phase hphase

end CubicFirstMoment
end
end

end OAI

	import Mathlib
	import OAI.NumberTheory.CubicGram.CommonFactors

	namespace OAI

	/-! Finite cubic sieve operators, bilinear norms and coprime completion. -/

	section

	noncomputable section
	open scoped BigOperators ContDiff
	open Set Filter MeasureTheory Topology
	attribute [local instance] Classical.propDecidable
	namespace CubicFirstMoment

	def primarySmoothedSieveMass (S : Finset Eisenstein) (u : Eisenstein → ℂ)
	(W : ℝ → ℂ) (M : ℝ) : ℂ :=
	∑' x : Eisenstein, if primary x then
	W (norm x / M) * (‖∑ n ∈ S, u n * cubicSymbol n x‖^2 : ℝ) else 0

	lemma primarySmoothedSieveMass_gram (S : Finset Eisenstein)
	(hS : ∀ n ∈ S, primary n) (u : Eisenstein → ℂ)
	(W : ℝ → ℂ) (hW : HasCompactSupport W) (hW' : ContDiff ℝ ∞ W)
	{M : ℝ} (hM : 0 < M) :
	primarySmoothedSieveMass S u W M =
	∑ a ∈ S, ∑ b ∈ S, u a * star (u b) * primaryCharacterGram a b W M := by
	let F (a b x : Eisenstein) : ℂ := u a * star (u b) *
	(if primary x then W (norm x / M) * cubicSymbol a x * star (cubicSymbol b x) else 0)
	have hF (a : Eisenstein) (ha : a ∈ S) (b : Eisenstein) (hb : b ∈ S) :
	Summable (F a b) := by
	apply Summable.mul_left
	simpa only [and_true] using
	summable_gram_restricted (hS a ha) (hS b hb) W hW hW' hM (fun _ => True)
	have hpoint (x : Eisenstein) :
	(if primary x then W (norm x / M) *
	(‖∑ n ∈ S, u n * cubicSymbol n x‖^2 : ℝ) else 0) =
	∑ a ∈ S, ∑ b ∈ S, F a b x := by
	by_cases hx : primary x
	· simp only [ite_eq_left hx, ← Complex.normSq_eq_norm_sq, ← Complex.mul_conj,
	map_sum, map_mul, F]
	simp_rw [Finset.sum_mul, Finset.mul_sum]
	apply Finset.sum_congr rfl
	intro a ha
	apply Finset.sum_congr rfl
	intro b hb
	simp only [starRingEnd_apply]
	ring
	· simp [hx, F]
	unfold primarySmoothedSieveMass
	simp_rw [hpoint]
	rw [Summable.tsum_finsetSum (fun a ha => summable_sum (fun b hb => hF a ha b hb))]
	apply Finset.sum_congr rfl
	intro a ha
	rw [Summable.tsum_finsetSum (hF a ha)]
	apply Finset.sum_congr rfl
	intro b hb
	exact tsum_mul_left

	end CubicFirstMoment

	open UniqueFactorizationMonoid
	namespace CubicFirstMoment

	lemma primaryPrime_mem_factors {n p : Eisenstein} (hn : primary n)
	(hp : primaryPrime p) (hd : p ∣ n) : p ∈ primaryPrimeFactors n := by
	obtain ⟨q,hq,hpq⟩ := exists_mem_normalizedFactors_of_dvd
	(primary_ne_zero hn) hp.2.irreducible hd
	apply Finset.mem_image.mpr
	refine ⟨q, Multiset.mem_toFinset.mpr hq, ?_⟩
	apply primary_associated_eq
	(primaryNormalize_primary (unit_residue_of_dvd_primary hn (dvd_of_mem_normalizedFactors hq))) hp.1
	exact (primaryNormalize_associated q).symm.trans hpq.symm

	lemma primaryPrime_mem_factors_iff {n p : Eisenstein} (hn : primary n) :
	p ∈ primaryPrimeFactors n ↔ primaryPrime p ∧ p ∣ n :=
	⟨primaryPrimeFactor_spec hn, fun h => primaryPrime_mem_factors hn h.1 h.2⟩

	lemma primeSet_subset_factors_of_prod_dvd {n : Eisenstein} (hn : primary n)
	{s : Finset Eisenstein} (hs : ∀ p ∈ s, primaryPrime p)
	(hd : (∏ p ∈ s, p) ∣ n) : s ⊆ primaryPrimeFactors n := by
	intro p hp
	exact primaryPrime_mem_factors hn (hs p hp)
	((Finset.dvd_prod_of_mem (fun p : Eisenstein => p) hp).trans hd)

	lemma primary_moebius_universal_indicator {U : Finset Eisenstein}
	(hU : ∀ p ∈ U, primaryPrime p) {a b : Eisenstein}
	(ha : primary a) (hsa : Squarefree a) (haU : primaryPrimeFactors a ⊆ U) :
	(if IsCoprime a b then (1 : ℂ) else 0) =
	∑ s ∈ U.powerset, (idealMoebius (∏ p ∈ s, p) : ℂ) *
	(if (∏ p ∈ s, p) ∣ a ∧ (∏ p ∈ s, p) ∣ b then (1 : ℂ) else 0) := by
	rw [primary_moebius_coprimality ha hsa]
	calc
	_ = ∑ s ∈ (primaryPrimeFactors a).powerset,
	(idealMoebius (∏ p ∈ s, p) : ℂ) *
	(if (∏ p ∈ s, p) ∣ a ∧ (∏ p ∈ s, p) ∣ b then (1 : ℂ) else 0) := by
	apply Finset.sum_congr rfl
	intro s hs
	have hd : (∏ p ∈ s, p) ∣ a :=
	(primary_subsets_prod_dvd ha (Finset.mem_powerset.mp hs) a).mpr
	(fun p hp => (primaryPrimeFactor_spec ha (Finset.mem_powerset.mp hs hp)).2)
	simp only [hd, true_and]
	_ = _ := by
	apply Finset.sum_subset (Finset.powerset_mono.mpr haU)
	intro s hs hn
	have hnd : ¬(∏ p ∈ s, p) ∣ a := by
	intro hd
	apply hn
	exact Finset.mem_powerset.mpr (primeSet_subset_factors_of_prod_dvd ha
	(fun p hp => hU p (Finset.mem_powerset.mp hs hp)) hd)
	simp [hnd]

	theorem coprime_bilinear_moebius (S T U : Finset Eisenstein)
	(hS : ∀ a ∈ S, primary a ∧ Squarefree a)
	(hU : ∀ p ∈ U, primaryPrime p)
	(hSU : ∀ a ∈ S, primaryPrimeFactors a ⊆ U) (v w : Eisenstein → ℂ) :
	(∑ a ∈ S, ∑ b ∈ T, if IsCoprime a b then v a * star (w b) else 0) =
	∑ s ∈ U.powerset, (idealMoebius (∏ p ∈ s, p) : ℂ) *
	(∑ a ∈ S, if (∏ p ∈ s, p) ∣ a then v a else 0) *
	star (∑ b ∈ T, if (∏ p ∈ s, p) ∣ b then w b else 0) := by
	have hpoint (a : Eisenstein) (ha : a ∈ S) (b : Eisenstein) :
	(if IsCoprime a b then v a * star (w b) else 0) =
	∑ s ∈ U.powerset, (idealMoebius (∏ p ∈ s, p) : ℂ) *
	((if (∏ p ∈ s, p) ∣ a then v a else 0) *
	star (if (∏ p ∈ s, p) ∣ b then w b else 0)) := by
	have hi := congrArg (fun z : ℂ => z * (v a * star (w b)))
	(primary_moebius_universal_indicator hU (hS a ha).1 (hS a ha).2 (hSU a ha)
	(b := b))
	rw [Finset.sum_mul] at hi
	convert hi using 1
	· split_ifs <;> simp
	· apply Finset.sum_congr rfl
	intro s hs
	split_ifs <;> simp_all
	simp_rw [Finset.sum_congr rfl (fun a ha => Finset.sum_congr rfl
	(fun b _ => hpoint a ha b))]

	simp_rw [Finset.sum_comm (s := T) (t := U.powerset)]
	rw [Finset.sum_comm (s := S) (t := U.powerset)]
	apply Finset.sum_congr rfl
	intro s hs
	simp only [star_sum]
	simp only [Finset.sum_mul, Finset.mul_sum]
	rw [Finset.sum_comm (s := T) (t := S)]
	apply Finset.sum_congr rfl
	intro a ha
	apply Finset.sum_congr rfl
	intro b hb
	ring

	end CubicFirstMoment

	namespace CubicFirstMoment

	def finiteCubicOperator (S H : Finset Eisenstein) :
	EuclideanSpace ℂ S →L[ℂ] EuclideanSpace ℂ H :=
	(Matrix.toEuclideanLin (fun (h : H) (a : S) => cubicSymbol a.val h.val)).toContinuousLinearMap

	def finiteCubicBound (S H : Finset Eisenstein) : ℝ := ‖finiteCubicOperator S H‖^2

	lemma finiteCubicBound_nonneg (S H : Finset Eisenstein) : 0 ≤ finiteCubicBound S H :=
	sq_nonneg _

	lemma finiteCubicOperator_apply (S H : Finset Eisenstein) (u : Eisenstein → ℂ)
	(h : H) :
	finiteCubicOperator S H (WithLp.toLp 2 (fun a : S => u a.val)) h =
	∑ a ∈ S, u a * cubicSymbol a h.val := by
	change (∑ a : S, cubicSymbol a.val h.val * u a.val) = _
	rw [← Finset.sum_subtype S (fun _ => Iff.rfl) (fun a => cubicSymbol a h.val * u a)]
	apply Finset.sum_congr rfl
	intro a ha
	ring

	lemma finiteCubicBound_controls_mass (S H : Finset Eisenstein) (u : Eisenstein → ℂ) :
	(∑ h ∈ H, ‖∑ a ∈ S, u a * cubicSymbol a h‖^2) ≤
	finiteCubicBound S H * ∑ a ∈ S, ‖u a‖^2 := by
	let v : EuclideanSpace ℂ S := WithLp.toLp 2 (fun a : S => u a.val)
	have hb := (finiteCubicOperator S H).le_opNorm v
	have hs := pow_le_pow_left₀ (_root_.norm_nonneg _) hb 2
	rw [mul_pow, EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq] at hs
	change (∑ h : H, ‖finiteCubicOperator S H v h‖^2) ≤
	finiteCubicBound S H * ∑ a : S, ‖u a.val‖^2 at hs
	dsimp only [v] at hs
	simp_rw [finiteCubicOperator_apply] at hs
	simpa only [← Finset.sum_subtype H (fun _ => Iff.rfl)
	(fun h => ‖∑ a ∈ S, u a * cubicSymbol a h‖^2),
	← Finset.sum_subtype S (fun _ => Iff.rfl) (fun a => ‖u a‖^2)] using hs

	end CubicFirstMoment

	namespace CubicFirstMoment

	private def euclideanConj {ι : Type*} [Fintype ι] (v : EuclideanSpace ℂ ι) :
	EuclideanSpace ℂ ι := WithLp.toLp 2 (fun i => star (v i))

	private lemma euclideanConj_norm {ι : Type*} [Fintype ι] (v : EuclideanSpace ℂ ι) :
	‖euclideanConj v‖ = ‖v‖ := by
	apply (sq_eq_sq₀ (_root_.norm_nonneg _) (_root_.norm_nonneg _)).mp
	simp [EuclideanSpace.norm_sq_eq, euclideanConj]

	private def matrixHilbertOperator {ι κ : Type*} [Fintype ι] [Fintype κ]
	(A : Matrix ι κ ℂ) : EuclideanSpace ℂ κ →L[ℂ] EuclideanSpace ℂ ι :=
	A.toEuclideanLin.toContinuousLinearMap

	private lemma matrixHilbertOperator_conj {ι κ : Type*} [Fintype ι] [Fintype κ]
	(A : Matrix ι κ ℂ) (v : EuclideanSpace ℂ κ) :
	matrixHilbertOperator (A.map star) v =
	euclideanConj (matrixHilbertOperator A (euclideanConj v)) := by
	apply PiLp.ext
	intro i
	change (∑ j, star (A i j) * v j) = star (∑ j, A i j * star (v j))
	simp [star_sum, star_mul, mul_comm]

	private lemma matrixHilbertOperator_conj_norm_le {ι κ : Type*} [Fintype ι] [Fintype κ]
	(A : Matrix ι κ ℂ) :
	‖matrixHilbertOperator (A.map star)‖ ≤ ‖matrixHilbertOperator A‖ := by
	apply (ContinuousLinearMap.opNorm_le_iff (f := matrixHilbertOperator (A.map star))
	(_root_.norm_nonneg (matrixHilbertOperator A))).mpr
	intro v
	rw [matrixHilbertOperator_conj, euclideanConj_norm]
	simpa only [euclideanConj_norm] using
	(matrixHilbertOperator A).le_opNorm (euclideanConj v)

	private lemma matrixHilbertOperator_conj_norm {ι κ : Type*} [Fintype ι] [Fintype κ]
	(A : Matrix ι κ ℂ) :
	‖matrixHilbertOperator (A.map star)‖ = ‖matrixHilbertOperator A‖ := by
	apply le_antisymm (matrixHilbertOperator_conj_norm_le A)
	have he : (A.map star).map star = A := by ext i j; simp
	have h := matrixHilbertOperator_conj_norm_le (A.map star)
	rwa [he] at h

	private lemma matrixHilbertOperator_transpose_norm {ι κ : Type*}
	[Fintype ι] [Fintype κ] (A : Matrix ι κ ℂ) :
	‖matrixHilbertOperator A.transpose‖ = ‖matrixHilbertOperator A‖ := by
	have hc : A.transpose = (A.conjTranspose).map star := by
	ext i j
	simp
	rw [hc, matrixHilbertOperator_conj_norm]
	change ‖LinearMap.toContinuousLinearMap A.conjTranspose.toEuclideanLin‖ = _
	rw [Matrix.toEuclideanLin_conjTranspose_eq_adjoint, LinearMap.adjoint_toContinuousLinearMap]
	exact ContinuousLinearMap.adjoint.norm_map _

	theorem finiteCubicBound_symm (S H : Finset Eisenstein)
	(hS : ∀ a ∈ S, primary a) (hH : ∀ h ∈ H, primary h) :
	finiteCubicBound S H = finiteCubicBound H S := by
	let A : Matrix H S ℂ := fun h a => cubicSymbol a.val h.val
	have hmat : (fun (a : S) (h : H) => cubicSymbol h.val a.val) = A.transpose := by
	funext a h
	exact cubic_reciprocity (hH h.val h.property) (hS a.val a.property)
	unfold finiteCubicBound finiteCubicOperator
	rw [hmat]
	exact congrArg (fun x : ℝ => x^2) (matrixHilbertOperator_transpose_norm A).symm

	end CubicFirstMoment

	namespace CubicFirstMoment

	theorem finiteCubicBound_bilinear (S H : Finset Eisenstein)
	(v w phase : Eisenstein → ℂ) (hphase : ∀ h ∈ H, ‖phase h‖ ≤ 1) :
	‖∑ h ∈ H, phase h * (∑ a ∈ S, v a * cubicSymbol a h) *
	star (∑ a ∈ S, w a * cubicSymbol a h)‖ ≤
	finiteCubicBound S H *
	((∑ a ∈ S, ‖v a‖^2) + (∑ a ∈ S, ‖w a‖^2)) / 2 := by
	let V (h : Eisenstein) := ∑ a ∈ S, v a * cubicSymbol a h
	let W (h : Eisenstein) := ∑ a ∈ S, w a * cubicSymbol a h
	calc
	_ ≤ ∑ h ∈ H, ‖phase h * V h * star (W h)‖ := norm_sum_le _ _
	_ ≤ ∑ h ∈ H, (‖V h‖^2+‖W h‖^2)/2 := by
	apply Finset.sum_le_sum
	intro h hh
	rw [norm_mul, norm_mul, norm_star]
	calc
	_ ≤ 1 * ‖V h‖ * ‖W h‖ :=
	mul_le_mul_of_nonneg_right
	(mul_le_mul_of_nonneg_right (hphase h hh) (_root_.norm_nonneg _))
	(_root_.norm_nonneg _)
	_ ≤ _ := by nlinarith [sq_nonneg (‖V h‖-‖W h‖)]
	_ = ((∑ h ∈ H, ‖V h‖^2) + (∑ h ∈ H, ‖W h‖^2))/2 := by
	rw [← Finset.sum_div, Finset.sum_add_distrib]
	_ ≤ _ := by
	have hv := finiteCubicBound_controls_mass S H v
	have hw := finiteCubicBound_controls_mass S H w
	change (∑ h ∈ H, ‖V h‖^2) ≤ _ at hv
	change (∑ h ∈ H, ‖W h‖^2) ≤ _ at hw
	nlinarith

	theorem finiteCubicBound_conjugate_bilinear (S H : Finset Eisenstein)
	(v w phase : Eisenstein → ℂ) (hphase : ∀ h ∈ H, ‖phase h‖ ≤ 1) :
	‖∑ h ∈ H, phase h * (∑ a ∈ S, v a * star (cubicSymbol a h)) *
	star (∑ a ∈ S, w a * star (cubicSymbol a h))‖ ≤
	finiteCubicBound S H *
	((∑ a ∈ S, ‖v a‖^2) + (∑ a ∈ S, ‖w a‖^2)) / 2 := by
	have h := finiteCubicBound_bilinear S H (fun a => star (w a))
	(fun a => star (v a)) phase hphase
	simp only [norm_star] at h
	have he : (∑ h ∈ H, phase h * (∑ a ∈ S, v a * star (cubicSymbol a h)) *
	star (∑ a ∈ S, w a * star (cubicSymbol a h))) =
	∑ h ∈ H, phase h * (∑ a ∈ S, star (w a) * cubicSymbol a h) *
	star (∑ a ∈ S, star (v a) * cubicSymbol a h) := by
	apply Finset.sum_congr rfl
	intro h hh
	simp [star_sum, star_mul, mul_comm, mul_left_comm, mul_assoc]
	rw [he]
	simpa only [add_comm] using h

	end CubicFirstMoment

	namespace CubicFirstMoment

	lemma coefficientVector_norm_sq (S : Finset Eisenstein) (u : Eisenstein → ℂ) :
	‖(WithLp.toLp 2 (fun a : S => u a.val) : EuclideanSpace ℂ S)‖^2 =
	∑ a ∈ S, ‖u a‖^2 := by
	rw [EuclideanSpace.norm_sq_eq]
	exact (Finset.sum_subtype S (fun _ => Iff.rfl) (fun a => ‖u a‖^2)).symm

	lemma finiteCubicOperator_norm_sq (S H : Finset Eisenstein) (u : Eisenstein → ℂ) :
	‖finiteCubicOperator S H (WithLp.toLp 2 (fun a : S => u a.val))‖^2 =
	∑ h ∈ H, ‖∑ a ∈ S, u a * cubicSymbol a h‖^2 := by
	rw [EuclideanSpace.norm_sq_eq]
	change (∑ h : H, ‖finiteCubicOperator S H (WithLp.toLp 2 (fun a : S => u a.val)) h‖^2) = _
	simp_rw [finiteCubicOperator_apply]
	exact (Finset.sum_subtype H (fun _ => Iff.rfl)
	(fun h => ‖∑ a ∈ S, u a * cubicSymbol a h‖^2)).symm

	lemma finiteCubicBound_le_iff (S H : Finset Eisenstein) {C : ℝ} (hC : 0 ≤ C) :
	finiteCubicBound S H ≤ C ↔ ∀ u : Eisenstein → ℂ,
	(∑ h ∈ H, ‖∑ a ∈ S, u a * cubicSymbol a h‖^2) ≤
	C * ∑ a ∈ S, ‖u a‖^2 := by
	constructor
	· intro h u
	exact (finiteCubicBound_controls_mass S H u).trans
	(mul_le_mul_of_nonneg_right h (Finset.sum_nonneg (fun _ _ => sq_nonneg _)))
	· intro h
	have hn : ‖finiteCubicOperator S H‖ ≤ Real.sqrt C := by
	apply (ContinuousLinearMap.opNorm_le_iff (f := finiteCubicOperator S H)
	(Real.sqrt_nonneg C)).mpr
	intro v
	let u (a : Eisenstein) : ℂ := if ha : a ∈ S then v ⟨a, ha⟩ else 0
	have hv : WithLp.toLp 2 (fun a : S => u a.val) = v := by
	apply PiLp.ext
	intro a
	simp [u]
	have hh : ‖finiteCubicOperator S H v‖^2 ≤ C*‖v‖^2 := by
	rw [← hv, finiteCubicOperator_norm_sq, coefficientVector_norm_sq]
	exact h u
	apply (sq_le_sq₀ (_root_.norm_nonneg _) (mul_nonneg (Real.sqrt_nonneg _) (_root_.norm_nonneg _))).mp
	simpa only [mul_pow, Real.sq_sqrt hC] using hh
	exact (pow_le_pow_left₀ (_root_.norm_nonneg _) hn 2).trans_eq (Real.sq_sqrt hC)

	end CubicFirstMoment

	namespace CubicFirstMoment

	lemma finiteCubicBound_mono {S T H J : Finset Eisenstein} (hST : S ⊆ T) (hHJ : H ⊆ J) :
	finiteCubicBound S H ≤ finiteCubicBound T J := by
	apply (finiteCubicBound_le_iff S H (finiteCubicBound_nonneg T J)).mpr
	intro u
	let v (a : Eisenstein) := if a ∈ S then u a else 0
	have hi (h : Eisenstein) : (∑ a ∈ T, v a * cubicSymbol a h) =
	∑ a ∈ S, u a * cubicSymbol a h := by
	calc
	_ = ∑ a ∈ S, v a * cubicSymbol a h :=
	(Finset.sum_subset hST (fun a _ ha => by simp [v, ha])).symm
	_ = _ := Finset.sum_congr rfl (fun a ha => by simp [v, ha])
	have hv : (∑ a ∈ T, ‖v a‖^2) = ∑ a ∈ S, ‖u a‖^2 := by
	calc
	_ = ∑ a ∈ S, ‖v a‖^2 :=
	(Finset.sum_subset hST (fun a _ ha => by simp [v, ha])).symm
	_ = _ := Finset.sum_congr rfl (fun a ha => by simp [v, ha])
	have hb := finiteCubicBound_controls_mass T J v
	simp_rw [hi, hv] at hb
	exact (Finset.sum_le_sum_of_subset_of_nonneg hHJ (fun _ _ _ => sq_nonneg _)).trans hb

	lemma finiteCubicBound_mul_support_le (Q H : Finset Eisenstein)
	{m : Eisenstein} (hm : primary m) (hQ : ∀ b ∈ Q, primary b) :
	finiteCubicBound (Q.image (fun b => m*b)) H ≤ finiteCubicBound Q H := by
	apply (finiteCubicBound_le_iff _ _ (finiteCubicBound_nonneg Q H)).mpr
	intro u
	have hinj : ∀ a ∈ Q, ∀ b ∈ Q, ma=mb → a=b := by
	intro a ha b hb he
	exact mul_left_cancel₀ (primary_ne_zero hm) he
	have he (h : Eisenstein) :
	(∑ a ∈ Q.image (fun b => mb), u a cubicSymbol a h) =
	cubicSymbol m h * ∑ b ∈ Q, u (mb) cubicSymbol b h := by
	rw [Finset.sum_image hinj, Finset.mul_sum]
	apply Finset.sum_congr rfl
	intro b hb
	rw [cubicSymbol_mul_lower (primary_ne_zero hm) (primary_ne_zero (hQ b hb))]
	ring
	have hn (h : Eisenstein) :
	‖∑ a ∈ Q.image (fun b => mb), u a cubicSymbol a h‖^2 ≤
	‖∑ b ∈ Q, u (mb) cubicSymbol b h‖^2 := by
	rw [he, norm_mul]
	apply pow_le_pow_left₀ (by positivity)
	simpa only [one_mul] using mul_le_mul_of_nonneg_right
	(norm_cubicSymbol_le_one hm h) (_root_.norm_nonneg _)
	calc
	_ ≤ ∑ h ∈ H, ‖∑ b ∈ Q, u (mb) cubicSymbol b h‖^2 :=
	Finset.sum_le_sum (fun h _ => hn h)
	_ ≤ finiteCubicBound Q H * ∑ b ∈ Q, ‖u (m*b)‖^2 :=
	finiteCubicBound_controls_mass Q H (fun b => u (m*b))
	_ = _ := by rw [Finset.sum_image hinj]

	end CubicFirstMoment

	namespace CubicFirstMoment

	lemma norm_idealMoebius_le_one (m : Eisenstein) : ‖(idealMoebius m : ℂ)‖ ≤ 1 := by
	have hs : (idealMoebius m : ℝ)^2 ≤ 1 := by
	rw [idealMoebius_sq]
	split_ifs <;> norm_num
	have hn : ‖(idealMoebius m : ℂ)‖ = \|(idealMoebius m : ℝ)\| :=
	Complex.norm_intCast _
	rw [hn]
	nlinarith [sq_abs (idealMoebius m : ℝ), abs_nonneg (idealMoebius m : ℝ)]

	theorem finiteCubicBound_coprime_bilinear (S H U : Finset Eisenstein)
	(hS : ∀ a ∈ S, primary a ∧ Squarefree a)
	(hU : ∀ p ∈ U, primaryPrime p)
	(hSU : ∀ a ∈ S, primaryPrimeFactors a ⊆ U)
	(v w phase : Eisenstein → ℂ) (hphase : ∀ h ∈ H, ‖phase h‖ ≤ 1) :
	‖∑ h ∈ H, phase h * ∑ a ∈ S, ∑ b ∈ S,
	if IsCoprime a b then
	(v a * star (cubicSymbol a h)) * star (w b * star (cubicSymbol b h)) else 0‖ ≤
	∑ s ∈ U.powerset,
	finiteCubicBound (S.filter (fun a => (∏ p ∈ s, p) ∣ a)) H *
	((∑ a ∈ S.filter (fun a => (∏ p ∈ s, p) ∣ a), ‖v a‖^2) +
	(∑ a ∈ S.filter (fun a => (∏ p ∈ s, p) ∣ a), ‖w a‖^2)) / 2 := by
	let V (s : Finset Eisenstein) (h : Eisenstein) : ℂ :=
	∑ a ∈ S.filter (fun a => (∏ p ∈ s, p) ∣ a), v a * star (cubicSymbol a h)
	let W (s : Finset Eisenstein) (h : Eisenstein) : ℂ :=
	∑ a ∈ S.filter (fun a => (∏ p ∈ s, p) ∣ a), w a * star (cubicSymbol a h)
	have he (h : Eisenstein) := coprime_bilinear_moebius S S U hS hU hSU
	(fun a => v a * star (cubicSymbol a h)) (fun a => w a * star (cubicSymbol a h))
	have he' (h : Eisenstein) :
	(∑ a ∈ S, ∑ b ∈ S, if IsCoprime a b then
	(v a * star (cubicSymbol a h)) * star (w b * star (cubicSymbol b h)) else 0) =
	∑ s ∈ U.powerset, (idealMoebius (∏ p ∈ s, p) : ℂ) * V s h * star (W s h) := by
	simpa only [V, W, Finset.sum_filter] using he h
	simp_rw [he', Finset.mul_sum]
	rw [Finset.sum_comm]
	have hfactor (s : Finset Eisenstein) :
	(∑ h ∈ H, phase h * ((idealMoebius (∏ p ∈ s, p) : ℂ) * V s h * star (W s h))) =
	(idealMoebius (∏ p ∈ s, p) : ℂ) * ∑ h ∈ H, phase h * V s h * star (W s h) := by
	rw [Finset.mul_sum]
	apply Finset.sum_congr rfl
	intro h hh
	ring
	simp_rw [hfactor]
	apply (norm_sum_le _ _).trans
	apply Finset.sum_le_sum
	intro s hs
	rw [norm_mul]
	calc
	_ ≤ 1 * ‖∑ h ∈ H, phase h * V s h * star (W s h)‖ :=
	mul_le_mul_of_nonneg_right (norm_idealMoebius_le_one _) (_root_.norm_nonneg _)
	_ ≤ _ := by
	rw [one_mul]
	exact finiteCubicBound_conjugate_bilinear _ H v w phase hphase

	end CubicFirstMoment
	end
	end

	end OAI