lean/OAI/Geometry/PinnedDistances/ArithmeticSequence.lean

import OAI.Geometry.PinnedDistances.ArithmeticContradiction

namespace OAI

universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11

section
noncomputable section
namespace WeakPinned
open Filter
open scoped Topology BigOperators

/-- Diagonalized extraction of genuine configurations. The errors tend to zero
and vertex numbers tend to infinity. -/
theorem graph_sequence_of_failure (hfail : ¬ MainStatement) :
    ∃ c s γ : ℝ, 0<c ∧ c<γ ∧ γ<s ∧ s<1 ∧ 0<Theta s ∧
      ∃ (P : ℕ → Finset Plane) (C : ∀ j, Finset (P j)),
        (∀ j, j<(C j).card) ∧
        ∀ j, PreparedGraph (C j) ((planarFibers (P j) s).Active (cutoff s (P j).card) (C j))
          (coreGraph (P j) s (C j)) (Theta s) γ c (Theta s/(j+2)) := by
  classical
  obtain ⟨c,s,γ,hc,hc3,hcs,hs,hcγ,hγs,hct,hθ⟩ := extremal_parameters hfail
  have hcrit := critical_le_one hfail
  have hsc : (1-c)*criticalExponent<s := (le_max_right _ _).trans_lt hcs
  have hc1 : c<1 := by linarith
  have hγ0 : 0<γ := hc.trans hcγ
  have hex (j : ℕ) := arbitrarily_large_graph hfail hγ0 hγs hc hc1 hsc hct
    (Theta s/(j+2)) (div_pos hθ (by positivity)) j
  choose P C hC hG using hex
  exact ⟨c,s,γ,hc,hcγ,hγs,hs.trans_le hcrit,hθ,P,C,hC,hG⟩

lemma graph_card_tendsto {P : ℕ → Finset Plane} {C : ∀ j, Finset (P j)}
    (hC : ∀ j, j<(C j).card) : Tendsto (fun j => (C j).card) atTop atTop :=
  tendsto_atTop_mono (fun j => (hC j).le) tendsto_id

lemma graph_error_tendsto (θ : ℝ) : Tendsto (fun j : ℕ => θ/((j:ℝ)+2)) atTop (𝓝 0) := by
  have ht : Tendsto (fun j : ℕ => (j:ℝ)+2) atTop atTop := tendsto_atTop_add_const_right _ _ tendsto_natCast_atTop_atTop
  exact tendsto_const_nhds.div_atTop ht

end WeakPinned

/- Algebraic representation of the finite rational polynomial pattern of a
   real configuration in a number field. -/

noncomputable section

namespace WeakPinned.Transfer
open MvPolynomial

/-- A system of rational polynomial equations with a complex solution has an
    algebraic solution. No finite set assumption on the equations is needed. -/
theorem equations {ι : Type u1} [Finite ι]
    (S : Set (MvPolynomial ι ℚ)) (x : ι → ℂ)
    (hx : ∀ p ∈ S, aeval x p = 0) :
    ∃ y : ι → AlgebraicClosure ℚ, ∀ p ∈ S, aeval y p = 0 := by
  let I : Ideal (MvPolynomial ι ℚ) := vanishingIdeal ℚ {x}
  have hI : I ≠ ⊤ := by
    intro h
    have h1 : (1 : MvPolynomial ι ℚ) ∈ I := by rw [h]; trivial
    have hz := (mem_vanishingIdeal_singleton_iff x 1).mp h1
    simp at hz
  obtain ⟨M, hM, hIM⟩ := Ideal.exists_le_maximal I hI
  obtain ⟨y, hy⟩ := eq_vanishingIdeal_singleton_of_isMaximal (AlgebraicClosure ℚ) hM
  refine ⟨y, fun p hp => ?_⟩
  apply (mem_vanishingIdeal_singleton_iff y p).mp
  rw [← hy]
  exact hIM ((mem_vanishingIdeal_singleton_iff x p).mpr (hx p hp))

/-- Simultaneous equations and finitely many inequations transfer to algebraic
    coordinates. The inverse-variable equation enforces every inequation. -/
theorem equations_and_inequations {ι : Type u2} [Finite ι]
    (S : Set (MvPolynomial ι ℚ)) (T : Finset (MvPolynomial ι ℚ)) (x : ι → ℂ)
    (hx : ∀ p ∈ S, aeval x p = 0) (hT : ∀ p ∈ T, aeval x p ≠ 0) :
    ∃ y : ι → AlgebraicClosure ℚ,
      (∀ p ∈ S, aeval y p = 0) ∧ (∀ p ∈ T, aeval y p ≠ 0) := by
  let q : MvPolynomial ι ℚ := ∏ p ∈ T, p
  let r : MvPolynomial (Option ι) ℚ := X none * rename some q - 1
  let S' : Set (MvPolynomial (Option ι) ℚ) := (rename some) '' S ∪ {r}
  let x' : Option ι → ℂ := fun j => j.elim (aeval x q)⁻¹ x
  have hq : aeval x q ≠ 0 := by
    simpa [q] using Finset.prod_ne_zero_iff.mpr hT
  have hl (p : MvPolynomial ι ℚ) : aeval x' (rename some p) = aeval x p := by
    simp [aeval_rename, Function.comp_def, x']
  have hx' : ∀ p ∈ S', aeval x' p = 0 := by
    intro p hp
    rcases hp with ⟨p, hp, rfl⟩ | rfl
    · rw [hl]
      exact hx p hp
    · simp [r, hl, x', hq]
  obtain ⟨y', hy'⟩ := equations S' x' hx'
  let y : ι → AlgebraicClosure ℚ := fun i => y' (some i)
  have hl' (p : MvPolynomial ι ℚ) : aeval y' (rename some p) = aeval y p := by
    simp [aeval_rename, Function.comp_def, y]
  refine ⟨y, ?_, ?_⟩
  · intro p hp
    rw [← hl']
    exact hy' _ (Or.inl ⟨p, hp, rfl⟩)
  · have hr := hy' r (Or.inr rfl)
    have hq' : aeval y q ≠ 0 := by
      intro hz
      simp [r, hl', hz] at hr
    have hprod : (∏ p ∈ T, aeval y p) ≠ 0 := by simpa [q] using hq'
    exact Finset.prod_ne_zero_iff.mp hprod

/-- The algebraic realization lies in a single number field generated by its
    finitely many coordinates. The field is not fixed in advance. -/
theorem numberField_pattern {ι : Type u3} [Finite ι]
    (S : Set (MvPolynomial ι ℚ)) (T : Finset (MvPolynomial ι ℚ)) (x : ι → ℂ)
    (hx : ∀ p ∈ S, aeval x p = 0) (hT : ∀ p ∈ T, aeval x p ≠ 0) :
    ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
      ∃ y : ι → K, (∀ p ∈ S, aeval y p = 0) ∧ (∀ p ∈ T, aeval y p ≠ 0) := by
  obtain ⟨y, hy, hTy⟩ := equations_and_inequations S T x hx hT
  let K := IntermediateField.adjoin ℚ (Set.range y)
  have : Finite (Set.range y) := (Set.finite_range y).to_subtype
  have : Algebra.IsAlgebraic ℚ (AlgebraicClosure ℚ) := AlgebraicClosure.isAlgebraic ℚ
  have : Algebra.IsIntegral ℚ (AlgebraicClosure ℚ) := Algebra.IsAlgebraic.isIntegral
  have hfd : FiniteDimensional ℚ K :=
    IntermediateField.finiteDimensional_adjoin (fun z _ => Algebra.IsIntegral.isIntegral z)
  have hnf : NumberField K := { to_charZero := inferInstance, to_finiteDimensional := hfd }
  let z : ι → K := fun i => ⟨y i, IntermediateField.subset_adjoin ℚ (Set.range y) ⟨i, rfl⟩⟩
  have he (p : MvPolynomial ι ℚ) : K.val (aeval z p) = aeval y p :=
    comp_aeval_apply z K.val p
  refine ⟨K, hnf, z, ?_, ?_⟩
  · intro p hp
    apply K.val.injective
    change K.val (aeval z p) = K.val 0
    rw [he, map_zero, hy p hp]
  · intro p hp hz
    apply hTy p hp
    rw [← he, hz, map_zero]

/-- Preservation of an arbitrary finite zero-pattern, simultaneously, in one
    number field. In particular this preserves all selected equalities exactly. -/
theorem numberField_zero_pattern {ι : Type u4} [Finite ι]
    (T : Finset (MvPolynomial ι ℚ)) (x : ι → ℂ) :
    ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
      ∃ y : ι → K, ∀ p ∈ T, aeval y p = 0 ↔ aeval x p = 0 := by
  classical
  let S : Set (MvPolynomial ι ℚ) := {p | p ∈ T ∧ aeval x p = 0}
  let U := T.filter (fun p => aeval x p ≠ 0)
  obtain ⟨K, hK, y, hy, hUy⟩ := numberField_pattern S U x
    (fun _ hp => hp.2) (fun p hp => (Finset.mem_filter.mp hp).2)
  refine ⟨K, hK, y, fun p hp => ⟨?_, ?_⟩⟩
  · intro h
    by_contra h'
    exact hUy p (Finset.mem_filter.mpr ⟨hp, h'⟩) h
  · intro h
    exact hy p ⟨hp, h⟩

abbrev Plane := EuclideanSpace ℝ (Fin 2)

/-- Complexified variables; the extra variable forces a square root of -1
    into the field, so every archimedean place is complex. -/
abbrev Variables (ι : Type u5) := Option (Bool × ι)

def secondCoord (b : Bool) (z : Plane) : ℂ :=
  if b then ⟨z 0, -z 1⟩ else ⟨z 0, z 1⟩

def complexAssignment {ι : Type u6} (x : ι → Plane) : Variables ι → ℂ :=
  fun j => j.elim Complex.I (fun bi => secondCoord bi.1 (x bi.2))

def distancePoly {ι : Type u7} (a b : ι) : MvPolynomial (Variables ι) ℚ :=
  (X (some (false, a)) - X (some (false, b))) *
  (X (some (true, a)) - X (some (true, b)))

lemma eval_distancePoly {ι : Type u8} (x : ι → Plane) (a b : ι) :
    aeval (complexAssignment x) (distancePoly a b) = ((dist (x a) (x b)) ^ 2 : ℝ) := by
  simp only [distancePoly, map_mul, map_sub, aeval_X, complexAssignment,
    Option.elim_some, secondCoord, Bool.false_eq_true, reduceIte]
  rw [EuclideanSpace.dist_sq_eq]
  simp only [Fin.sum_univ_two, Real.dist_eq, sq_abs]
  apply Complex.ext <;> simp [-Complex.ofReal_pow] <;> ring

/-- Every finite real planar configuration has a faithful number-field model
    for BOTH distance factors. The real configuration is not restricted or
    replaced: the exact equality pattern of ALL its distances is retained.
    The factorization at every place does not require conjugacy in the new field. -/
theorem planar_factor_model {ι : Type u9} [Finite ι]
    (x : ι → Plane) (hx : Function.Injective x) :
    ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
      ∃ (ii : K) (Z₁ Z₂ : ι → K), ii ^ 2 = -1 ∧
        Function.Injective Z₁ ∧ Function.Injective Z₂ ∧
        ∀ a b c d, (Z₁ a - Z₁ b) * (Z₂ a - Z₂ b) =
          (Z₁ c - Z₁ d) * (Z₂ c - Z₂ d) ↔ dist (x a) (x b) = dist (x c) (x d) := by
  classical
  let := Fintype.ofFinite ι
  let D := @distancePoly ι
  let root : MvPolynomial (Variables ι) ℚ := X none ^ 2 + 1
  let pairs := Finset.univ.image (fun t : ι × ι => D t.1 t.2)
  let quadruples := Finset.univ.image (fun t : (ι × ι) × (ι × ι) =>
    D t.1.1 t.1.2 - D t.2.1 t.2.2)
  let T := pairs ∪ quadruples ∪ {root}
  obtain ⟨K, hK, y, hy⟩ := numberField_zero_pattern T (complexAssignment x)
  let Z₁ : ι → K := fun a => y (some (false, a))
  let Z₂ : ι → K := fun a => y (some (true, a))
  have he (a b : ι) : aeval y (D a b) = (Z₁ a - Z₁ b) * (Z₂ a - Z₂ b) := by
    simp [D, distancePoly, Z₁, Z₂]
  have hp (a b : ι) : D a b ∈ T := by
    apply Finset.mem_union.mpr
    left
    apply Finset.mem_union.mpr
    left
    exact Finset.mem_image.mpr ⟨(a,b), Finset.mem_univ _, rfl⟩
  have h0 (a b : ι) : (Z₁ a - Z₁ b) * (Z₂ a - Z₂ b) = 0 ↔ a = b := by
    rw [← he, hy _ (hp a b), eval_distancePoly]
    simp [hx.eq_iff]
  have hdist (a b c d : ι) : D a b - D c d ∈ T := by
    apply Finset.mem_union.mpr
    left
    apply Finset.mem_union.mpr
    right
    exact Finset.mem_image.mpr ⟨((a,b),(c,d)), Finset.mem_univ _, rfl⟩
  have hroot : y none ^ 2 + 1 = 0 := by
    have hm : root ∈ T := Finset.mem_union.mpr (Or.inr (Finset.mem_singleton_self _))
    have hc : aeval (complexAssignment x) root = 0 := by
      simp [root, complexAssignment, Complex.I_sq]
    simpa [root] using (hy root hm).mpr hc
  refine ⟨K, hK, y none, Z₁, Z₂, eq_neg_of_add_eq_zero_left hroot, ?_, ?_, ?_⟩
  · intro a b hab
    apply (h0 a b).mp
    rw [hab, sub_self, zero_mul]
  · intro a b hab
    apply (h0 a b).mp
    rw [hab, sub_self, mul_zero]
  · intro a b c d
    rw [← he, ← he]
    have h := hy _ (hdist a b c d)
    rw [map_sub, sub_eq_zero, map_sub, sub_eq_zero, eval_distancePoly,
      eval_distancePoly, Complex.ofReal_inj] at h
    exact h.trans (sq_eq_sq₀ dist_nonneg dist_nonneg)

end WeakPinned.Transfer

namespace WeakPinned
open scoped BigOperators

/-- The exact arithmetic model for each extracted real graph. No point is
assumed algebraic; the original graph's complete distance-equality pattern
is preserved by the proved finite specialization theorem. -/
theorem core_arithmetic_model (P : Finset Plane) (s : ℝ) (C : Finset P)
    (hne : ∀ e, ((coreGraph P s C).targets e).Nonempty) :
    ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
      ∃ (ii : K) (Z₁ Z₂ : C → K) (t : (planarFibers P s).Active (cutoff s P.card) C → K),
        ii^2 = -1 ∧ Function.Injective Z₁ ∧ Function.Injective Z₂ ∧
        (∀ e, t e ≠ 0) ∧
        (∀ e y, y ∈ (coreGraph P s C).targets e →
          (Z₁ ((coreGraph P s C).source e)-Z₁ y)*(Z₂ ((coreGraph P s C).source e)-Z₂ y)=t e) ∧
        (∀ a b c d : C, (Z₁ a-Z₁ b)*(Z₂ a-Z₂ b)=(Z₁ c-Z₁ d)*(Z₂ c-Z₂ d) ↔
          dist a.val.val b.val.val=dist c.val.val d.val.val) := by
  classical
  let x : C → Plane := fun a => a.val.val
  have hx : Function.Injective x := Subtype.val_injective.comp Subtype.val_injective
  obtain ⟨K,hK,ii,Z₁,Z₂,hii,h₁,h₂,hpat⟩ := Transfer.planar_factor_model x hx
  choose y hy using hne
  let G := coreGraph P s C
  let t := fun e => (Z₁ (G.source e)-Z₁ (y e))*(Z₂ (G.source e)-Z₂ (y e))
  refine ⟨K,hK,ii,Z₁,Z₂,t,hii,h₁,h₂,?_,?_,hpat⟩
  · intro e
    have hxy : G.source e ≠ y e := by intro he; exact G.no_loop e (he ▸ hy e)
    exact mul_ne_zero (sub_ne_zero.mpr (h₁.ne hxy)) (sub_ne_zero.mpr (h₂.ne hxy))
  · intro e z hz
    apply (hpat _ _ _ _).mpr
    have hz' := (Finset.mem_filter.mp hz).2
    have hy' := (Finset.mem_filter.mp (hy e)).2
    have hzdist := (planar_target_iff P s e.val z.val).mp hz'
    have hydist := (planar_target_iff P s e.val (y e).val).mp hy'
    change dist e.val.val.1 z.val.val=dist e.val.val.1 (y e).val.val
    rw [dist_comm,dist_comm e.val.val.1,hzdist,hydist]

lemma PreparedGraph.targets_nonempty {V : Type u10} {I : Type u11} [Fintype V] [Fintype I] [DecidableEq V]
    {G : Core.Fibers V I} {θ γ c η : ℝ} (h : PreparedGraph V I G θ γ c η) :
    ∀ e, (G.targets e).Nonempty := by
  intro e
  have hn : (0:ℝ)<Fintype.card V := by exact_mod_cast (show 0<Fintype.card V by have := h.card_ge; omega)
  have hh : (0:ℝ)<(G.targets e).card := (Real.rpow_pos_of_pos hn γ).trans (h.fiber e)
  exact Finset.card_pos.mp (by exact_mod_cast hh)

/-- Specialization is available independently for EVERY graph in the genuine
counterexample sequence, over number fields of unrestricted degree. -/
theorem sequence_arithmetic_model (hfail : ¬ MainStatement) :
    ∃ c s γ : ℝ, 0<c ∧ c<γ ∧ γ<s ∧ s<1 ∧ 0<Theta s ∧
      ∃ (P : ℕ → Finset Plane) (C : ∀ j, Finset (P j)),
        (∀ j, j<(C j).card) ∧
        (∀ j, PreparedGraph (C j) ((planarFibers (P j) s).Active (cutoff s (P j).card) (C j))
          (coreGraph (P j) s (C j)) (Theta s) γ c (Theta s/(j+2))) ∧
        ∀ j, ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
          ∃ (ii : K) (Z₁ Z₂ : C j → K) (t : (planarFibers (P j) s).Active (cutoff s (P j).card) (C j) → K),
            ii^2 = -1 ∧ Function.Injective Z₁ ∧ Function.Injective Z₂ ∧
            (∀ e, t e ≠ 0) ∧
            (∀ e y, y ∈ (coreGraph (P j) s (C j)).targets e →
              (Z₁ ((coreGraph (P j) s (C j)).source e)-Z₁ y)*(Z₂ ((coreGraph (P j) s (C j)).source e)-Z₂ y)=t e) := by
  obtain ⟨c,s,γ,hc,hcγ,hγs,hs,hθ,P,C,hC,hG⟩ := graph_sequence_of_failure hfail
  refine ⟨c,s,γ,hc,hcγ,hγs,hs,hθ,P,C,hC,hG,?_⟩
  intro j
  obtain ⟨K,hK,ii,Z₁,Z₂,t,hii,h₁,h₂,ht,hfiber,_⟩ :=
    core_arithmetic_model (P j) s (C j) (hG j).targets_nonempty
  exact ⟨K,hK,ii,Z₁,Z₂,t,hii,h₁,h₂,ht,hfiber⟩

end WeakPinned

end
end
end

end OAI