| 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 |
| |
