import OAI.Geometry.PinnedDistances.ArithmeticGraphs namespace OAI universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 section noncomputable section open NumberField open scoped BigOperators noncomputable section open MeasureTheory open scoped Topology noncomputable section open MeasureTheory open scoped Topology ENNReal noncomputable section open MeasureTheory NumberField open scoped BigOperators noncomputable section open scoped BigOperators open MeasureTheory noncomputable section open Filter open scoped Topology noncomputable section open scoped BigOperators noncomputable section open Filter open scoped Topology namespace WeakPinned /-- The scalar limiting step for small overlap, with fixed positive τ and H. -/ theorem small_overlap_rate (n η U A D : ℕ → ℝ) (θ : ℝ) (hθ : 0 < θ) (hn : Tendsto n atTop atTop) (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0 ≤ η j) (hU : ∀ j, 0 ≤ U j) (hA : ∀ j, 0 ≤ A j) (hD : ∀ j, 0 ≤ D j) (hbound : ∀ τ H : ℝ, 0 < τ → τ ≤ 1 → 0 ≤ H → ∀ᶠ j in atTop, D j ≤ (2*(1/Real.log (n j))/τ^2+2*(1/Real.log (n j))/(θ-η j)+((θ+η j)/(θ-η j)-1))*U j + (2*τ/(θ-η j)+(2-2*((θ-η j)/(θ+η j/8))))*A j + 4*(1/Real.log (n j))*H + 32*Real.exp (-H/2)) : Tendsto (fun j => D j/(U j+A j+1)) atTop (𝓝 0) := by have hlamT : Tendsto (fun j => 1/Real.log (n j)) atTop (𝓝 0) := tendsto_const_nhds.div_atTop (Real.tendsto_log_atTop.comp hn) apply tendsto_order.mpr constructor · intro a ha exact Filter.Eventually.of_forall fun j => ha.trans_le (div_nonneg (hD j) (by linarith [hU j,hA j])) · intro ε hε let τ := min 1 (ε*θ/16) have hτ : 0 < τ := lt_min (by norm_num) (by positivity) have hτ1 : τ ≤ 1 := min_le_left _ _ have hτb : 2*τ/θ ≤ ε/8 := by apply (div_le_iff₀ hθ).mpr have := min_le_right (1:ℝ) (ε*θ/16) dsimp [τ] at ⊢ linarith have htExp : Tendsto (fun H : ℝ => 32*Real.exp (-H/2)) atTop (𝓝 0) := by have hd : Tendsto (fun H : ℝ => H/2) atTop atTop := by simpa only [div_eq_mul_inv, mul_comm, id_eq, one_mul] using (tendsto_const_mul_atTop_of_pos (by norm_num : (0:ℝ)<1/2)).mpr tendsto_id simpa only [Function.comp_def,neg_div,mul_zero] using (Real.tendsto_exp_neg_atTop_nhds_zero.comp hd).const_mul 32 obtain ⟨H,hH,hHb⟩ := (eventually_ge_atTop (0:ℝ) |>.and (htExp.eventually (eventually_lt_nhds (show (0:ℝ)<ε/8 by linarith)))).exists let a := fun j => 2*(1/Real.log (n j))/τ^2+2*(1/Real.log (n j))/(θ-η j)+((θ+η j)/(θ-η j)-1) let b := fun j => 2*τ/(θ-η j)+(2-2*((θ-η j)/(θ+η j/8))) let e := fun j => 4*(1/Real.log (n j))*H+32*Real.exp (-H/2) have hθm : Tendsto (fun j => θ-η j) atTop (𝓝 θ) := by simpa using (tendsto_const_nhds (x:=θ)).sub hη have hθp : Tendsto (fun j => θ+η j) atTop (𝓝 θ) := by simpa using (tendsto_const_nhds (x:=θ)).add hη have hθp8 : Tendsto (fun j => θ+η j/8) atTop (𝓝 θ) := by simpa using (tendsto_const_nhds (x:=θ)).add (hη.div_const 8) have ha : Tendsto a atTop (𝓝 0) := by have ha1 : Tendsto (fun j => 2*(1/Real.log (n j))/τ^2) atTop (𝓝 0) := by simpa using (hlamT.const_mul 2).div_const (τ^2) have ha2 : Tendsto (fun j => 2*(1/Real.log (n j))/(θ-η j)) atTop (𝓝 0) := by convert (hlamT.const_mul 2).div hθm hθ.ne' using 1 simp have ha3 : Tendsto (fun j => (θ+η j)/(θ-η j)-1) atTop (𝓝 0) := by simpa [hθ.ne',Pi.div_apply] using (hθp.div hθm hθ.ne').sub (tendsto_const_nhds (x:=1)) simpa [a] using (ha1.add ha2).add ha3 have hb : Tendsto b atTop (𝓝 (2*τ/θ)) := by have hb1 := (tendsto_const_nhds (x:=2*τ)).div hθm hθ.ne' have hb2 : Tendsto (fun j => 2-2*((θ-η j)/(θ+η j/8))) atTop (𝓝 0) := by simpa [hθ.ne',Pi.div_apply] using (tendsto_const_nhds (x:= (2:ℝ))).sub ((hθm.div hθp8 hθ.ne').const_mul 2) simpa [b,Pi.div_apply] using hb1.add hb2 have he : Tendsto e atTop (𝓝 (32*Real.exp (-H/2))) := by simpa [e] using ((hlamT.const_mul 4).mul_const H).add (tendsto_const_nhds (x:=32*Real.exp (-H/2))) have hlt : 2*τ/θ+32*Real.exp (-H/2)<ε := by linarith have hsum := (ha.add hb |>.add he).eventually (eventually_lt_nhds (by simpa using hlt)) filter_upwards [hbound τ H hτ hτ1 hH,hsum,hη.eventually (eventually_lt_nhds hθ), (Real.tendsto_log_atTop.comp hn).eventually (eventually_gt_atTop (0:ℝ))] with j hj hjlt hjη hjlog have ha0 : 0 ≤ a j := by have hr : 1 ≤ (θ+η j)/(θ-η j) := (one_le_div (sub_pos.mpr hjη)).mpr (by linarith [hη0 j]) dsimp [a] positivity have hb0 : 0 ≤ b j := by have hr : (θ-η j)/(θ+η j/8) ≤ 1 := (div_le_one (by linarith [hη0 j] : 0<θ+η j/8)).mpr (by linarith [hη0 j]) have hr0 : 0 ≤ 2-2*((θ-η j)/(θ+η j/8)) := by linarith dsimp [b] positivity have he0 : 0 ≤ e j := by dsimp [e]; positivity have hden : 0 < U j+A j+1 := by linarith [hU j,hA j] have hupper : D j ≤ (a j+b j+e j)*(U j+A j+1) := by have hj' : D j ≤ a j*U j+b j*A j+e j := by simpa only [a,b,e,add_assoc] using hj nlinarith only [hj',hU j,hA j,ha0,hb0,he0,mul_nonneg ha0 (hA j),mul_nonneg hb0 (hU j),mul_nonneg he0 (hU j),mul_nonneg he0 (hA j)] exact (div_le_iff₀ hden).mpr hupper |>.trans_lt hjlt end WeakPinned namespace WeakPinned open Filter open scoped Topology open Global variable (V : ℕ → Type u1) (I : ℕ → Type u2) (K : ℕ → Type u3) [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)] [∀ j, Field (K j)] [∀ j, NumberField (K j)] variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ) (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j)) /-- Application of the scalar rate to the genuinely prepared arithmetic graphs, with no degree bound on the sequence of number fields. -/ theorem arithmetic_small_overlap_rate (hθ : 0<θ) (hn : Tendsto (fun j => (Fintype.card (V j) : ℝ)) atTop atTop) (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j)) (t : ∀ j, I j → K j) (ht : ∀ j e y, y ∈ (G j).targets e → (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) : Tendsto (fun j => smallOverlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) c / (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) := by apply small_overlap_rate (fun j => (Fintype.card (V j):ℝ)) η _ _ _ θ hθ hn hη hη0 (fun j => overlapScale_nonneg _ _ _ _) (fun j => potentialScale_nonneg _ _ _ _) (fun j => smallOverlapScale_nonneg _ _ _ _ c) intro τ H hτ hτ1 hH filter_upwards [hη.eventually (eventually_lt_nhds hθ), (Real.tendsto_log_atTop.comp hn).eventually (eventually_ge_atTop (2:ℝ)), hn.eventually (eventually_ge_atTop (2/τ))] with j hjη hjlog hjn have hlog : 0 ≤ 1/Real.log (Fintype.card (V j):ℝ) := by positivity have hlog2 : 1/Real.log (Fintype.card (V j):ℝ) ≤ 1/2 := by exact one_div_le_one_div_of_le (by norm_num) hjlog have hτn : 2≤τ*(Fintype.card (V j):ℝ) := by have := (div_le_iff₀ hτ).mp hjn nlinarith exact (hG j).arithmetic_small_overlap (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) hjη (hη0 j) (t j) (ht j) τ H hlog hlog2 hτ hτ1 hτn hH end WeakPinned namespace WeakPinned.Core.Fibers open scoped BigOperators open Variance variable {V : Type u4} {I : Type u5} [Fintype V] [Fintype I] [DecidableEq V] variable (G : Fibers V I) def retainedWeight (T : ℝ) (y : V) : ℝ := if T≤((G.incoming y).card:ℝ) then G.incomingWeight y else 0 def retainedSet (T : ℝ) (y : V) : Finset V := if T≤((G.incoming y).card:ℝ) then G.incoming y else Finset.univ omit [Fintype I] in lemma retainedWeight_nonneg (T : ℝ) (y : V) : 0≤G.retainedWeight T y := by unfold retainedWeight incomingWeight edgeCount split_ifs <;> positivity omit [Fintype I] in lemma retainedWeight_le (T : ℝ) (y : V) : G.retainedWeight T y≤G.incomingWeight y := by unfold retainedWeight split_ifs · exact le_rfl · unfold incomingWeight edgeCount; positivity omit [Fintype I] in lemma sum_retainedWeight (T : ℝ) (hG : 0 G.retainedWeight_le T y) omit [Fintype I] in lemma retainedSet_size (T : ℝ) (hT : T≤Fintype.card V) (y : V) : T≤((G.retainedSet T y).card:ℝ) := by unfold retainedSet split_ifs with h · exact h · simpa using hT omit [Fintype I] in lemma retained_domination (T M : ℝ) (hdom : ∀ x y, G.edgeLaw x y≤M*pairProb Finset.univ x y) (hn : 2≤Fintype.card V) (x : V) : (∑ y, G.retainedWeight T y*prob (G.retainedSet T y) x)≤M*prob Finset.univ x := by refine le_trans (Finset.sum_le_sum (fun y _ => ?_)) (G.incoming_domination M hdom hn x) unfold retainedWeight retainedSet split_ifs · exact le_rfl · simp only [zero_mul] unfold incomingWeight edgeCount prob split_ifs <;> positivity end WeakPinned.Core.Fibers namespace WeakPinned open Filter Variance Global open scoped Topology BigOperators variable {V : ℕ → Type u6} {I : ℕ → Type u7} {K : ℕ → Type u8} [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)] [∀ j, Field (K j)] [∀ j, NumberField (K j)] variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ) (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j)) lemma PreparedGraph.uniform_domination {V : Type u9} {I : Type u10} [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) (hθ : 0<θ) (hη : η≤θ/2) (x y : V) : G.edgeLaw x y≤(2/θ)*pairProb Finset.univ x y := by have hh : η<θ := by linarith refine (h.edge_domination hh x y).trans (mul_le_mul_of_nonneg_right ?_ (by unfold pairProb split_ifs · have hn : (2:ℝ)≤Fintype.card V := by exact_mod_cast h.card_ge simp only [Finset.card_univ] exact div_nonneg (by norm_num) (mul_nonneg (by positivity) (by linarith)) · exact le_rfl)) apply (div_le_div_iff₀ (by linarith : 0<θ-η) hθ).mpr linarith def fiberVariance (j : ℕ) : ℝ := weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)) (varianceCell (G j).fiberWeight (G j).targets ((Fintype.card (V j):ℝ)^(1-c))) def incomingVariance (j : ℕ) : ℝ := weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)) (varianceCell ((G j).retainedWeight ((Fintype.card (V j):ℝ)^γ)) ((G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) ((Fintype.card (V j):ℝ)^(1-c))) /-- Both source sampling families satisfy the proved vanishing arithmetic variance estimate, on every sequence of prepared models. No uniform degree bound on the number fields is used. -/ theorem sampling_variance_rates (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hcγ : c<γ) (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop) (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) (hηb : ∀ j, η j≤θ/2) (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j)) (t : ∀ j, I j → K j) (ht : ∀ j e y, y ∈ (G j).targets e → (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) : Tendsto (fun j => fiberVariance G c Z₁ Z₂ h₁ h₂ j / (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) ∧ Tendsto (fun j => incomingVariance G γ c Z₁ Z₂ h₁ h₂ j / (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) := by have hsmall := arithmetic_small_overlap_rate V I K G θ γ c η Z₁ Z₂ h₁ h₂ hθ hn hη hη0 hG t ht have hηlt (j) : η j<θ := by linarith [hηb j] have hpos (j) : 0<(G j).edgeCount := (hG j).edgeCount_pos (hηlt j) have hdom (j x y) := (hG j).uniform_domination hθ (hηb j) x y constructor · exact arithmetic_variance_rate Z₁ Z₂ h₁ h₂ (fun j => (G j).fiberWeight) (fun j => (G j).targets) γ c (2/θ) hγ hγ1 hcγ (hcγ.trans hγ1) (by positivity) hn (fun j => (G j).fiberWeight_nonneg) (fun j => (G j).sum_fiberWeight (hpos j) |>.le) (Eventually.of_forall fun j e => (hG j).fiber e |>.le) (Eventually.of_forall fun j x => (G j).fiber_domination _ (hdom j) (hG j).card_ge x) hsmall · apply arithmetic_variance_rate Z₁ Z₂ h₁ h₂ (fun j => (G j).retainedWeight ((Fintype.card (V j):ℝ)^γ)) (fun j => (G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) γ c (2/θ) hγ hγ1 hcγ (hcγ.trans hγ1) (by positivity) hn (fun j y => (G j).retainedWeight_nonneg _ y) (fun j => (G j).sum_retainedWeight _ (hpos j)) _ _ hsmall · apply Eventually.of_forall intro j y apply (G j).retainedSet_size have hn1 : (1:ℝ)≤Fintype.card (V j) := by exact_mod_cast (by have := (hG j).card_ge; omega : 1≤Fintype.card (V j)) simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn1 hγ1.le · exact Eventually.of_forall fun j x => (G j).retained_domination _ _ (hdom j) (hG j).card_ge x end WeakPinned end end end end end end end end end end OAI