| import Mathlib |
| import OAI.Analysis.CoulombIonization.Ionization.PricedCoulombTail |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.ObservedFieldScale |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| section |
| open MeasureTheory Set |
|
|
| lemma annular_cut_cost_innerScale {D h u : ℝ} (hD : 0 ≤ D) (hh : 0 < h) (hu : h ≤ u) : |
| (annularOffsetMass D u)^2/u/h ≤ 3*innerScale D h := by |
| have hup : 0 < u := hh.trans_le hu |
| calc |
| _ ≤ (3*(1/u^6+1+D*u))/u/h := |
| div_le_div_of_nonneg_right (div_le_div_of_nonneg_right (annularOffsetMass_sq_le hD hup) hup.le) hh.le |
| _ = 3*((1/u^7)/h+(1/u)/h+D/h) := by field_simp |
| _ ≤ 3*((1/h^7)/h+(1/h)/h+D/h) := by gcongr |
| _ = _ := by unfold innerScale; field_simp |
|
|
| lemma finite_annular_excess_at_innerScale {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevFermion psi) (hm : formMass psi = 1) {Z lam alpha beta u D h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 < lam) (ha : 0 < alpha) (hD : 0 ≤ D) (hh : 0 < h) (hu : h ≤ u) |
| (he : max (corePriceExcess Z lam psi) 0 ≤ D) (ys : List Space) |
| (hys : ∀ y ∈ ys, alpha*u ≤ ‖y‖ ∧ ‖y‖ ≤ beta*u) : |
| max (((CoreObservationEnsemble.initial psi hpsi).observeCells ys).excess Z lam) 0/h ≤ |
| ((radialCutExcessConstant actualCellMomentConstant+1)^ys.length* |
| (1+3*annularCellCostConstant alpha beta))*innerScale D h := by |
| have hbase := actual_finite_annular_cut_excess hpsi hm hZ hlam ha (hh.trans_le hu) hD he ys hys |
| have hcost := annular_cut_cost_innerScale hD hh hu |
| have hC := (annularCellCostConstant_pos (beta := beta) ha).le |
| have hK : 0 ≤ (radialCutExcessConstant actualCellMomentConstant+1)^ys.length := by |
| have := radialCutExcessConstant_one_le actualCellMomentConstant_one_le |
| positivity |
| have hD' : D/h ≤ innerScale D h := by |
| unfold innerScale |
| have h1 : 0 ≤ 1/h^8 := by positivity |
| have h2 : 0 ≤ 1/h^2 := by positivity |
| linarith |
| calc |
| _ ≤ ((radialCutExcessConstant actualCellMomentConstant+1)^ys.length* |
| (D+annularCellCostConstant alpha beta*(annularOffsetMass D u)^2/u))/h := |
| div_le_div_of_nonneg_right hbase hh.le |
| _ = (radialCutExcessConstant actualCellMomentConstant+1)^ys.length* |
| (D/h+annularCellCostConstant alpha beta*((annularOffsetMass D u)^2/u/h)) := by ring |
| _ ≤ (radialCutExcessConstant actualCellMomentConstant+1)^ys.length* |
| (innerScale D h+annularCellCostConstant alpha beta*(3*innerScale D h)) := |
| mul_le_mul_of_nonneg_left (add_le_add hD' (mul_le_mul_of_nonneg_left hcost hC)) hK |
| _ = _ := by ring |
|
|
| lemma actual_observed_local_count {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevFermion psi) (hm : formMass psi = 1) {Z lam D : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 < lam) (he : max (corePriceExcess Z lam psi) 0 ≤ D) |
| (ys : List Space) {z : Space} (hz : z ≠ 0) : |
| ((CoreObservationEnsemble.initial psi hpsi).observeCells ys).countMoment z (8*localCellRadius z) ≤ |
| actualCellMomentConstant*(localOffsetMass D z)^2 := by |
| let E := CoreObservationEnsemble.initial psi hpsi |
| have hr := localCellRadius_pos hz |
| calc |
| _ ≤ E.countMoment z (8*localCellRadius z) := E.observeCells_countMoment ys _ _ |
| _ ≤ E.countMoment z (32*localCellRadius z) := E.countMoment_le_of_radius z z |
| (by simp only [sub_self,norm_zero,zero_add]; linarith) |
| _ = rawCountMoment psi z (32*localCellRadius z) := CoreObservationEnsemble.initial_countMoment .. |
| _ ≤ actualCellMomentConstant*(localOffsetMass (max (corePriceExcess Z lam psi) 0) z)^2 := |
| priced_enlarged_cell_count hpsi hm hZ hlam hz |
| _ ≤ _ := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ |
| (zero_le_one.trans (localOffsetMass_one_le _ z)) (localOffsetMass_mono he z) 2) |
| (zero_le_one.trans actualCellMomentConstant_one_le) |
|
|
| theorem exists_actual_observed_innerCap_constant {alpha beta : ℝ} (ha : 0 < alpha) (L : ℕ) : |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ (Z lam : ℝ) (N : ℕ) (psi : FormVector N) (hpsi : SobolevFermion psi), |
| formMass psi = 1 → 0 ≤ Z → 0 < lam → ∀ (D h u : ℝ), 0 ≤ D → 0 < h → h ≤ u → |
| max (corePriceExcess Z lam psi) 0 ≤ D → ∀ ys : List Space, ys.length = L → |
| (∀ y ∈ ys, alpha*u ≤ ‖y‖ ∧ ‖y‖ ≤ beta*u) → |
| ((CoreObservationEnsemble.initial psi hpsi).observeCells ys).innerCapAverage Z lam h ≤ |
| C*(innerScale D h+lam^2) := by |
| obtain ⟨T,hT,hTail⟩ := exists_priced_coulomb_tail_constant |
| let A := (radialCutExcessConstant actualCellMomentConstant+1)^L*(1+3*annularCellCostConstant alpha beta) |
| let F := localFieldUniversalConstant*(100000*A+(5+actualCellMomentConstant)*3*(annularCellScaleConstant 1 3*100000)^2) |
| let K := 3*(packetDensityConstant*64*(36*Real.pi))^2*F+3*(16*packetPotentialConstant)^2*T |
| let C := max K 3 |
| have hA : 0 ≤ A := by |
| have := radialCutExcessConstant_one_le actualCellMomentConstant_one_le |
| have := annularCellCostConstant_pos (beta := beta) ha |
| dsimp [A]; positivity |
| have hF : 0 ≤ F := by |
| have := localFieldUniversalConstant_pos |
| have := actualCellMomentConstant_one_le |
| dsimp [F]; positivity |
| refine ⟨C,(by dsimp [C]; exact (by norm_num : (1:ℝ) ≤ 3).trans (le_max_right _ _)),?_⟩ |
| intro Z lam N psi hpsi hm hZ hlam D h u hD hh hu he ys hlen hys |
| let E := (CoreObservationEnsemble.initial psi hpsi).observeCells ys |
| have hmE : E.mass = 1 := (CoreObservationEnsemble.observeCells_mass ..).trans ((CoreObservationEnsemble.initial_mass ..).trans hm) |
| have hEx : max (E.excess Z lam) 0/h ≤ A*innerScale D h := by |
| simpa only [E,A,hlen] using finite_annular_excess_at_innerScale hpsi hm hZ hlam ha hD hh hu he ys hys |
| have hf : ∀ z ∈ innerFieldTestAnnulus h, E.fieldMoment Z lam z ≤ F*innerScale D h := by |
| intro z hz |
| exact ensemble_field_annular_bound E hZ hlam hD hh hmE hEx |
| (fun z hz => actual_observed_local_count hpsi hm hZ hlam he ys (norm_pos_iff.mp (hh.trans_le hz.1))) hz |
| have ht : E.weightedMoment (coulombTailWeight h) ≤ T*innerScale D h := by |
| calc |
| _ ≤ (CoreObservationEnsemble.initial psi hpsi).weightedMoment (coulombTailWeight h) := |
| CoreObservationEnsemble.observeCells_weightedMoment _ ys (coulombTailWeight_measurable h) |
| (coulombTailWeight_nonneg h) (coulombTailWeight_le hh) |
| _ = rawWeightedMoment psi (coulombTailWeight h) := CoreObservationEnsemble.initial_weightedMoment .. |
| _ ≤ _ := hTail Z lam N psi hpsi hm hZ hlam D h hD hh he |
| have hb := innerCapAverage_scale E hZ hlam.le hD hh hmE hF hf ht |
| calc |
| _ ≤ K*innerScale D h+3*lam^2 := hb |
| _ ≤ C*innerScale D h+C*lam^2 := add_le_add |
| (mul_le_mul_of_nonneg_right (le_max_left _ _) (innerScale_nonneg hD hh)) |
| (mul_le_mul_of_nonneg_right (le_max_right _ _) (sq_nonneg _)) |
| _ = _ := by ring |
|
|
| end |
| open MeasureTheory Filter |
| open scoped BigOperators |
|
|
| structure FullStatistic where |
| value : ∀ N, Configuration N → ℝ |
| measurable : ∀ N, Measurable (value N) |
| bounded : ∀ N, ∃ B : ℝ, ∀ x, ‖value N x‖ ≤ B |
| reindex : ∀ {M N} (e : Fin M ≃ Fin N) (x : Configuration M), |
| value N (x ∘ e.symm) = value M x |
|
|
| lemma rawFormPair_reindex {M N : ℕ} (e : Fin M ≃ Fin N) (psi : FormVector N) |
| (F : FullStatistic) : rawFormPair (reindexForm e psi) (F.value M) = rawFormPair psi (F.value N) := by |
| have he (s : Spins M) : |
| (∫ x, F.value M x*‖(reindexForm e psi).value s x‖^2) = |
| ∫ x, F.value N x*‖psi.value (s ∘ e.symm) x‖^2 := by |
| simpa only [reindexForm,F.reindex] using |
| integral_reindex e (fun x => F.value N x*‖psi.value (s ∘ e.symm) x‖^2) |
| unfold rawFormPair |
| simp_rw [he] |
| exact sum_spin_reindex e (fun s : Spins N => ∫ x, F.value N x*‖psi.value s x‖^2) |
|
|
| lemma rawFormPair_nextCoreObservation {N M : ℕ} |
| (p : Fin 2 → SmoothMultiplier spaceDirections) (hp : ∀ x, ∑ a, (p a).value x^2=1) |
| (psi : FormVector (N+M)) (c : Fin N → Fin 2) (F : FullStatistic) : |
| rawFormPair (nextCoreObservation p hp psi c) (F.value _) = |
| rawFormPair (multiplyForm (coreLiftMultiplier M (spatialProduct p hp c)) psi) (F.value _) := by |
| unfold nextCoreObservation repeatedCutForm |
| rw [rawFormPair_reindex,rawFormPair_reindex] |
|
|
| lemma sum_rawFormPair_nextCoreObservation {N M : ℕ} {psi : FormVector (N+M)} |
| (hpsi : SobolevVector psi) (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ a, (p a).value x^2=1) (F : FullStatistic) : |
| (∑ c : Fin N → Fin 2, rawFormPair (nextCoreObservation p hp psi c) (F.value _)) = |
| rawFormPair psi (F.value _) := by |
| simp_rw [rawFormPair_nextCoreObservation] |
| obtain ⟨B,hB⟩ := F.bounded (N+M) |
| unfold rawFormPair |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro s _ |
| have hi (c : Fin N → Fin 2) : Integrable (fun x => F.value (N+M) x* |
| ‖(multiplyForm (coreLiftMultiplier M (spatialProduct p hp c)) psi).value s x‖^2) := |
| (((hpsi.multiply _).1 s).norm.integrable_sq).bdd_mul (F.measurable _).aestronglyMeasurable |
| (ae_of_all _ hB) |
| rw [←integral_finsetSum _ (fun c _ => hi c)] |
| apply integral_congr_ae |
| apply ae_of_all |
| intro x |
| have hs : ∑ c : Fin N → Fin 2, (coreLiftMultiplier M (spatialProduct p hp c)).value x^2 = 1 := by |
| exact spatial_square_partition p hp (leftList x) |
| simp only [multiplyForm,Complex.norm_mul,Complex.norm_real,Real.norm_eq_abs,mul_pow,sq_abs] |
| rw [←Finset.mul_sum,←Finset.sum_mul,hs] |
| ring |
|
|
| namespace CoreObservationGraph |
| def fullAverage (G : CoreObservationGraph) (F : FullStatistic) : ℝ := |
| rawFormPair G.vector (F.value _) |
| lemma observe_fullAverage (G : CoreObservationGraph) |
| (p : Fin 2 → SmoothMultiplier spaceDirections) (hp : ∀ x, ∑ a, (p a).value x^2=1) |
| (F : FullStatistic) : |
| (∑ c : Fin G.coreSize → Fin 2, (G.observe p hp c).fullAverage F) = G.fullAverage F := |
| sum_rawFormPair_nextCoreObservation G.sobolev p hp F |
| end CoreObservationGraph |
| namespace CoreObservationEnsemble |
| def fullAverage (E : CoreObservationEnsemble) (F : FullStatistic) : ℝ := |
| ∑ i, (E.graph i).fullAverage F |
| lemma initial_fullAverage {N : ℕ} (psi : FormVector N) (hpsi : SobolevFermion psi) (F : FullStatistic) : |
| (initial psi hpsi).fullAverage F = rawFormPair psi (F.value N) := by |
| change (∑ _i : Unit, rawFormPair psi (F.value N)) = _ |
| simp |
| lemma observe_fullAverage (E : CoreObservationEnsemble) |
| (p : Fin 2 → SmoothMultiplier spaceDirections) (hp : ∀ x, ∑ a, (p a).value x^2=1) |
| (F : FullStatistic) : (E.observe p hp).fullAverage F = E.fullAverage F := by |
| change (∑ ic : Σ i : E.Branch, Fin (E.graph i).coreSize → Fin 2, |
| ((E.graph ic.1).observe p hp ic.2).fullAverage F) = _ |
| rw [Fintype.sum_sigma] |
| exact Finset.sum_congr rfl (fun i _ => (E.graph i).observe_fullAverage p hp F) |
| lemma observeCells_fullAverage (E : CoreObservationEnsemble) (ys : List Space) (F : FullStatistic) : |
| (E.observeCells ys).fullAverage F = E.fullAverage F := by |
| induction ys generalizing E with |
| | nil => rfl |
| | cons y ys ih => |
| rw [observeCells,ih] |
| unfold observeCell |
| split_ifs |
| · rfl |
| · exact observe_fullAverage E _ _ F |
| end CoreObservationEnsemble |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
|
|
