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import OAI.Analysis.CoulombIonization.Localization.MasterExpandedAnnularQuery
namespace OAI
noncomputable section
namespace CoulombBarrier
open MeasureTheory ProbabilityTheory Filter Set Metric
open scoped BigOperators Topology ContDiff
open CoulombAtom CoulombObservation
lemma original_posterior_expanded_count_band_bound {N K : ℕ}
(μ : Measure (Configuration N)) [IsFiniteMeasure μ]
(ell : Fin K → ℝ) (hell : ∀ k, 0 ≤ ell k) (j : ℕ) (k : Fin K)
(hk : j ≤ k.val) {B r : ℝ} (hB : 1 ≤ B) (hr : 0 < r)
(hnoise : Real.sqrt 3*ell k ≤ r/4) :
∀ᵐ z ∂physicalObservationLaw μ K,
(∫ x, rawAnnularCount (r/2) (2*B*r) x ∂originalRawKernel μ ell j (originalDatum ell j z)) ≤
observedAnnularCount ell k (r/4) (3*B*r) z := by
filter_upwards [original_posterior_annular_count_le_observed μ ell hell j k hk (r/2) (2*B*r)] with z hz
apply hz.trans
exact rawAnnularCount_mono (by linarith) (by nlinarith) _
theorem exists_observed_posterior_expanded_density_constant {g : Space → ℝ} (hg : Continuous g)
(hgs : tsupport g ⊆ ball 0 1) {B c₁ : ℝ} (hB : 1 ≤ B) (hc : 0 < c₁)
(hcL : c₁ < (10*(100000:ℝ))⁻¹) :
∃ D : ℝ, 0 ≤ D ∧ ∀ {N K : ℕ} (μ : Measure (Configuration N)),
∀ [IsFiniteMeasure μ] (ell : Fin K → ℝ), (∀ k, 0 ≤ ell k) →
∀ (j : ℕ) (k : Fin K), j ≤ k.val → ∀ {r₀ s r T : ℝ},
0 < r₀ → 0 < s → s ≤ 1 → r₀ ≤ s → r₀ ≤ 2*r → r ≤ s →
Real.sqrt 3*ell k ≤ r/4 →
∀ᵐ z ∂physicalObservationLaw μ K,
observedAnnularCount ell k (r/4) (3*B*r) z ≤ T/r^3 →
∀ y : Space, r ≤ ‖y‖ → ‖y‖ ≤ B*r →
jointMasterPosterior μ ell j c₁ r₀ s g (originalDatum ell j z) y ≤
D*T*r^(-6-3*masterExponent) := by
obtain ⟨D,hD,hbound⟩ := exists_posterior_expanded_annular_density_constant hg hgs hB hc hcL
refine ⟨D,hD,?_⟩
intro N K μ hμ ell hell j k hk r₀ s r T hr₀ hs hs1 hr₀s hrr hrs hnoise
have hr : 0 < r := by linarith
filter_upwards [original_posterior_expanded_count_band_bound μ ell hell j k hk hB hr hnoise] with z hz
intro hcount y hy hy2
exact (hbound μ ell j hr₀ hs hs1 hrr hrs _ y hy hy2).trans
((mul_le_mul_of_nonneg_left (hz.trans hcount) (by positivity)).trans_eq
(master_density_count_scale hr D T))
theorem exists_observed_posterior_expanded_query_constant {g : Space → ℝ}
(hg : ContDiff ℝ ∞ g) (hgs : tsupport g ⊆ ball 0 1) {B c₁ : ℝ} (hB : 1 ≤ B)
(hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹) :
∃ C : ℝ, 0 ≤ C ∧ ∀ {N K : ℕ} (μ : Measure (Configuration N)),
∀ [IsFiniteMeasure μ] (ell : Fin K → ℝ), (∀ k, 0 ≤ ell k) →
∀ (j : ℕ) (k : Fin K), j ≤ k.val → ∀ {r₀ s r T : ℝ},
0 < r₀ → 0 < s → s ≤ 1 → r₀ ≤ s → r₀ ≤ 2*r → r ≤ s →
Real.sqrt 3*ell k ≤ r/4 →
∀ᵐ q ∂physicalObservationLaw μ K,
observedAnnularCount ell k (r/4) (3*B*r) q ≤ T/r^3 →
∀ y z : Space, r ≤ ‖y‖ → ‖y‖ ≤ B*r → r ≤ ‖z‖ → ‖z‖ ≤ B*r →
‖jointMasterPosterior μ ell j c₁ r₀ s g (originalDatum ell j q) y-
jointMasterPosterior μ ell j c₁ r₀ s g (originalDatum ell j q) z‖ ≤
C*T*r^(-7-4*masterExponent)*‖y-z‖ := by
obtain ⟨C,hC,hbound⟩ := exists_posterior_expanded_annular_query_constant hg hgs hB hc hcL
refine ⟨C,hC,?_⟩
intro N K μ hμ ell hell j k hk r₀ s r T hr₀ hs hs1 hr₀s hrr hrs hnoise
have hr : 0 < r := by linarith
filter_upwards [original_posterior_expanded_count_band_bound μ ell hell j k hk hB hr hnoise] with q hq
intro hcount y z hy hy2 hz hz2
exact (hbound μ ell j hr₀ hs hs1 hr₀s hrr hrs _ y z hy hy2 hz hz2).trans
((mul_le_mul_of_nonneg_left (hq.trans hcount) (by positivity)).trans_eq
(master_query_count_scale hr C T ‖y-z‖))
end CoulombBarrier
open MeasureTheory Filter Set
open scoped Topology ContDiff
namespace CoulombAtom
open CoulombAnalysis CoulombObservation CoulombBarrier
attribute [local irreducible] graphComponent graphFormVector fermionGraph weakGraph fermionGraphValue
lemma rpow_half_scale {u : ℝ} (hu : 0 ≤ u) (e : ℝ) :
(u/2)^e = (1/2:ℝ)^e*u^e := by
rw [show u/2 = (1/2:ℝ)*u by ring, Real.mul_rpow (by norm_num) hu]
theorem exists_actual_expanded_conditional_regularity_constants {B c₁ : ℝ}
(hB : 1 ≤ B) (hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹) :
∃ T D L : ℝ, 0 < T ∧ 0 ≤ D ∧ 0 ≤ L ∧ ∀ {ι : Type*} {l : Filter ι}
{r₀ s Z lam : ι → ℝ} {N K : ι → ℕ} {F : ∀ i, fermionGraph (N i)} {δ : ℝ},
Tendsto s l (𝓝 0) → (∀ᶠ i in l, 0 < r₀ i) →
(∀ᶠ i in l, 0 ≤ Z i ∧ 0 < lam i ∧
OwnProbabilityTailTiltState (Z i) (lam i) (r₀ i) (K i)
(fun k => tinyProbabilityFloor (Z i) ((2:ℝ)^k.val*r₀ i)) δ (F i)) →
∀ᶠ i in l, ∀ j : Fin (K i), (2:ℝ)^j.val*r₀ i ≤ s i →
let u := (2:ℝ)^j.val*r₀ i
let count := observedAnnularCount
(fun k : Fin (K i) => dyadicObservationWidth (r₀ i) k) j (u/8) (3*B*u)
Measurable[observationInformation
(fun k : Fin (K i) => dyadicObservationWidth (r₀ i) k) j] count ∧
(physicalObservationLaw (graphRawLaw (F i)) (K i)).real
{z | T/u^3 < count z} < u^40 ∧
∀ᵐ q ∂physicalObservationLaw (graphRawLaw (F i)) (K i),
count q ≤ T/u^3 →
(∀ y : Space, u/2 ≤ ‖y‖ → ‖y‖ ≤ B*u →
originalQueryDensity (F i) (r₀ i) j c₁ (r₀ i) (s i) q y ≤
D*u^(-6-3*masterExponent)) ∧
(∀ y z : Space, u/2 ≤ ‖y‖ → ‖y‖ ≤ B*u → u/2 ≤ ‖z‖ → ‖z‖ ≤ B*u →
‖originalQueryDensity (F i) (r₀ i) j c₁ (r₀ i) (s i) q y-
originalQueryDensity (F i) (r₀ i) j c₁ (r₀ i) (s i) q z‖ ≤
L*u^(-7-4*masterExponent)*‖y-z‖) := by
obtain ⟨T,hT,hcount⟩ := exists_actual_uniform_annular_good_constant
(by norm_num : (0:ℝ) < 1/8) (by linarith : (0:ℝ) < 3*B)
obtain ⟨D,hD,hdensity⟩ := exists_observed_posterior_expanded_density_constant
canonicalRealPacket_smooth.continuous canonicalRealPacket_support (by linarith : 1 ≤ 2*B) hc hcL
obtain ⟨L,hL,hquery⟩ := exists_observed_posterior_expanded_query_constant
canonicalRealPacket_smooth canonicalRealPacket_support (by linarith : 1 ≤ 2*B) hc hcL
refine ⟨T,D*T*(1/2:ℝ)^(-6-3*masterExponent),L*T*(1/2:ℝ)^(-7-4*masterExponent),
hT,by positivity,by positivity,?_⟩
intro ι l r₀ s Z lam N K F δ hs0 hr₀ hstate
have hn := uniform_observation_noise_eventually (by norm_num : (0:ℝ) < 1/8) hs0
filter_upwards [hcount hs0 hr₀ hstate,hr₀,hn,
hs0.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1))] with i hci hri hni hsi
intro j hjs
let u := (2:ℝ)^j.val*r₀ i
have hru : r₀ i ≤ u := le_mul_of_one_le_left hri.le (one_le_pow₀ (by norm_num))
have hu : 0 < u := hri.trans_le hru
have hs : 0 < s i := hu.trans_le hjs
have hnoise : Real.sqrt 3*dyadicObservationWidth (r₀ i) j ≤ (u/2)/4 := by
have hh := hni u hu hjs
simpa only [dyadicObservationWidth,u,show (1.01:ℝ) = 101/100 by norm_num,
one_div_mul_eq_div,div_div,show (2:ℝ)*4 = 8 by norm_num] using hh
have hr₀s := hru.trans hjs
have hrr : r₀ i ≤ 2*(u/2) := by linarith
have hrs : u/2 ≤ s i := by linarith
have henn : 0 ≤ u/2 := by positivity
have hshape : (u/2)/4 = u/8 := by ring
have hshape' : 3*(2*B)*(u/2) = 3*B*u := by ring
have hshape'' : (2*B)*(u/2) = B*u := by ring
refine ⟨observedAnnularCount_information_measurable _ j le_rfl _ _,?_,?_⟩
· simpa only [one_div_mul_eq_div] using hci j hjs
· filter_upwards [hdensity (graphRawLaw (F i)) _ (fun k => (dyadicObservationWidth_pos hri k).le)
j j le_rfl hri hs hsi.le hr₀s hrr hrs hnoise,
hquery (graphRawLaw (F i)) _ (fun k => (dyadicObservationWidth_pos hri k).le)
j j le_rfl hri hs hsi.le hr₀s hrr hrs hnoise] with q hd hq
intro hcut
have hcut' : observedAnnularCount (fun k : Fin (K i) => dyadicObservationWidth (r₀ i) k)
j ((u/2)/4) (3*(2*B)*(u/2)) q ≤ T/(u/2)^3 := by
rw [hshape,hshape']
exact hcut.trans (div_le_div_of_nonneg_left hT.le (by positivity)
(pow_le_pow_left₀ henn (by linarith : u/2 ≤ u) 3))
have hd' := hd hcut'
have hq' := hq hcut'
rw [hshape''] at hd' hq'
constructor
· intro y hyl hyh
simpa only [rpow_half_scale hu.le,mul_assoc,u,originalQueryDensity] using hd' y hyl hyh
· intro y z hyl hyh hzl hzh
simpa only [rpow_half_scale hu.le,mul_assoc,u,originalQueryDensity] using hq' y z hyl hyh hzl hzh
end CoulombAtom
end
end OAI
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