| import OAI.NumberTheory.TwoPoint.Walks.TupleSmoothCost |
| import OAI.NumberTheory.TwoPoint.Walks.CanonicalTupleSlices |
| import OAI.NumberTheory.TwoPoint.Walks.RetainedRowScale |
|
|
| namespace OAI |
|
|
| /-! Uniformly bounded smooth-dilation cost for every retained subtuple of |
| the actual canonical prime bands. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset Filter |
| open scoped Classical |
|
|
| lemma retained_tuple_factor_subset {J : ℕ} (P : Fin J → Finset ℕ) |
| (hp : ∀ i, ∀ p ∈ P i, p.Prime) (I : Finset (Fin J)) |
| (y : (i : {i // i ∉ I}) → P i) : |
| (∏ i, (y i).val).primeFactors ⊆ univ.image (fun i => (y i).val) := by |
| intro p hp' |
| have hpprime := Nat.prime_of_mem_primeFactors hp' |
| have hdiv := Nat.dvd_of_mem_primeFactors hp' |
| obtain ⟨i, _, hi⟩ := (hpprime.prime.dvd_finsetProd_iff (fun i => (y i).val)).mp hdiv |
| have he : p = (y i).val := (Nat.prime_dvd_prime_iff_eq hpprime |
| (hp i _ (y i).property)).mp hi |
| exact mem_image.mpr ⟨i, mem_univ _, he.symm⟩ |
|
|
| lemma retained_tuple_factor_card {J : ℕ} (P : Fin J → Finset ℕ) |
| (hp : ∀ i, ∀ p ∈ P i, p.Prime) (I : Finset (Fin J)) |
| (y : (i : {i // i ∉ I}) → P i) : (∏ i, (y i).val).primeFactors.card ≤ J := by |
| calc |
| _ ≤ (univ.image (fun i => (y i).val)).card := card_le_card (retained_tuple_factor_subset P hp I y) |
| _ ≤ (univ : Finset {i : Fin J // i ∉ I}).card := card_image_le |
| _ ≤ J := by simpa only [card_univ, Fintype.card_fin] using Fintype.card_subtype_le (fun i : Fin J => i ∉ I) |
|
|
| theorem eventually_canonical_retained_smooth_cost (E : Finset ℕ) (W : ℝ) (hW : 1 ≤ W) : |
| ∀ᶠ L : ℝ in atTop, |
| let J := primeSupplyCount W L |
| let P := centeredPrimeBands E (L ^ (199 / 200 : ℝ)) W J |
| ∀ I : Finset (Fin J), ∀ y : (i : {i // i ∉ I}) → P i, |
| smoothReciprocalProduct (∏ i, (y i).val).primeFactors (1 / 2 : ℝ) ≤ 2 := by |
| filter_upwards [eventually_canonical_smooth_half_cost, eventually_ge_atTop (1 : ℝ)] with L hcost hL |
| dsimp only |
| let J := primeSupplyCount W L |
| let A := L ^ (199 / 200 : ℝ) |
| let P := centeredPrimeBands E A W J |
| have hLp : 0 < L := zero_lt_one.trans_le hL |
| have hp : ∀ i, ∀ p ∈ P i, p.Prime := centeredPrimeBands_prime E A W J |
| intro I y |
| apply hcost |
| · exact (show ((∏ i, (y i).val).primeFactors.card : ℝ) ≤ J by |
| exact_mod_cast retained_tuple_factor_card P hp I y).trans |
| ((primeSupplyCount_le_log W L hW hL).trans |
| ((Real.log_le_sub_one_of_pos hLp).trans (by linarith))) |
| · intro p hpmem |
| obtain ⟨i, _, hi⟩ := mem_image.mp (retained_tuple_factor_subset P hp I y hpmem) |
| subst p |
| exact (centeredPrimeSupply_global_bounds (Real.rpow_nonneg hLp.le _) (zero_le_one.trans hW) i.val.isLt |
| (primeSupplyScale_endpoint W L (by linarith) hL) (y i).property).1.le |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
