| import OAI.NumberTheory.Jacobsthal.Paths.OutsideExceptionsWitness |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.GeometricTotalBoxMass |
| open FinitePathGeometry PrimeHistories PrimeBinMembership ActualSafeBoxMass |
| open GeometricBoxImages SafeSubsetBoxGeometry CanonicalSubsetBox FilteredBoxWordMass |
| open LogarithmicBinScale ErdosSubsetWord |
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| theorem uniform_geometric_box_mass (K : ℝ) (hK : 3 ≤ K) : |
| ∃ C B₀ w₀ : ℝ,0 < C ∧ 3 ≤ B₀ ∧ 1 < w₀ ∧ |
| ∀ B w top : ℝ,B₀ ≤ B → w₀ ≤ w → ∀ (hw : 1 < w) (htop : w < top), |
| ∀ xi Clen : ℝ,∀ hxi : 0 < xi,1 ≤ Clen → Real.log B ≤ 2*Real.log w → 2*Clen*xi ≤ 1 → |
| ∀ z : Node,z.side=.even → 199/100 ≤ z.ratio → z.ratio ≤ 23/10 → Consistent z → |
| z.cutoff=B → z.closed=true → w^B=top → ∀ L alpha beta : ℝ, |
| B^2*(∑ m∈geometricBoxes hw htop hxi Clen B K L alpha beta z,selectionMass (globalBins w top xi) m) ≤ C := by |
| obtain ⟨C,B₀,W,hC,hB₀,hW,hTotal⟩ := uniform_safe_box_mass K hK |
| refine ⟨C,B₀,max W (Real.exp 1),hC,hB₀,hW.trans_le (le_max_left _ _),?_⟩ |
| intro B w top hB hw₀ hw htop xi Clen hxi hClen hcomp hsmall z hi h199 h23 hz hcut hclosed hpower L alpha beta |
| have hlog : 1 ≤ Real.log w := by |
| have hh := Real.log_le_log (Real.exp_pos 1) ((le_max_right _ _).trans hw₀) |
| simpa only [Real.log_exp] using hh |
| have hmesh : 2*(xi/Real.log w) ≤ 6*(2*Clen*xi) := by |
| have hh := div_le_self hxi.le hlog |
| nlinarith |
| exact hTotal B w top hB ((le_max_left _ _).trans hw₀) hw htop xi Clen hxi |
| (by linarith) hcomp hsmall z hi h199 h23 hz hcut hclosed hpower |
| (geometricBoxes hw htop hxi Clen B K L alpha beta z) |
| (geometric_box_anchor hw htop hxi (by linarith) hcomp hmesh z) |
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| theorem unlisted_box_transfer {w top xi Cs C B K L alpha beta sigma A : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hsigma : 0 ≤ sigma) |
| (a : ℕ → ℕ) (z : Node) (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) |
| (F : Finset (List ℕ)) |
| (hTotal : B^2*(∑ m∈geometricBoxes hw htop hxi C B K L alpha beta z,selectionMass (globalBins w top xi) m) ≤ A) |
| (hBox : ∀ m∈geometricBoxes hw htop hxi C B K L alpha beta z, |
| selectedWordMass (globalBins w top xi) m |
| (ExpandedCompactPartition.unlistedGeometric hw htop hxi Cs C B K L alpha beta a z Q F) |
| ≤ sigma*selectionMass (globalBins w top xi) m) : |
| B^2*(∑ ps∈ExpandedCompactPartition.unlistedGeometric hw htop hxi Cs C B K L alpha beta a z Q F, |
| ErdosPrimeInputs.PrimePrefixMass.prefixWeight ps) ≤ sigma*A := by |
| rw [unlisted_geometric_box_mass hw htop hxi a z Q F] |
| calc |
| _ ≤ B^2*(∑ m∈geometricBoxes hw htop hxi C B K L alpha beta z,sigma*selectionMass (globalBins w top xi) m) := |
| mul_le_mul_of_nonneg_left (Finset.sum_le_sum hBox) (sq_nonneg B) |
| _ = sigma*(B^2*(∑ m∈geometricBoxes hw htop hxi C B K L alpha beta z,selectionMass (globalBins w top xi) m)) := by |
| rw [← Finset.mul_sum] |
| ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left hTotal hsigma |
| end NumberTheoryLean.GeometricTotalBoxMass |
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| end Erdos970 |
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| end OAI |
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