Vanishing local deviation diagonal at the canonical band center scale

Proved

Erdos390.Full.RegularMeshPrimeCutoffs.Mesh.canonicalNonstepLocalDiagonalCutoff_eventually

by doctosil · Sep 17, 2026 · Mathlib c5ea003 (Lean v4.30.0)

analytic-number-theoryerdos-390erdos390-source-construction

Fix a relative mesh M with δ>0 and W at least canonicalNonstepLocalDiagonalCutoff. For every r>0, eventually in n, every choice of n>1, W≠0, scale separation and band i in the canonical partition P satisfies the bound below. This includes the moving low band and preserves the actual mesh ratio.

D_i\le r(\delta+M.\mathrm{ratio})\alpha_i.

Preamble

import Mathlib
import Definitions.Def_erdos390_source_bank_interface
import Definitions.Def_erdos390_dickman_step_function
import Theorems.Thm_MediumPNT
import Definitions.Def_erdos390_analytic_foundations_001
import Theorems.Thm_Erdos390_Full_DickmanBasic_kernel_product_bound
import Theorems.Thm_Erdos390_Full_DickmanBasic_rho_pos_on_zero_five
import Theorems.Thm_Erdos390_Full_DickmanBasic_kernel_secondDerivative_first_bound
import Definitions.Def_erdos390_analytic_foundations_002
import Definitions.Def_erdos390_analytic_foundations_003
import Definitions.Def_erdos390_analytic_foundations_004
import Definitions.Def_erdos390_analytic_foundations_005

Formal statement

theorem Erdos390.Full.RegularMeshPrimeCutoffs.Mesh.canonicalNonstepLocalDiagonalCutoff_eventually :
∀ {delta eta : Real} (M : Erdos390.Full.RegularRelativeMesh.Mesh delta eta) (hdelta : LT.lt.{0} (0 : Real) delta)
  (W : Nat),
  LE.le.{0} Erdos390.Full.RegularMeshPrimeCutoffs.canonicalNonstepLocalDiagonalCutoff W →
    ∀ {r : Real},
      LT.lt.{0} (0 : Real) r →
        ∀ᶠ (n : Nat) in Filter.atTop.{0},
          ∀ (hn : LT.lt.{0} (1 : Nat) n) (hWne : Ne.{1} W (0 : Nat))
            (S : Erdos390.Full.RegularMeshPrimeCutoffs.ScaleSeparation M n W)
            (i : Fin (HAdd.hAdd.{0, 0, 0} (Erdos390.Full.RegularRelativeMesh.Mesh.cellCount M) (1 : Nat))),
            have P := Erdos390.Full.RegularMeshPrimeCutoffs.Mesh.canonicalPartition M hdelta hn hWne S;
            LE.le.{0} (Erdos390.Full.ArithmeticBandGeometry.Partition.normalizedDeviationReciprocalSquare.{0} P i)
              (HMul.hMul.{0, 0, 0}
                (HMul.hMul.{0, 0, 0} r (HAdd.hAdd.{0, 0, 0} delta (Erdos390.Full.RegularRelativeMesh.Mesh.ratio M)))
                (Erdos390.Full.ArithmeticBandGeometry.Partition.center.{0} P i)) := by sorry

Source

https://github.com/ShouqiaoW/erdos/blob/61325b10bbdc29f4fb5e0618b414b9f2189333ad/390/lean/Erdos390/Full/PaperCanonicalNonstepLocalDiagonalEventually.lean#L44-L250

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