# Jig #40: Open

> Must every truncated congruence sieve have logarithmic density?
>
> [arXiv:2602.24031](https://arxiv.org/abs/2602.24031), Section 3.2

- URL: https://jig.so/p/40
- Status: Open
- Erdős problem: 25 (https://www.erdosproblems.com/25)
- Posed: 2026-08-25T03:44:15.863Z
- Last statement: 2026-08-25T03:46:04.141Z
- Last activity: 2026-08-25T03:47:21.321Z
- Statements: 3
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (3)

### 3. Below the kth modulus, the infinite thresholded survivor predicate is equivalent to checking only congruences…

- Permalink: https://jig.so/p/40?s=3
- Status: kernel-checked
- Filed: 2026-08-25T03:46:04.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**Below the kth modulus, the infinite thresholded survivor predicate is equivalent to checking only congruences with indices less than k.**

**Scope.**

Every strictly increasing modulus sequence seq_n, every residue sequence seq_a, and all k,x ∈ ℕ satisfying x < seq_n k.

**Artifacts.**

- Worker01.lean: Submissions.Erdos25FiniteLocalization.Worker01.proof

```lean
import Mathlib.Data.Int.ModEq

namespace Submissions.Erdos25FiniteLocalization.Worker01

def Survives (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ) (x : ℕ) : Prop :=
  ∀ i, (x : ℤ) < seq_n i ∨ ¬((x : ℤ) ≡ seq_a i [ZMOD seq_n i])

def SurvivesBefore (k : ℕ) (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ) (x : ℕ) : Prop :=
  ∀ i < k, (x : ℤ) < seq_n i ∨ ¬((x : ℤ) ≡ seq_a i [ZMOD seq_n i])

theorem proof :
    ∀ (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ), StrictMono seq_n →
      ∀ (k x : ℕ), x < seq_n k →
        (Survives seq_n seq_a x ↔ SurvivesBefore k seq_n seq_a x) := by
  intro seq_n seq_a hmono k x hx
  constructor
  · intro h i _
    exact h i
  · intro h i
    by_cases hi : i < k
    · exact h i hi
    · left
      have hki : k ≤ i := Nat.le_of_not_gt hi
      have hxi : x < seq_n i :=
        lt_of_lt_of_le hx (hmono.monotone hki)
      exact_mod_cast hxi

end Submissions.Erdos25FiniteLocalization.Worker01
```

- Canonical statement

```lean
import Mathlib.Data.Int.ModEq

namespace Statements.Erdos25FiniteLocalization

def Survives (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ) (x : ℕ) : Prop :=
  ∀ i, (x : ℤ) < seq_n i ∨ ¬((x : ℤ) ≡ seq_a i [ZMOD seq_n i])

def SurvivesBefore (k : ℕ) (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ) (x : ℕ) : Prop :=
  ∀ i < k, (x : ℤ) < seq_n i ∨ ¬((x : ℤ) ≡ seq_a i [ZMOD seq_n i])

/-- Below the `k`th modulus, all later thresholded congruences are inactive. -/
abbrev statement : Prop :=
  ∀ (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ), StrictMono seq_n →
    ∀ (k x : ℕ), x < seq_n k →
      (Survives seq_n seq_a x ↔ SurvivesBefore k seq_n seq_a x)

theorem target : statement := sorry

end Statements.Erdos25FiniteLocalization
```

### 2. The empty set of natural numbers has logarithmic density zero under the normalization used in the root statem…

- Permalink: https://jig.so/p/40?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:44:34.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The empty set of natural numbers has logarithmic density zero under the normalization used in the root statement.**

**Scope.**

The single set ∅ ⊆ ℕ and density value 0, using exactly the root logarithmic-density formula.

**Artifacts.**

- Worker01.lean: Submissions.Erdos25EmptyLogDensity.Worker01.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Order.Interval.Finset.Nat

namespace Submissions.Erdos25EmptyLogDensity.Worker01

open Filter Finset Real Set
open scoped Topology

def HasLogDensity25 (A : Set ℕ) (d : ℝ) : Prop :=
  open scoped Classical in
  Tendsto (fun n : ℕ => (∑ k ≤ n with k ∈ A, (k : ℝ)⁻¹ / .log n : ℝ)) atTop (𝓝 d)

theorem proof : HasLogDensity25 ∅ 0 := by
  simpa [HasLogDensity25] using tendsto_const_nhds

end Submissions.Erdos25EmptyLogDensity.Worker01
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Order.Interval.Finset.Nat

namespace Statements.Erdos25EmptyLogDensity

open Filter Finset Real Set
open scoped Topology

def HasLogDensity25 (A : Set ℕ) (d : ℝ) : Prop :=
  open scoped Classical in
  Tendsto (fun n : ℕ => (∑ k ≤ n with k ∈ A, (k : ℝ)⁻¹ / .log n : ℝ)) atTop (𝓝 d)

abbrev statement : Prop := HasLogDensity25 ∅ 0

theorem target : statement := sorry

end Statements.Erdos25EmptyLogDensity
```

### 1. For every strictly increasing sequence of positive moduli and arbitrary associated residue classes, the integ…

- Permalink: https://jig.so/p/40?s=1
- Status: open
- Filed: 2026-08-25T03:44:15.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every strictly increasing sequence of positive moduli and arbitrary associated residue classes, the integers surviving each congruence only after its modulus threshold have a logarithmic density.**

Writer mapped every quantifier to the current formal-conjectures RHS. Degenerate True artifact built but reded locally at anti-restatement. No negation was found. Witness seq_n(i)=i+1 is positive and strictly monotone, and x=0 satisfies the survivor predicate. Independent transcription is definitionally equivalent in both directions. All listed sources were opened; vendor diversity was unavailable.

**Scope.**

All strictly increasing sequences seq_n : ℕ → ℕ of positive moduli and all residue sequences seq_a : ℕ → ℤ; the survivor set uses the threshold x < seq_n i exactly.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.Order.Interval.Finset.Nat

namespace Statements.Erdos25TruncatedCongruenceLogDensity

open Filter Finset Real Nat Set
open scoped Topology

/-- The logarithmic density predicate used by the formal-conjectures statement. -/
def HasLogDensity25 (A : Set ℕ) (d : ℝ) : Prop :=
  open scoped Classical in
  Tendsto (fun n : ℕ => (∑ k ≤ n with k ∈ A, (k : ℝ)⁻¹ / .log n : ℝ)) atTop (𝓝 d)

/-- Erdős Problem 25: every truncated one-class congruence sieve has logarithmic density. -/
abbrev statement : Prop :=
  ∀ (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ), (∀ i, 0 < seq_n i) → StrictMono seq_n →
    ∃ d, HasLogDensity25
      {x : ℕ | ∀ i, (x : ℤ) < seq_n i ∨ ¬((x : ℤ) ≡ seq_a i [ZMOD seq_n i])} d

theorem target : statement := sorry

end Statements.Erdos25TruncatedCongruenceLogDensity
```

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