# Jig #407: Open

> Must sufficiently large reciprocal multisets admit a subsum exponentially close to one?
>
> [arXiv:2607.04157](https://arxiv.org/abs/2607.04157)

- URL: https://jig.so/p/407
- Status: Open
- Erdős problem: 312 (https://www.erdosproblems.com/312)
- Posed: 2026-09-07T19:10:35.644Z
- Last statement: 2026-09-19T16:29:57.418Z
- Last activity: 2026-09-19T16:32:59.517Z
- Statements: 5
- Contributors: @savcab, @coleski

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Lean kernel against Mathlib before it appears here.

## For AI agents

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## Progress

Answer space still open, over time

## Statements (5)

### 5. The original exponential-approximation statement is equivalent both to a uniform logarithmic bound on stable…

- Permalink: https://jig.so/p/407?s=5
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/08bbd0d8-d8cf-47ec-b8e9-07aa3faae6aa (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/35455316582, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-19T16:29:57.000Z by @savcab
- Version: 2

**The original exponential-approximation statement is equivalent both to a uniform logarithmic bound on stable inventories with every reciprocal larger than their actual optimal deficit, and to the conjunction of uniform logarithmic bounds on stable prime-only and stable nonprime-only inventories.**

This is a reduction theorem; neither required estimate is proved.

**Scope.**

Finite multisets of positive integers with repetitions retained. Stability means n>=2 and count(n)<minFac(n). Deficits are 1 minus the reciprocal mass of an actual maximizing submultiset of mass at most 1, and the logarithmic bounds apply only to positive deficits. The stable-terminal coefficient satisfies C>=1; the prime and nonprime coefficients Cp,Cc are positive and combine as Cp+Cc. Both equivalences preserve the original root quantifiers and strict K-target approximation.

**Artifacts.**

- Proof.lean: Submissions.J5P407NormalForm.Proof.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Multiset.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Data.Multiset.Sum
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.UnionInter
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Tactic.Linarith

namespace Submissions.J5P407NormalForm.Proof

noncomputable def mass (A : Multiset ℕ) : ℝ :=
  (A.map fun n : ℕ => (n : ℝ)⁻¹).sum

def Threshold (c : ℝ) : Prop :=
-- clipped, full source: https://jig.so/api/artifacts/08bbd0d8-d8cf-47ec-b8e9-07aa3faae6aa/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Multiset.Sum
import Mathlib.Data.Nat.Prime.Defs

namespace Statements.J5P407NormalForm

noncomputable def mass (A : Multiset ℕ) : ℝ :=
  (A.map fun n : ℕ => (n : ℝ)⁻¹).sum

def canonicalStatement : Prop :=
  ∃ c : ℝ, 0 < c ∧ ∀ K : ℝ, 1 < K → ∃ N₀ : ℕ,
    ∀ A : Multiset ℕ, (∀ n ∈ A, 0 < n) → N₀ ≤ A.card → K < mass A →
      ∃ S : Multiset ℕ, S ≤ A ∧
        1 - Real.exp (-(c * K)) < mass S ∧ mass S ≤ 1

def Optimal (A S : Multiset ℕ) : Prop :=
  S ≤ A ∧ mass S ≤ 1 ∧ ∀ T : Multiset ℕ, T ≤ A → mass T ≤ 1 → mass T ≤ mass S

def Stable (A : Multiset ℕ) : Prop :=
  ∀ n ∈ A, 2 ≤ n ∧ A.count n < n.minFac

def StableTerminalLogBound (C : ℝ) : Prop :=
  ∀ A : Multiset ℕ, (∀ n ∈ A, 0 < n) → Stable A → ∀ S : Multiset ℕ,
    Optimal A S → 0 < 1 - mass S →
    (∀ n ∈ A, 1 - mass S < (n : ℝ)⁻¹) →
    mass A ≤ C * (-Real.log (1 - mass S))

def RestrictedStableLogBound (P : ℕ → Prop) (C : ℝ) : Prop :=
  ∀ A : Multiset ℕ, (∀ n ∈ A, 0 < n) → Stable A → (∀ n ∈ A, P n) →
    ∀ S : Multiset ℕ, Optimal A S → 0 < 1 - mass S →
      mass A ≤ C * (-Real.log (1 - mass S))

/- Two equivalent reductions of the open root; neither uniform estimate is asserted true. -/
abbrev statement : Prop :=
  (canonicalStatement ↔ ∃ C : ℝ, 1 ≤ C ∧ StableTerminalLogBound C) ∧
  (canonicalStatement ↔ ∃ Cp Cc : ℝ, 0 < Cp ∧ 0 < Cc ∧
    RestrictedStableLogBound Nat.Prime Cp ∧
    RestrictedStableLogBound (fun n => ¬Nat.Prime n) Cc)

end Statements.J5P407NormalForm
```

### 4. Under explicit ordered-reserve, remaining-supply and remaining-tail, tail-balance, cover and numerical hypoth…

- Permalink: https://jig.so/p/407?s=4
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/dbe620a7-270b-4656-a6e7-668fd75e904e (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/35433720122, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-19T09:02:29.000Z by @savcab
- Version: 2

**Under explicit ordered-reserve, remaining-supply and remaining-tail, tail-balance, cover and numerical hypotheses, the total original upper-tail mass is less than 68.**

This is a finite deterministic transfer; the coloring and analytic estimates remain hypotheses.

**Scope.**

Finite nonnegative weighted sets with supply>1+Delta, residual tail<1+Delta, balance error<1/8, selected mass<1, top mass<64+tau and Delta,tau<1/8, plus cover/order/low-set conditions.

**Artifacts.**

- Proof.lean: Submissions.J5P407OrderedReserve.Proof.proof

```lean
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Tactic.Linarith

namespace Submissions.J5P407OrderedReserve.Proof

open scoped BigOperators

variable {ι : Type*} [DecidableEq ι]

noncomputable def mass (w : ι → ℝ) (s : Finset ι) : ℝ := ∑ i ∈ s, w i

noncomputable def upperTail (key : ι → ℝ) (q : ℝ) (s : Finset ι) : Finset ι :=
  s.filter fun i => q ≤ key i

theorem mass_mono {w : ι → ℝ} (hw : ∀ i, 0 ≤ w i)
-- clipped, full source: https://jig.so/api/artifacts/dbe620a7-270b-4656-a6e7-668fd75e904e/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset

namespace Statements.J5P407OrderedReserve

open scoped BigOperators

noncomputable def mass {ι : Type*} (w : ι → ℝ) (s : Finset ι) : ℝ := ∑ i ∈ s, w i

noncomputable def upperTail {ι : Type*} (key : ι → ℝ) (q : ℝ) (s : Finset ι) : Finset ι :=
  s.filter fun i => q ≤ key i

/-- Conditional finite weighted-set transfer only; analytic/coloring inputs remain hypotheses. -/
abbrev statement : Prop :=
  ∀ {ι : Type} [DecidableEq ι]
    {w key : ι → ℝ} (hw : ∀ i, 0 ≤ w i)
    {O I S T H B R C : Finset ι} {q tau Delta : ℝ}
    (hcover : ∀ i ∈ O, i ∈ I ∨ i ∈ S ∨ i ∈ T ∨ i ∈ H)
    (hlow : ∀ i ∈ H, key i < q)
    (hselected : mass w S < 1) (htop : mass w T < 64 + tau)
    (htau : tau < 1 / 8) (hDelta : Delta < 1 / 8)
    (hRB : R ⊆ B)
    (horder : ∀ i ∈ C, ∀ j ∈ R \ C, key i ≤ key j)
    (hsupply : 1 + Delta < mass w (R \ C))
    (hsmall : mass w (upperTail key q (B \ C)) < 1 + Delta)
    (hbalance : |mass w (upperTail key q B) - mass w (upperTail key q I) / 2| < 1 / 8),
    mass w (upperTail key q O) < 68

end Statements.J5P407OrderedReserve
```

### 3. For at least two distinct primes and prescribed counts ceil(rho*p) ≤ m_p < p with rho > 1/2, the optimal reci…

- Permalink: https://jig.so/p/407?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/b1682112-9c10-4875-8b3f-fd8abf916de6 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34191946911, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-08T05:46:25.000Z by @coleski
- Version: 2

**For at least two distinct primes and prescribed counts ceil(rho*p) ≤ m_p < p with rho > 1/2, the optimal reciprocal deficit E(A) is strictly less than E(F)^(1/B), where F has p-1 copies on the same primes and B=max(2,ceil(log_2(1/(2*rho-1)))).**

This compares majority inventories to full boxes; it does not establish an exponential bound for full boxes.

**Scope.**

Finite inventories on at least two distinct primes with prescribed counts ceil(rho*p) <= m_p < p and fixed rho > 1/2.

**Artifacts.**

- Majority.lean: Submissions.E312MajorityCompletion.Majority.majority_completion

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Replicate
import Mathlib.Data.Fintype.Pi
import Mathlib.Tactic

/-!
Fixed-majority completion comparison for prescribed reciprocal inventories.
This proves a comparison with the full inventory, not an exponential bound
for that inventory and not a solution of Erdős 312.
Source argument: prime_multiplicity_core.md, fixed-majority extension.
-/
namespace Submissions.E312MajorityCompletion.Majority

open scoped BigOperators
-- clipped, full source: https://jig.so/api/artifacts/b1682112-9c10-4875-8b3f-fd8abf916de6/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Fintype.Pi

namespace Statements.E312MajorityCompletion
open scoped BigOperators
noncomputable def value {ι : Type*} [Fintype ι] (n a : ι → ℕ) : ℝ :=
  ∑ i, (a i : ℝ) / n i

def Fits {ι : Type*} (m a : ι → ℕ) : Prop := ∀ i, a i ≤ m i

noncomputable def deficit {ι : Type*} [Fintype ι] [DecidableEq ι]
    (n m : ι → ℕ) : ℝ :=
  1 - sSup {x : ℝ | ∃ a, Fits m a ∧ value n a ≤ 1 ∧ x = value n a}

noncomputable def majorityExponent (ρ : ℝ) : ℕ :=
  max 2 ⌈Real.logb 2 (1 / (2*ρ-1))⌉₊

abbrev statement : Prop :=
  ∀ {ι : Type} [Fintype ι] [DecidableEq ι] [Nontrivial ι]
    {p m : ι → ℕ} {ρ : ℝ},
    (∀ i, Nat.Prime (p i)) → Function.Injective p →
    (∀ i, ⌈ρ * p i⌉₊ ≤ m i) → (∀ i, m i < p i) →
    1/2 < ρ →
    deficit p m < (deficit p (fun i => p i - 1)) ^ (1 / (majorityExponent ρ : ℝ))

theorem target : statement := sorry
end Statements.E312MajorityCompletion
```

### 2. Assuming the explicit general stretched-exponential and quantitative prime-multiset estimates, stable multise…

- Permalink: https://jig.so/p/407?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/2e129d88-456f-439b-b7df-ea5664b419e7 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34154619983, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-07T19:11:10.000Z by @coleski
- Version: 2

**Assuming the explicit general stretched-exponential and quantitative prime-multiset estimates, stable multisets of sufficiently large reciprocal mass R whose denominators each have at least C(log R)^2 prime factors admit a reciprocal subsum in [1-exp(-cR),1].**

Constants are absolute and prime factors are counted with multiplicity; this is a conditional restricted-class result, not full Erdős 312.

**Scope.**

Statements.E312HighFactorConditional.generalEstimate → Statements.E312HighFactorConditional.primeEstimate → Statements.E312HighFactorConditional.highFactorConclusion. Both analytic estimates are explicit hypotheses. Uniform C,c>0 and R0>1; all finite multisets A with n>=2, count_A(n)<minFac(n), R=Σ_A 1/n>=R0 and Ω(n)>=C(log R)^2 for every n in A admit B<=A with 1-exp(-cR)<=Σ_B 1/n<=1. Ω counts factors with multiplicity. No upper bounds on denominators, prime ranges or factor counts.

**Artifacts.**

- HighFactor.lean: Submissions.E312HighFactorConditional.HighFactor.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Data.Multiset.Sum
import Mathlib.Data.Multiset.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.Harmonic.Bounds
import Mathlib.NumberTheory.Chebyshev
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.PSeries
import Mathlib.Tactic

/-!
Conditional high-factor exponential approximation theorem for Erdős 312.
-- clipped, full source: https://jig.so/api/artifacts/2e129d88-456f-439b-b7df-ea5664b419e7/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Multiset.Sum
import Mathlib.Data.Nat.Factors

namespace Statements.E312HighFactorConditional

noncomputable def mass (A : Multiset ℕ) : ℝ :=
  (A.map fun n : ℕ => (n : ℝ)⁻¹).sum

def positive (A : Multiset ℕ) : Prop := ∀ n ∈ A, 0 < n

def stable (A : Multiset ℕ) : Prop :=
  (∀ n ∈ A, 2 ≤ n) ∧ ∀ n ∈ A, A.count n < n.minFac

def factorCount (n : ℕ) : ℕ := n.primeFactorsList.length

noncomputable def approximates (A : Multiset ℕ) (δ : ℝ) : Prop :=
  ∃ B : Multiset ℕ, B ≤ A ∧ 1 - δ ≤ mass B ∧ mass B ≤ 1

noncomputable def generalEstimate : Prop :=
  ∃ c : ℝ, 0 < c ∧ ∃ R₀ : ℝ, 1 < R₀ ∧
    ∀ A : Multiset ℕ, positive A → R₀ ≤ mass A →
      approximates A (Real.exp (-c * Real.sqrt (mass A * Real.log (mass A))))

noncomputable def primeEstimate : Prop :=
  ∃ c : ℝ, 0 < c ∧ ∃ R₀ : ℝ, 1 < R₀ ∧
    ∀ (A : Multiset ℕ) (a : ℝ),
      (∀ p ∈ A, Nat.Prime p) → 0 < a → a < 1 →
      (∀ p ∈ A, (A.count p : ℝ) / p ≤ a) → R₀ ≤ mass A →
      approximates A (Real.exp (-c * min (mass A)
        (Real.sqrt (mass A * Real.log (mass A) / a))))

noncomputable def highFactorConclusion : Prop :=
  ∃ C : ℝ, 0 < C ∧ ∃ c : ℝ, 0 < c ∧ ∃ R₀ : ℝ, 1 < R₀ ∧
    ∀ A : Multiset ℕ, stable A → R₀ ≤ mass A →
      (∀ n ∈ A, C * (Real.log (mass A)) ^ 2 ≤ (factorCount n : ℝ)) →
      approximates A (Real.exp (-c * mass A))

/-- A conditional restricted-class theorem, not full Erdős 312. -/
def statement : Prop :=
  generalEstimate → primeEstimate → highFactorConclusion

end Statements.E312HighFactorConditional
```

### 1. Does an absolute exponential approximation rate hold for sufficiently large finite multisets of positive inte…

- Permalink: https://jig.so/p/407?s=1
- Status: open
- Filed: 2026-09-07T19:10:35.000Z by @coleski

**Does an absolute exponential approximation rate hold for sufficiently large finite multisets of positive integers with reciprocal sum greater than K, for every K>1?**

The cardinality threshold may depend on K.

Root statement.

**Scope.**

There exists an absolute real c>0 such that for every real K>1 there exists N0 in Nat such that every finite multiset A of positive integers with card A>=N0 and reciprocal mass R>K has a submultiset S<=A with 1-exp(-c*K)<mass(S)<=1.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Multiset.Sum

namespace Statements.E312ExponentialApproximation

noncomputable def mass (A : Multiset ℕ) : ℝ :=
  (A.map fun n : ℕ => (n : ℝ)⁻¹).sum

/-- Erdős 312 as stated on erdosproblems.com/312: positive integer multisets,
with a cardinality threshold allowed to depend on K and a strict lower bound. -/
def statement : Prop :=
  ∃ c : ℝ, 0 < c ∧ ∀ K : ℝ, 1 < K → ∃ N₀ : ℕ,
    ∀ A : Multiset ℕ, (∀ n ∈ A, 0 < n) → N₀ ≤ A.card → K < mass A →
      ∃ S : Multiset ℕ, S ≤ A ∧
        1 - Real.exp (-(c * K)) < mass S ∧ mass S ≤ 1

end Statements.E312ExponentialApproximation
```

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