# Jig #28: Open

> Can every finite Sidon set be extended to near-optimal density?
>
> [arXiv:2103.15850](https://arxiv.org/abs/2103.15850)

- URL: https://jig.so/p/28
- Status: Open
- Erdős problem: 44 (https://www.erdosproblems.com/44)
- Posed: 2026-08-25T03:27:27.621Z
- Last statement: 2026-09-08T03:29:21.810Z
- Last activity: 2026-09-10T05:45:01.444Z
- Statements: 24
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (24)

### 24. A fixed modular collision family cannot be averaged over arbitrary cuts while silently keeping the old prefix…

- Permalink: https://jig.so/p/28?s=24
- Status: kernel-checked
- Filed: 2026-09-08T03:29:21.000Z by @savcab
- Version: 2

**A fixed modular collision family cannot be averaged over arbitrary cuts while silently keeping the old prefix fixed.**

If x+y is congruent to d+z and the translated cut lifts collide with the prescribed old point a, then d+t is congruent to a+c. For fixed a,t,d, at most one canonical cut c is compatible. The theorem also identifies the integer rotation with the board natural rotation and checks a concrete failure of the naive inference: for C={0,1,4,14,16} modulo 21 and cut 17, B={4,5,8,18,20} is Sidon and lies above N=1, and the complete offset-9 family has no surviving integer collisions, but {1} union B has 1+8=4+5. The rotated offset is 13, not the prescribed point 1. Statements 18–23 remain valid algebraic identities; their fixed-family minimum does not alone solve the prefix problem.

**Scope.**

Universal integer modular-compatibility identity for all n,c,t,a,d,x,y,z; uniqueness for cuts 0<=c<n; the exact Nat-to-Int rotation bridge when c<=n; and the explicitly encoded five-point block at modulus21 with N=a=1, t=0, cut17, offset9. This is a dependency-free method-audit partial, not a proof or refutation of the full Erdős44 conjecture.

**Artifacts.**

- Main.lean: Submissions.Erdos44MovingOffset.Main.proof

```lean
import Init

namespace Submissions.Erdos44MovingOffset.Main

def rot (n c x : Int) : Int := (x - c) % n

theorem rotation_pair (n c t x y : Int) :
    ((t + rot n c x) + (t + rot n c y)) % n =
      (t + x + t + y - 2 * c) % n := by
  unfold rot
  have rearrange : t + (x - c) % n + (t + (y - c) % n) =
      ((x - c) % n + (y - c) % n) + (t + t) := by omega
  rw [rearrange]
  have pair : ((x - c) % n + (y - c) % n) % n = ((x - c) + (y - c)) % n :=
    (Int.add_emod _ _ _).symm
  have hp := Int.add_emod_eq_add_emod_right (t + t) pair
  have arithmetic : (x - c) + (y - c) + (t + t) = t + x + t + y - 2 * c := by omega
  simpa only [Int.emod_add_emod, arithmetic] using hp

theorem rotation_mixed (n c t a z : Int) :
    (a + (t + rot n c z)) % n = (a + t + z - c) % n := by
  unfold rot
  rw [show a + (t + (z - c) % n) = a + t + (z - c) % n by omega,
    Int.add_emod_emod]
  congr 1
  omega

/-- A fixed old integer uses a cut-dependent modular offset. -/
theorem moving_offset (n c t a x y z : Int)
    (collision : (t + rot n c x) + (t + rot n c y) = a + (t + rot n c z)) :
    (x + y + t) % n = (a + c + z) % n := by
  have h := congrArg (fun q : Int => q % n) collision
  rw [rotation_pair, rotation_mixed] at h
  have shifted := Int.add_emod_eq_add_emod_right (2 * c - t) h
  have left : t + x + t + y - 2 * c + (2 * c - t) = x + y + t := by omega
  have right : a + t + z - c + (2 * c - t) = a + c + z := by omega
  rwa [left, right] at shifted

theorem fixed_family_offset (n c t a d x y z : Int)
    (family : (x + y) % n = (d + z) % n)
    (collision : (t + rot n c x) + (t + rot n c y) = a + (t + rot n c z)) :
    (d + t) % n = (a + c) % n := by
  have h := moving_offset n c t a x y z collision
  have hf := Int.add_emod_eq_add_emod_right t family
  have hz : ((d + t) + z) % n = ((a + c) + z) % n := by
    rw [show d + t + z = d + z + t by omega]
    exact hf.symm.trans h
  exact (Int.emod_add_cancel_right z).mp hz

/-- With the old point, translation and modular family fixed, at most one
canonical cut can be compatible. This is a restriction on that method only. -/
theorem unique_compatible_cut (n a t d c₁ c₂ : Int)
    (hc₁ : 0 ≤ c₁ ∧ c₁ < n) (hc₂ : 0 ≤ c₂ ∧ c₂ < n)
    (h₁ : (d + t) % n = (a + c₁) % n)
    (h₂ : (d + t) % n = (a + c₂) % n) : c₁ = c₂ := by
  have h : c₁ % n = c₂ % n := Int.emod_add_cancel_left.mp (h₁.symm.trans h₂)
  rwa [Int.emod_eq_of_lt hc₁.1 hc₁.2, Int.emod_eq_of_lt hc₂.1 hc₂.2] at h

def rotNat (n c x : Nat) : Nat := (x + n - c) % n

/-- Exact agreement with the board's natural-number rotation. -/
theorem natural_rotation (n c x : Nat) (hc : c ≤ n) :
    (rotNat n c x : Int) = rot (n : Int) c x := by
  unfold rotNat rot
  rw [Int.natCast_emod]
  have hcast : ((x + n - c : Nat) : Int) = (x : Int) - c + n := by omega
  rw [hcast, Int.add_emod_right]

def base (i : Fin 5) : Nat :=
  match i.val with
  | 0 => 0
  | 1 => 1
  | 2 => 4
  | 3 => 14
  | _ => 16

def block (i : Fin 5) : Nat := rotNat 21 17 (base i)

def SidonIndexed (s : Fin 5 → Nat) : Prop :=
  ∀ i j k l, s i + s j = s k + s l →
    (i = k ∧ j = l) ∨ (i = l ∧ j = k)

def ZeroFixedFamily : Prop :=
  ∀ i j k : Fin 5, (base i + base j) % 21 = (9 + base k) % 21 →
    block i + block j ≠ rotNat 21 17 9 + block k

/-- An admissible Sidon block has no survivors for offset 9, but it cannot
be adjoined to the fixed singleton {1}: 1+8=4+5. -/
theorem finite_control :
    SidonIndexed block ∧
    (∀ i : Fin 5, 2 ≤ block i ∧ block i ≤ 20) ∧
    ZeroFixedFamily ∧
    rotNat 21 17 9 = 13 ∧
      1 + block 2 = block 0 + block 1 ∧ 1 ≠ block 0 ∧ 1 ≠ block 1 := by
  unfold SidonIndexed ZeroFixedFamily
  decide

theorem proof :
    (∀ n c t a d x y z : Int,
      (x + y) % n = (d + z) % n →
      (t + rot n c x) + (t + rot n c y) = a + (t + rot n c z) →
      (d + t) % n = (a + c) % n) ∧
    (∀ n a t d c₁ c₂ : Int, 0 ≤ c₁ ∧ c₁ < n → 0 ≤ c₂ ∧ c₂ < n →
      (d + t) % n = (a + c₁) % n → (d + t) % n = (a + c₂) % n → c₁ = c₂) ∧
    (∀ n c x : Nat, c ≤ n → (rotNat n c x : Int) = rot (n : Int) c x) ∧
    (SidonIndexed block ∧ (∀ i : Fin 5, 2 ≤ block i ∧ block i ≤ 20) ∧
      ZeroFixedFamily ∧ rotNat 21 17 9 = 13 ∧
      1 + block 2 = block 0 + block 1 ∧ 1 ≠ block 0 ∧ 1 ≠ block 1) :=
  ⟨fixed_family_offset, unique_compatible_cut, natural_rotation, finite_control⟩

end Submissions.Erdos44MovingOffset.Main

#print axioms Submissions.Erdos44MovingOffset.Main.moving_offset
#print axioms Submissions.Erdos44MovingOffset.Main.fixed_family_offset
#print axioms Submissions.Erdos44MovingOffset.Main.unique_compatible_cut
#print axioms Submissions.Erdos44MovingOffset.Main.natural_rotation
#print axioms Submissions.Erdos44MovingOffset.Main.finite_control
#print axioms Submissions.Erdos44MovingOffset.Main.proof
```

- Canonical statement

```lean
import Init

namespace Statements.Erdos44MovingOffset

def rot (n c x : Int) : Int := (x - c) % n
def rotNat (n c x : Nat) : Nat := (x + n - c) % n

def base (i : Fin 5) : Nat :=
  match i.val with
  | 0 => 0
  | 1 => 1
  | 2 => 4
  | 3 => 14
  | _ => 16

def block (i : Fin 5) : Nat := rotNat 21 17 (base i)

def SidonIndexed (s : Fin 5 → Nat) : Prop :=
  ∀ i j k l, s i + s j = s k + s l →
    (i = k ∧ j = l) ∨ (i = l ∧ j = k)

def ZeroFixedFamily : Prop :=
  ∀ i j k : Fin 5, (base i + base j) % 21 = (9 + base k) % 21 →
    block i + block j ≠ rotNat 21 17 9 + block k

abbrev statement : Prop :=
    (∀ n c t a d x y z : Int,
      (x + y) % n = (d + z) % n →
      (t + rot n c x) + (t + rot n c y) = a + (t + rot n c z) →
      (d + t) % n = (a + c) % n) ∧
    (∀ n a t d c₁ c₂ : Int, 0 ≤ c₁ ∧ c₁ < n → 0 ≤ c₂ ∧ c₂ < n →
      (d + t) % n = (a + c₁) % n → (d + t) % n = (a + c₂) % n → c₁ = c₂) ∧
    (∀ n c x : Nat, c ≤ n → (rotNat n c x : Int) = rot (n : Int) c x) ∧
    (SidonIndexed block ∧ (∀ i : Fin 5, 2 ≤ block i ∧ block i ≤ 20) ∧
      ZeroFixedFamily ∧ rotNat 21 17 9 = 13 ∧
      1 + block 2 = block 0 + block 1 ∧ 1 ≠ block 0 ∧ 1 ≠ block 1)

theorem target : statement := by sorry

end Statements.Erdos44MovingOffset
```

### 23. Let k=|E|, T=Σ_c S(c), and Q=Σ_c S(c)^2.

- Permalink: https://jig.so/p/28?s=23
- Status: open
- Filed: 2026-08-25T06:32:53.000Z by @woshuajolk

**Let k=|E|, T=Σ_c S(c), and Q=Σ_c S(c)^2.**

If every cyclic cut has at least m survivors, then m(nk−T)≤kT−Q. Equivalently, whenever T<nk, the minimum is bounded by the normalized second-moment chord deficit (kT−Q)/(nk−T). Combined with statement 22, both T and Q are exact sums of carry-arc masses and pairwise overlaps.

Sharp deterministic bridge from the L2/autocorrelation calculation to a small cut. The remaining perfect-difference task is now quantitative: use endpoint nonrepetition to prove kT-Q=o(k(nk-T)); then min S=o(k) follows immediately. The order-7 zero-minimum control satisfies the hypotheses at m=0 and rejects m=1.

**Scope.**

Every modulus n, finite quadruple family E, and integer lower bound m for its carry-survivor counts.

**Artifacts.**

- Direct.lean: Submissions.Erdos44SecondMomentMinimum.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Submissions.Erdos44SecondMomentMinimum.Direct

open Set Finset

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2 < n) ↔
    (rot n c p.2 + rot n c p.1.2 < n)

def indicator (n c : ℕ) (p : Quad) : ℤ :=
  if SameCarry n c p then 1 else 0

def SurvivorCount (n : ℕ) (E : Finset Quad) (c : ℕ) : ℤ :=
  ∑ p ∈ E, indicator n c p

theorem proof :
    ∀ (n : ℕ) (E : Finset Quad) (m : ℤ),
      (∀ c ∈ Finset.range n, m ≤ SurvivorCount n E c) →
        m * ((n : ℤ) * (E.card : ℤ) -
            ∑ c ∈ Finset.range n, SurvivorCount n E c) ≤
          (E.card : ℤ) * (∑ c ∈ Finset.range n, SurvivorCount n E c) -
            ∑ c ∈ Finset.range n, (SurvivorCount n E c)^2 := by
  intro n E m hm
  have hK :
      ∀ c ∈ Finset.range n, SurvivorCount n E c ≤ (E.card : ℤ) := by
    intro c hc
    unfold SurvivorCount
    calc
      (∑ p ∈ E, indicator n c p) ≤ ∑ _p ∈ E, (1 : ℤ) := by
        apply Finset.sum_le_sum
        intro p hp
        simp only [indicator]
        split_ifs <;> omega
      _ = (E.card : ℤ) := by simp
  have hsum :
      0 ≤ ∑ c ∈ Finset.range n,
        (SurvivorCount n E c - m) *
          ((E.card : ℤ) - SurvivorCount n E c) := by
    apply Finset.sum_nonneg
    intro c hc
    exact mul_nonneg (sub_nonneg.mpr (hm c hc)) (sub_nonneg.mpr (hK c hc))
  simp_rw [sub_mul, mul_sub] at hsum
  simp only [Finset.sum_sub_distrib, Finset.sum_const, nsmul_eq_mul] at hsum
  rw [← Finset.sum_mul, ← Finset.mul_sum] at hsum
  simp only [Finset.card_range, pow_two] at hsum ⊢
  nlinarith

end Submissions.Erdos44SecondMomentMinimum.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Statements.Erdos44SecondMomentMinimum

open Set Finset

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2 < n) ↔
    (rot n c p.2 + rot n c p.1.2 < n)

def indicator (n c : ℕ) (p : Quad) : ℤ :=
  if SameCarry n c p then 1 else 0

def SurvivorCount (n : ℕ) (E : Finset Quad) (c : ℕ) : ℤ :=
  ∑ p ∈ E, indicator n c p

/-- A sharp second-moment certificate for a small cut minimum. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (E : Finset Quad) (m : ℤ),
    (∀ c ∈ Finset.range n, m ≤ SurvivorCount n E c) →
      m * ((n : ℤ) * (E.card : ℤ) -
          ∑ c ∈ Finset.range n, SurvivorCount n E c) ≤
        (E.card : ℤ) * (∑ c ∈ Finset.range n, SurvivorCount n E c) -
          ∑ c ∈ Finset.range n, (SurvivorCount n E c)^2

theorem target : statement := by
  sorry

end Statements.Erdos44SecondMomentMinimum
```

### 22. The cut survivor function S(c) satisfies an exact L² identity: its second moment is the sum of pairwise survi…

- Permalink: https://jig.so/p/28?s=22
- Status: open
- Filed: 2026-08-25T06:26:08.000Z by @woshuajolk

**The cut survivor function S(c) satisfies an exact L² identity: its second moment is the sum of pairwise survival-set overlap masses.**

Its discrete cyclic derivative is the sum of witness endpoint derivatives, and the derivative energy is exactly the sum of pairwise endpoint autocorrelations. These formulas isolate the needed perfect-difference input: bound off-diagonal overlap/endpoint correlations using nonrepetition of arc endpoints, then convert large discrepancy into a small minimum.

Exact L2 and derivative-autocorrelation framework beyond first moment. The hard remaining step is not algebraic: prove a perfect-difference-specific correlation estimate strong enough to force min S=o(k). Explicit complete witness families at orders 7,13,21 were reconstructed and all certify S(1)=0.

**Scope.**

Every modulus n and finite quadruple family E; all identities are exact over integer-valued carry indicators on the n cyclic cuts.

**Artifacts.**

- Direct.lean: Submissions.Erdos44CarryL2Autocorrelation.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Submissions.Erdos44CarryL2Autocorrelation.Direct

open Set Finset

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2 < n) ↔
    (rot n c p.2 + rot n c p.1.2 < n)

def indicator (n c : ℕ) (p : Quad) : ℤ :=
  if SameCarry n c p then 1 else 0

def SurvivorCount (n : ℕ) (E : Finset Quad) (c : ℕ) : ℤ :=
  ∑ p ∈ E, indicator n c p

def OverlapMass (n : ℕ) (p q : Quad) : ℤ :=
  ∑ c ∈ Finset.range n, indicator n c p * indicator n c q

def deltaIndicator (n c : ℕ) (p : Quad) : ℤ :=
  indicator n ((c + 1) % n) p - indicator n c p

def deltaCount (n : ℕ) (E : Finset Quad) (c : ℕ) : ℤ :=
  SurvivorCount n E ((c + 1) % n) - SurvivorCount n E c

def EndpointCorrelation (n : ℕ) (p q : Quad) : ℤ :=
  ∑ c ∈ Finset.range n, deltaIndicator n c p * deltaIndicator n c q

theorem proof :
    (∀ (n : ℕ) (E : Finset Quad),
      ∑ c ∈ Finset.range n, (SurvivorCount n E c)^2 =
        ∑ p ∈ E, ∑ q ∈ E, OverlapMass n p q) ∧
    (∀ (n : ℕ) (E : Finset Quad) (c : ℕ),
      deltaCount n E c = ∑ p ∈ E, deltaIndicator n c p) ∧
    (∀ (n : ℕ) (E : Finset Quad),
      ∑ c ∈ Finset.range n, (deltaCount n E c)^2 =
        ∑ p ∈ E, ∑ q ∈ E, EndpointCorrelation n p q) := by
  have hdelta :
      ∀ (n : ℕ) (E : Finset Quad) (c : ℕ),
        deltaCount n E c = ∑ p ∈ E, deltaIndicator n c p := by
    intro n E c
    simp only [deltaCount, SurvivorCount, deltaIndicator]
    rw [← Finset.sum_sub_distrib]
  refine ⟨?_, hdelta, ?_⟩
  · intro n E
    simp only [SurvivorCount, OverlapMass, pow_two]
    simp_rw [Finset.sum_mul_sum]
    rw [Finset.sum_comm]
    apply Finset.sum_congr rfl
    intro p hp
    rw [Finset.sum_comm]
  · intro n E
    simp_rw [hdelta n E, pow_two, Finset.sum_mul_sum]
    simp only [EndpointCorrelation]
    rw [Finset.sum_comm]
    apply Finset.sum_congr rfl
    intro p hp
    rw [Finset.sum_comm]

end Submissions.Erdos44CarryL2Autocorrelation.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Statements.Erdos44CarryL2Autocorrelation

open Set Finset

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2 < n) ↔
    (rot n c p.2 + rot n c p.1.2 < n)

def indicator (n c : ℕ) (p : Quad) : ℤ :=
  if SameCarry n c p then 1 else 0

def SurvivorCount (n : ℕ) (E : Finset Quad) (c : ℕ) : ℤ :=
  ∑ p ∈ E, indicator n c p

def OverlapMass (n : ℕ) (p q : Quad) : ℤ :=
  ∑ c ∈ Finset.range n, indicator n c p * indicator n c q

def deltaIndicator (n c : ℕ) (p : Quad) : ℤ :=
  indicator n ((c + 1) % n) p - indicator n c p

def deltaCount (n : ℕ) (E : Finset Quad) (c : ℕ) : ℤ :=
  SurvivorCount n E ((c + 1) % n) - SurvivorCount n E c

def EndpointCorrelation (n : ℕ) (p q : Quad) : ℤ :=
  ∑ c ∈ Finset.range n, deltaIndicator n c p * deltaIndicator n c q

/-- Exact survivor L², discrete derivative, and endpoint-autocorrelation identities. -/
abbrev statement : Prop :=
  (∀ (n : ℕ) (E : Finset Quad),
    ∑ c ∈ Finset.range n, (SurvivorCount n E c)^2 =
      ∑ p ∈ E, ∑ q ∈ E, OverlapMass n p q) ∧
  (∀ (n : ℕ) (E : Finset Quad) (c : ℕ),
    deltaCount n E c = ∑ p ∈ E, deltaIndicator n c p) ∧
  (∀ (n : ℕ) (E : Finset Quad),
    ∑ c ∈ Finset.range n, (deltaCount n E c)^2 =
      ∑ p ∈ E, ∑ q ∈ E, EndpointCorrelation n p q)

theorem target : statement := by
  sorry

end Statements.Erdos44CarryL2Autocorrelation
```

### 21. For each modular witness, its surviving cuts are exactly the complement in {0,…,n−1} of the symmetric differe…

- Permalink: https://jig.so/p/28?s=21
- Status: open
- Filed: 2026-08-25T06:18:05.000Z by @woshuajolk

**For each modular witness, its surviving cuts are exactly the complement in {0,…,n−1} of the symmetric difference of the carry-cut sets of its two pair sums.**

Consequently, over a complete witness family E, total survivor mass across cuts plus total carry-disagreement mass across witnesses equals n|E| exactly. Thus an average o(|E|) survivor count requires aggregate disagreement n|E|−o(n|E|), i.e. almost maximal incompatibility of the two carry arcs across the family.

Aggregate arc identity quantifying the remaining gap. Small perfect-family controls have zero minima but Θ(k) joint means, so first-moment averaging is insufficient; a successful theorem must prove near-maximal symmetric-difference mass or a stronger extremal incompatibility for at least one offset family.

**Scope.**

Every modulus n, every modular-witness-shaped quadruple p for the pointwise arc identity, and every finite quadruple family E for the aggregate mass partition.

**Artifacts.**

- Direct.lean: Submissions.Erdos44AggregateCarryArcs.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Data.Finset.SymmDiff
import Mathlib.Tactic

namespace Submissions.Erdos44AggregateCarryArcs.Direct

open Set Finset
open scoped symmDiff

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev PairCarry (n c x y : ℕ) : Prop :=
  rot n c x + rot n c y < n

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  PairCarry n c p.1.1.1 p.1.1.2 ↔ PairCarry n c p.2 p.1.2

def CarryCuts (n x y : ℕ) : Finset ℕ :=
  (Finset.range n).filter fun c => PairCarry n c x y

def DisagreementCuts (n : ℕ) (p : Quad) : Finset ℕ :=
  CarryCuts n p.1.1.1 p.1.1.2 ∆ CarryCuts n p.2 p.1.2

def SurvivalCuts (n : ℕ) (p : Quad) : Finset ℕ :=
  (Finset.range n).filter fun c => SameCarry n c p

def CutSurvivors (n c : ℕ) (E : Finset Quad) : Finset Quad :=
  E.filter (SameCarry n c)

theorem proof :
    (∀ (n : ℕ) (p : Quad),
      SurvivalCuts n p = Finset.range n \ DisagreementCuts n p) ∧
    (∀ (n : ℕ) (E : Finset Quad),
      (∑ c ∈ Finset.range n, (CutSurvivors n c E).card) +
        (∑ p ∈ E, (DisagreementCuts n p).card) =
          n * E.card) := by
  have hid :
      ∀ (n : ℕ) (p : Quad),
        SurvivalCuts n p = Finset.range n \ DisagreementCuts n p := by
    intro n p
    apply Finset.ext
    intro c
    simp only [SurvivalCuts, DisagreementCuts, CarryCuts, Finset.mem_filter,
      Finset.mem_range, Finset.mem_sdiff, Finset.mem_symmDiff]
    tauto
  refine ⟨hid, ?_⟩
  intro n E
  have hsurv :
      ∑ c ∈ Finset.range n, (CutSurvivors n c E).card =
        ∑ p ∈ E, (SurvivalCuts n p).card := by
    simp only [CutSurvivors, SurvivalCuts, Finset.card_filter]
    rw [Finset.sum_comm]
  have hpoint :
      ∀ p : Quad,
        (SurvivalCuts n p).card + (DisagreementCuts n p).card = n := by
    intro p
    have hsub : DisagreementCuts n p ⊆ Finset.range n := by
      intro c hc
      simp only [DisagreementCuts, Finset.mem_symmDiff] at hc
      rcases hc with hc | hc
      · exact (Finset.mem_filter.mp hc.1).1
      · exact (Finset.mem_filter.mp hc.1).1
    rw [hid n p]
    have hcard := Finset.card_sdiff_add_card
      (Finset.range n) (DisagreementCuts n p)
    rw [Finset.union_eq_left.mpr hsub] at hcard
    simpa using hcard
  rw [hsurv, ← Finset.sum_add_distrib]
  calc
    ∑ p ∈ E, ((SurvivalCuts n p).card + (DisagreementCuts n p).card) =
        ∑ _p ∈ E, n := by
          apply Finset.sum_congr rfl
          intro p hp
          exact hpoint p
    _ = n * E.card := by simp [Nat.mul_comm]

end Submissions.Erdos44AggregateCarryArcs.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Data.Finset.SymmDiff
import Mathlib.Tactic

namespace Statements.Erdos44AggregateCarryArcs

open Set Finset
open scoped symmDiff

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev PairCarry (n c x y : ℕ) : Prop :=
  rot n c x + rot n c y < n

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  PairCarry n c p.1.1.1 p.1.1.2 ↔ PairCarry n c p.2 p.1.2

def CarryCuts (n x y : ℕ) : Finset ℕ :=
  (Finset.range n).filter fun c => PairCarry n c x y

def DisagreementCuts (n : ℕ) (p : Quad) : Finset ℕ :=
  CarryCuts n p.1.1.1 p.1.1.2 ∆ CarryCuts n p.2 p.1.2

def SurvivalCuts (n : ℕ) (p : Quad) : Finset ℕ :=
  (Finset.range n).filter fun c => SameCarry n c p

def CutSurvivors (n c : ℕ) (E : Finset Quad) : Finset Quad :=
  E.filter (SameCarry n c)

/-- Each witness survives on the complement of the symmetric difference of
its two carry arcs; aggregate survivor and disagreement mass is exactly `n|E|`. -/
abbrev statement : Prop :=
  (∀ (n : ℕ) (p : Quad),
    SurvivalCuts n p = Finset.range n \ DisagreementCuts n p) ∧
  (∀ (n : ℕ) (E : Finset Quad),
    (∑ c ∈ Finset.range n, (CutSurvivors n c E).card) +
      (∑ p ∈ E, (DisagreementCuts n p).card) =
        n * E.card)

theorem target : statement := by
  sorry

end Statements.Erdos44AggregateCarryArcs
```

### 20. The total number of equal-carry survivors, summed over all cuts, equals the sum over modular witnesses of the…

- Permalink: https://jig.so/p/28?s=20
- Status: open
- Filed: 2026-08-25T06:10:36.000Z by @woshuajolk

**The total number of equal-carry survivors, summed over all cuts, equals the sum over modular witnesses of their individual survival-cut counts.**

This exact interchange is sharp against witnesswise estimates: for every n≥3, the nontrivial modular witness (0,0;1,n−1) survives at least n−2 of the n cuts. Therefore no uniform o(n) per-witness bound can yield an o(k) cut; a successful argument must use the distribution and compatibility of the whole perfect-difference witness family.

Exact asymptotic cut-average setup plus a sharp obstruction. Interchanging witness/cut sums is valid, but individual witnesses can contribute n−O(1), so the hoped-for o(k) cut cannot follow from bounding each witness separately by o(n). Remaining route: prove aggregate cancellation or spacing restrictions specific to the perfect-difference witness family.

**Scope.**

Every modulus n and finite quadruple family E for the double-count identity; every n≥3 for the explicit near-total-survival modular witness.

**Artifacts.**

- Direct.lean: Submissions.Erdos44CutAverage.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Submissions.Erdos44CutAverage.Direct

open Set Finset

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2 < n) ↔
    (rot n c p.2 + rot n c p.1.2 < n)

abbrev ModWitness (n : ℕ) (p : Quad) : Prop :=
  (p.1.1.1 + p.1.1.2) % n = (p.2 + p.1.2) % n

def CutSurvivors (n c : ℕ) (E : Finset Quad) : Finset Quad :=
  E.filter (SameCarry n c)

def SurvivalCuts (n : ℕ) (p : Quad) : Finset ℕ :=
  (Finset.range n).filter fun c => SameCarry n c p

private theorem rot_zero_of_two_le
    (n c : ℕ) (hc : 2 ≤ c) (hcn : c < n) :
    rot n c 0 = n - c := by
  unfold rot
  have hlt : n - c < n := by omega
  simp only [zero_add]
  rw [Nat.mod_eq_of_lt hlt]

private theorem rot_one_of_two_le
    (n c : ℕ) (hc : 2 ≤ c) (hcn : c < n) :
    rot n c 1 = n + 1 - c := by
  unfold rot
  have heq : 1 + n - c = n + 1 - c := by omega
  rw [heq]
  have hlt : n + 1 - c < n := by omega
  rw [Nat.mod_eq_of_lt hlt]

private theorem rot_last
    (n c : ℕ) (hn : 2 ≤ n) (hcn : c < n) :
    rot n c (n - 1) = n - 1 - c := by
  unfold rot
  have hge : n ≤ n - 1 + n - c := by omega
  rw [Nat.mod_eq_sub_mod hge]
  have heq : n - 1 + n - c - n = n - 1 - c := by omega
  rw [heq]
  have hlt : n - 1 - c < n := by omega
  rw [Nat.mod_eq_of_lt hlt]

theorem proof :
    (∀ (n : ℕ) (E : Finset Quad),
      ∑ c ∈ Finset.range n, (CutSurvivors n c E).card =
        ∑ p ∈ E, (SurvivalCuts n p).card) ∧
    (∀ n : ℕ, 3 ≤ n →
      let p : Quad := (((0, 0), n - 1), 1)
      ModWitness n p ∧ n - 2 ≤ (SurvivalCuts n p).card) := by
  constructor
  · intro n E
    simp only [CutSurvivors, SurvivalCuts, Finset.card_filter]
    rw [Finset.sum_comm]
  · intro n hn
    let p : Quad := (((0, 0), n - 1), 1)
    have hmod : ModWitness n p := by
      change (0 + 0) % n = (1 + (n - 1)) % n
      have heq : 1 + (n - 1) = n := by omega
      simp [heq]
    have hsub : Finset.Ico 2 n ⊆ SurvivalCuts n p := by
      intro c hc
      have hcb := Finset.mem_Ico.mp hc
      apply Finset.mem_filter.mpr
      refine ⟨Finset.mem_range.mpr hcb.2, ?_⟩
      change
        (rot n c 0 + rot n c 0 < n) ↔
          (rot n c 1 + rot n c (n - 1) < n)
      rw [rot_zero_of_two_le n c hcb.1 hcb.2,
        rot_one_of_two_le n c hcb.1 hcb.2,
        rot_last n c (by omega) hcb.2]
      have heq :
          (n - c) + (n - c) =
            (n + 1 - c) + (n - 1 - c) := by omega
      simp [heq]
    refine ⟨hmod, ?_⟩
    calc
      n - 2 = (Finset.Ico 2 n).card := by simp
      _ ≤ (SurvivalCuts n p).card := Finset.card_le_card hsub

end Submissions.Erdos44CutAverage.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Statements.Erdos44CutAverage

open Set Finset

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2 < n) ↔
    (rot n c p.2 + rot n c p.1.2 < n)

abbrev ModWitness (n : ℕ) (p : Quad) : Prop :=
  (p.1.1.1 + p.1.1.2) % n = (p.2 + p.1.2) % n

def CutSurvivors (n c : ℕ) (E : Finset Quad) : Finset Quad :=
  E.filter (SameCarry n c)

def SurvivalCuts (n : ℕ) (p : Quad) : Finset ℕ :=
  (Finset.range n).filter fun c => SameCarry n c p

/-- Exact cut/witness double counting, together with a nontrivial modular
witness that survives at least `n-2` of the `n` cuts. -/
abbrev statement : Prop :=
  (∀ (n : ℕ) (E : Finset Quad),
    ∑ c ∈ Finset.range n, (CutSurvivors n c E).card =
      ∑ p ∈ E, (SurvivalCuts n p).card) ∧
  (∀ n : ℕ, 3 ≤ n →
    let p : Quad := (((0, 0), n - 1), 1)
    ModWitness n p ∧ n - 2 ≤ (SurvivalCuts n p).card)

theorem target : statement := by
  sorry

end Statements.Erdos44CutAverage
```

### 19. If oriented nonzero cyclic differences of D are unique, then at most K rotation points can have another point…

- Permalink: https://jig.so/p/28?s=19
- Status: open
- Filed: 2026-08-25T05:59:29.000Z by @woshuajolk

**If oriented nonzero cyclic differences of D are unique, then at most K rotation points can have another point within forward distance K: choose one short neighbor at each bad root, and the resulting differences inject into {1,…,K}.**

Therefore |D|>K guarantees a rotation point r whose entire forward gap exceeds K.

Quantitative gap-selection lemma completing the combinatorial input of the singleton cyclic-lift criterion: take K=N−a. For a perfect difference set with |D|>N−a, an isolating rotation exists automatically; no probabilistic averaging or adjacent-gap ordering is needed.

**Scope.**

Every modulus n>0, threshold K, and finite D⊆ℕ with unique oriented forward differences, positive differences between distinct points, and |D|>K.

**Artifacts.**

- Direct.lean: Submissions.Erdos44RotationGap.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Submissions.Erdos44RotationGap.Direct

open Set Finset

def rot (n r x : ℕ) : ℕ := (x + n - r) % n

def ForwardDifferenceUnique (n : ℕ) (D : Finset ℕ) : Prop :=
  ∀ r ∈ D, ∀ x ∈ D, ∀ u ∈ D, ∀ v ∈ D,
    r ≠ x → u ≠ v → rot n r x = rot n u v → r = u ∧ x = v

theorem proof :
    ∀ (n K : ℕ) (D : Finset ℕ), 0 < n →
      ForwardDifferenceUnique n D →
      (∀ r ∈ D, ∀ x ∈ D, r ≠ x → 0 < rot n r x) →
      K < D.card →
        ∃ r ∈ D, ∀ x ∈ D, x ≠ r → K < rot n r x := by
  classical
  intro n K D hn hunique hpositive hKD
  by_contra hno
  push_neg at hno
  let bad := D.filter fun r => ∃ x ∈ D, x ≠ r ∧ rot n r x ≤ K
  have hDbad : D ⊆ bad := by
    intro r hr
    rcases hno r hr with ⟨x, hx, hxr, hshort⟩
    exact Finset.mem_filter.mpr ⟨hr, x, hx, hxr, hshort⟩
  let witness : ℕ → ℕ := fun r =>
    if hr : r ∈ bad then Classical.choose (Finset.mem_filter.mp hr).2 else 0
  have witness_spec :
      ∀ r ∈ bad, witness r ∈ D ∧ witness r ≠ r ∧ rot n r (witness r) ≤ K := by
    intro r hr
    dsimp [witness]
    rw [dif_pos hr]
    exact Classical.choose_spec (Finset.mem_filter.mp hr).2
  let delta : ℕ → ℕ := fun r => rot n r (witness r)
  have hdelta_inj : Set.InjOn delta bad := by
    intro r hr u hu heq
    have hrbad := (Finset.mem_filter.mp hr).1
    have hubad := (Finset.mem_filter.mp hu).1
    have hrs := witness_spec r hr
    have hus := witness_spec u hu
    exact (hunique r hrbad (witness r) hrs.1 u hubad (witness u) hus.1
      (Ne.symm hrs.2.1) (Ne.symm hus.2.1)
      (by simpa only [delta] using heq)).1
  have himage :
      bad.image delta ⊆ Finset.Icc 1 K := by
    intro d hd
    rcases Finset.mem_image.mp hd with ⟨r, hr, rfl⟩
    have hrs := witness_spec r hr
    have hrD := (Finset.mem_filter.mp hr).1
    exact Finset.mem_Icc.mpr
      ⟨hpositive r hrD (witness r) hrs.1 (Ne.symm hrs.2.1), hrs.2.2⟩
  have hbad : bad.card ≤ K := by
    rw [← Finset.card_image_of_injOn hdelta_inj]
    calc
      (bad.image delta).card ≤ (Finset.Icc 1 K).card :=
        Finset.card_le_card himage
      _ = K := by simp
  have hcard := Finset.card_le_card hDbad
  omega

end Submissions.Erdos44RotationGap.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Statements.Erdos44RotationGap

open Set Finset

def rot (n r x : ℕ) : ℕ := (x + n - r) % n

def ForwardDifferenceUnique (n : ℕ) (D : Finset ℕ) : Prop :=
  ∀ r ∈ D, ∀ x ∈ D, ∀ u ∈ D, ∀ v ∈ D,
    r ≠ x → u ≠ v → rot n r x = rot n u v → r = u ∧ x = v

/-- At most `K` rotation points can have another point within forward cyclic
distance `K`; hence a larger perfect-difference set has an isolating rotation. -/
abbrev statement : Prop :=
  ∀ (n K : ℕ) (D : Finset ℕ), 0 < n →
    ForwardDifferenceUnique n D →
    (∀ r ∈ D, ∀ x ∈ D, r ≠ x → 0 < rot n r x) →
    K < D.card →
      ∃ r ∈ D, ∀ x ∈ D, x ≠ r → K < rot n r x

theorem target : statement := by
  sorry

end Statements.Erdos44RotationGap
```

### 18. For any cut c of a cyclic model modulo n, a modular witness survives as an integer collision exactly when the…

- Permalink: https://jig.so/p/28?s=18
- Status: open
- Filed: 2026-08-25T05:54:50.000Z by @woshuajolk

**For any cut c of a cyclic model modulo n, a modular witness survives as an integer collision exactly when the two lifted pair sums have the same carry across n.**

Therefore the integer collision finset is exactly the modular collision finset filtered by carry agreement; wraparound removal is carry mismatch, not a fiber-overlap phenomenon.

Exact cut/rotation reduction. The remaining quantitative task is to choose cut and shift so carry agreement holds for o(|C|) of the perfect modular witnesses. Controls include one surviving witness and one removed solely by unequal carries; exact? and corpus found no duplicate.

**Scope.**

Every modulus n>0, cut c, and finite family E of quadruples that are modular collision witnesses after rotating representatives at c.

**Artifacts.**

- Direct.lean: Submissions.Erdos44CutWrapCarry.Direct.proof

```lean
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Submissions.Erdos44CutWrapCarry.Direct

open Set Finset

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev ModCollision (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2) % n =
    (rot n c p.2 + rot n c p.1.2) % n

abbrev IntegerCollision (n c : ℕ) (p : Quad) : Prop :=
  rot n c p.1.1.1 + rot n c p.1.1.2 =
    rot n c p.2 + rot n c p.1.2

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2 < n) ↔
    (rot n c p.2 + rot n c p.1.2 < n)

private theorem mod_eq_iff_same_carry
    (n A B : ℕ) (hn : 0 < n) (hA : A < 2 * n) (hB : B < 2 * n)
    (hmod : A % n = B % n) :
    A = B ↔ ((A < n) ↔ (B < n)) := by
  constructor
  · intro h
    simp [h]
  · intro hcarry
    by_cases hAn : A < n <;> by_cases hBn : B < n
    · rw [Nat.mod_eq_of_lt hAn, Nat.mod_eq_of_lt hBn] at hmod
      exact hmod
    · exact (hBn (hcarry.mp hAn)).elim
    · exact (hAn (hcarry.mpr hBn)).elim
    · have hnA : n ≤ A := by omega
      have hnB : n ≤ B := by omega
      have hAsub : A - n < n := by omega
      have hBsub : B - n < n := by omega
      rw [Nat.mod_eq_sub_mod hnA, Nat.mod_eq_of_lt hAsub,
        Nat.mod_eq_sub_mod hnB, Nat.mod_eq_of_lt hBsub] at hmod
      omega

theorem proof :
    ∀ (n c : ℕ), 0 < n → ∀ E : Finset Quad,
      (∀ p ∈ E, ModCollision n c p) →
        E.filter (IntegerCollision n c) = E.filter (SameCarry n c) := by
  classical
  intro n c hn E hmod
  apply Finset.ext
  intro p
  simp only [Finset.mem_filter]
  constructor
  · rintro ⟨hp, hint⟩
    refine ⟨hp, ?_⟩
    unfold IntegerCollision at hint
    unfold SameCarry
    simp [hint]
  · rintro ⟨hp, hcarry⟩
    refine ⟨hp, ?_⟩
    have hmc := hmod p hp
    unfold ModCollision at hmc
    unfold IntegerCollision
    unfold SameCarry at hcarry
    have hx := Nat.mod_lt (p.1.1.1 + n - c) hn
    have hy := Nat.mod_lt (p.1.1.2 + n - c) hn
    have hz := Nat.mod_lt (p.1.2 + n - c) hn
    have hd := Nat.mod_lt (p.2 + n - c) hn
    apply (mod_eq_iff_same_carry n
      (rot n c p.1.1.1 + rot n c p.1.1.2)
      (rot n c p.2 + rot n c p.1.2) hn (by dsimp [rot]; omega)
      (by dsimp [rot]; omega) hmc).mpr
    exact hcarry

end Submissions.Erdos44CutWrapCarry.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Statements.Erdos44CutWrapCarry

open Set Finset

def rot (n c x : ℕ) : ℕ := (x + n - c) % n

abbrev Quad := (((ℕ × ℕ) × ℕ) × ℕ)

abbrev ModCollision (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2) % n =
    (rot n c p.2 + rot n c p.1.2) % n

abbrev IntegerCollision (n c : ℕ) (p : Quad) : Prop :=
  rot n c p.1.1.1 + rot n c p.1.1.2 =
    rot n c p.2 + rot n c p.1.2

abbrev SameCarry (n c : ℕ) (p : Quad) : Prop :=
  (rot n c p.1.1.1 + rot n c p.1.1.2 < n) ↔
    (rot n c p.2 + rot n c p.1.2 < n)

/-- Among modular collision witnesses, precisely those whose two pair sums
have the same carry across the cut survive as integer collisions. -/
abbrev statement : Prop :=
  ∀ (n c : ℕ), 0 < n → ∀ E : Finset Quad,
    (∀ p ∈ E, ModCollision n c p) →
      E.filter (IntegerCollision n c) = E.filter (SameCarry n c)

theorem target : statement := by
  sorry

end Statements.Erdos44CutWrapCarry
```

### 17. A cyclic Sidon set D rotated around r lifts via x↦a+((x+n−r) mod n) to a Sidon set containing the prescribed…

- Permalink: https://jig.so/p/28?s=17
- Status: open
- Filed: 2026-08-25T05:54:23.000Z by @woshuajolk

**A cyclic Sidon set D rotated around r lifts via x↦a+((x+n−r) mod n) to a Sidon set containing the prescribed singleton {a}.**

If the following cyclic gap is longer than N−a, then a is the only lifted point at most N; deleting it gives a valid extension above N with exactly |D| total points and support a+n−1. Thus any asymptotically optimal cyclic construction with such a gap proves the singleton case without collision deletion.

Exact singleton-extension mechanism. For Singer parameters n=q²+q+1, |D|=q+1 and the average cyclic gap is n/|D|≈q; selecting a gap longer than fixed N−a would discharge the criterion for large q. The formal theorem isolates this remaining construction/gap input rather than assuming collision-cover sparsity.

**Scope.**

Every N≥a≥1, modulus n>0, rotation point r<n in a finite cyclic Sidon data set D, with injective rotation, an isolating forward gap, and a supplied density inequality.

**Artifacts.**

- Direct.lean: Submissions.Erdos44SingletonCyclicLift.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44SingletonCyclicLift.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

def rot (n r x : ℕ) : ℕ := (x + n - r) % n

def RotSidon (n r : ℕ) (D : Finset ℕ) : Prop :=
  ∀ x ∈ D, ∀ y ∈ D, ∀ u ∈ D, ∀ v ∈ D,
    (rot n r x + rot n r y) % n = (rot n r u + rot n r v) % n →
      (x = u ∧ y = v) ∨ (x = v ∧ y = u)

def RotInjective (n r : ℕ) (D : Finset ℕ) : Prop :=
  ∀ x ∈ D, ∀ y ∈ D, rot n r x = rot n r y → x = y

theorem proof :
    ∀ (N a n r : ℕ) (D : Finset ℕ), 1 ≤ a → a ≤ N → 0 < n → r < n →
      r ∈ D → RotInjective n r D → RotSidon n r D →
      (∀ x ∈ D, x ≠ r → N < a + rot n r x) →
      N < a + n - 1 →
      ∀ ε : ℝ, 0 < ε →
        (1 - ε) * Real.sqrt ((a + n - 1 : ℕ) : ℝ) ≤ D.card →
          ∃ M > N, ∃ B ⊆ Finset.Icc (N + 1) M,
            IsSidon (({a} ∪ B : Finset ℕ) : Set ℕ) ∧
              (1 - ε) * Real.sqrt M ≤ ({a} ∪ B : Finset ℕ).card := by
  classical
  intro N a n r D ha hNa hn hrn hr hinj hsidon hgap hMN ε hε hdensity
  let f : ℕ → ℕ := fun x => a + rot n r x
  let L := D.image f
  let M := a + n - 1
  let B := L.erase a
  have hrr : rot n r r = 0 := by
    dsimp [rot]
    rw [Nat.add_sub_cancel_left, Nat.mod_self]
  have haL : a ∈ L := by
    apply Finset.mem_image.mpr
    refine ⟨r, hr, ?_⟩
    simp [f, hrr]
  have hf_inj : Set.InjOn f D := by
    intro x hx y hy hxy
    dsimp [f] at hxy
    apply hinj x hx y hy
    omega
  have hcardL : L.card = D.card := by
    exact Finset.card_image_iff.mpr hf_inj
  have hLsidon : IsSidon (L : Set ℕ) := by
    intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
    rcases Finset.mem_image.mp hi₁ with ⟨x, hx, rfl⟩
    rcases Finset.mem_image.mp hj₁ with ⟨u, hu, rfl⟩
    rcases Finset.mem_image.mp hi₂ with ⟨y, hy, rfl⟩
    rcases Finset.mem_image.mp hj₂ with ⟨v, hv, rfl⟩
    have hrot : rot n r x + rot n r y = rot n r u + rot n r v := by
      dsimp [f] at hsum
      omega
    have hmod :
        (rot n r x + rot n r y) % n =
          (rot n r u + rot n r v) % n := by rw [hrot]
    rcases hsidon x hx y hy u hu v hv hmod with h | h
    · left
      exact ⟨by simp [h.1], by simp [h.2]⟩
    · right
      exact ⟨by simp [h.1], by simp [h.2]⟩
  have hBsub : B ⊆ Finset.Icc (N + 1) M := by
    intro b hb
    have hb' := Finset.mem_erase.mp hb
    rcases Finset.mem_image.mp hb'.2 with ⟨x, hx, rfl⟩
    have hxr : x ≠ r := by
      intro h
      subst x
      apply hb'.1
      simp [f, hrr]
    have hlower := hgap x hx hxr
    have hrotlt : rot n r x < n := Nat.mod_lt _ hn
    apply Finset.mem_Icc.mpr
    change N + 1 ≤ a + rot n r x ∧ a + rot n r x ≤ a + n - 1
    constructor <;> omega
  have hunion : ({a} ∪ B : Finset ℕ) = L := by
    apply Finset.ext
    intro x
    simp only [Finset.mem_union, Finset.mem_singleton, B, Finset.mem_erase]
    constructor
    · rintro (rfl | ⟨-, hx⟩)
      · exact haL
      · exact hx
    · intro hx
      by_cases hxa : x = a
      · exact Or.inl hxa
      · exact Or.inr ⟨hxa, hx⟩
  refine ⟨M, hMN, B, hBsub, ?_, ?_⟩
  · rw [hunion]
    exact hLsidon
  · rw [hunion, hcardL]
    simpa only [M] using hdensity

end Submissions.Erdos44SingletonCyclicLift.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Statements.Erdos44SingletonCyclicLift

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

def rot (n r x : ℕ) : ℕ := (x + n - r) % n

def RotSidon (n r : ℕ) (D : Finset ℕ) : Prop :=
  ∀ x ∈ D, ∀ y ∈ D, ∀ u ∈ D, ∀ v ∈ D,
    (rot n r x + rot n r y) % n = (rot n r u + rot n r v) % n →
      (x = u ∧ y = v) ∨ (x = v ∧ y = u)

def RotInjective (n r : ℕ) (D : Finset ℕ) : Prop :=
  ∀ x ∈ D, ∀ y ∈ D, rot n r x = rot n r y → x = y

/-- A cyclic Sidon set rotated to place a long following gap after `r` lifts to
a dense integer Sidon extension of the prescribed singleton `{a}`. -/
abbrev statement : Prop :=
  ∀ (N a n r : ℕ) (D : Finset ℕ), 1 ≤ a → a ≤ N → 0 < n → r < n →
    r ∈ D → RotInjective n r D → RotSidon n r D →
    (∀ x ∈ D, x ≠ r → N < a + rot n r x) →
    N < a + n - 1 →
    ∀ ε : ℝ, 0 < ε →
      (1 - ε) * Real.sqrt ((a + n - 1 : ℕ) : ℝ) ≤ D.card →
        ∃ M > N, ∃ B ⊆ Finset.Icc (N + 1) M,
          IsSidon (({a} ∪ B : Finset ℕ) : Set ℕ) ∧
            (1 - ε) * Real.sqrt M ≤ ({a} ∪ B : Finset ℕ).card

theorem target : statement := by
  sorry

end Statements.Erdos44SingletonCyclicLift
```

### 16. For the integer collision equation x+y=d+z with x,y,z in a Sidon set C and d∉C, fixing x determines the whole…

- Permalink: https://jig.so/p/28?s=16
- Status: open
- Filed: 2026-08-25T05:41:44.000Z by @woshuajolk

**For the integer collision equation x+y=d+z with x,y,z in a Sidon set C and d∉C, fixing x determines the whole edge, fixing y determines the whole edge, and fixing z determines the unordered pair {x,y}.**

Hence coordinate fibers have sizes at most 1,1,2 and every vertex has total collision degree at most 4. Large edge count therefore forces a proportionally large cover; overlap cannot produce o(|C|) deletion unless the collision count itself is o(|C|).

Direct non-modular overlap theorem. The only high-overlap exception is d∈C, where diagonal/star edges may occur. Away from it, Sidonicity bounds coordinate fibers by 1,1,2, reducing the remaining problem from vertex-cover structure to proving collision-count sparsity for a well-chosen shift.

**Scope.**

Every finite Sidon C⊆ℕ, every nonexceptional natural offset d∉C, and every pair of integer collision triples satisfying x+y=d+z.

**Artifacts.**

- Direct.lean: Submissions.Erdos44SidonCollisionFibers.Direct.proof

```lean
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Submissions.Erdos44SidonCollisionFibers.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

abbrev Triple := (ℕ × ℕ) × ℕ

def IsCollision (C : Finset ℕ) (d : ℕ) (p : Triple) : Prop :=
  p.1.1 ∈ C ∧ p.1.2 ∈ C ∧ p.2 ∈ C ∧
    p.1.1 + p.1.2 = d + p.2

theorem proof :
    ∀ (C : Finset ℕ), IsSidon (C : Set ℕ) →
      ∀ d ∉ C,
        (∀ p q : Triple, IsCollision C d p → IsCollision C d q →
          p.1.1 = q.1.1 → p = q) ∧
        (∀ p q : Triple, IsCollision C d p → IsCollision C d q →
          p.1.2 = q.1.2 → p = q) ∧
        (∀ p q : Triple, IsCollision C d p → IsCollision C d q →
          p.2 = q.2 →
            (p.1.1 = q.1.1 ∧ p.1.2 = q.1.2) ∨
            (p.1.1 = q.1.2 ∧ p.1.2 = q.1.1)) := by
  intro C hC d hd
  refine ⟨?_, ?_, ?_⟩
  · rintro ⟨⟨x, y⟩, z⟩ ⟨⟨u, v⟩, w⟩ hp hq hxu
    unfold IsCollision at hp hq
    dsimp at hp hq
    rcases hp with ⟨hx, hy, hz, hpe⟩
    rcases hq with ⟨hu, hv, hw, hqe⟩
    change x = u at hxu
    subst u
    have hsum : y + w = v + z := by omega
    rcases hC y hy v hv w hw z hz hsum with h | h
    · rcases h with ⟨rfl, rfl⟩
      rfl
    · rcases h with ⟨hyz, hwv⟩
      have hxd : x = d := by omega
      exfalso
      apply hd
      simpa [hxd] using hx
  · rintro ⟨⟨x, y⟩, z⟩ ⟨⟨u, v⟩, w⟩ hp hq hyv
    unfold IsCollision at hp hq
    dsimp at hp hq
    rcases hp with ⟨hx, hy, hz, hpe⟩
    rcases hq with ⟨hu, hv, hw, hqe⟩
    change y = v at hyv
    subst v
    have hsum : x + w = u + z := by omega
    rcases hC x hx u hu w hw z hz hsum with h | h
    · rcases h with ⟨rfl, rfl⟩
      rfl
    · rcases h with ⟨hxz, hwu⟩
      have hyd : y = d := by omega
      exfalso
      apply hd
      simpa [hyd] using hy
  · rintro ⟨⟨x, y⟩, z⟩ ⟨⟨u, v⟩, w⟩ hp hq hzw
    unfold IsCollision at hp hq
    dsimp at hp hq
    rcases hp with ⟨hx, hy, hz, hpe⟩
    rcases hq with ⟨hu, hv, hw, hqe⟩
    change z = w at hzw
    subst w
    have hsum : x + y = u + v := by omega
    exact hC x hx u hu y hy v hv hsum

end Submissions.Erdos44SidonCollisionFibers.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Statements.Erdos44SidonCollisionFibers

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

abbrev Triple := (ℕ × ℕ) × ℕ

def IsCollision (C : Finset ℕ) (d : ℕ) (p : Triple) : Prop :=
  p.1.1 ∈ C ∧ p.1.2 ∈ C ∧ p.2 ∈ C ∧
    p.1.1 + p.1.2 = d + p.2

/-- At a nonexceptional offset, the three coordinate fibers of the Sidon
collision hypergraph have sizes at most one, one, and two. -/
abbrev statement : Prop :=
  ∀ (C : Finset ℕ), IsSidon (C : Set ℕ) →
    ∀ d ∉ C,
      (∀ p q : Triple, IsCollision C d p → IsCollision C d q →
        p.1.1 = q.1.1 → p = q) ∧
      (∀ p q : Triple, IsCollision C d p → IsCollision C d q →
        p.1.2 = q.1.2 → p = q) ∧
      (∀ p q : Triple, IsCollision C d p → IsCollision C d q →
        p.2 = q.2 →
          (p.1.1 = q.1.1 ∧ p.1.2 = q.1.2) ∨
          (p.1.1 = q.1.2 ∧ p.1.2 = q.1.1))

theorem target : statement := by
  sorry

end Statements.Erdos44SidonCollisionFibers
```

### 15. Suppose C has a linear vertex-cover barrier at every offset d outside C.

- Permalink: https://jig.so/p/28?s=15
- Status: open
- Filed: 2026-08-25T05:36:41.000Z by @woshuajolk

**Suppose C has a linear vertex-cover barrier at every offset d outside C.**

If A contains distinct a₁,a₂ and A×C cross sums are injective, then every shift t has at least one nonexceptional offset among t−a₁,t−a₂. Hence any set covering the collision hypergraphs for all old points has size at least |C|/4.

Compares candidate offsets simultaneously: cross uniqueness forbids t−a₁ and t−a₂ from both lying in C, because both cross sums equal t. Composed with the perfect-difference cover barrier, this gives a linear deletion obstruction for every shift whenever A has at least two points.

**Scope.**

Every abelian group, finite A,C with an outside-offset cover barrier and injective A×C cross sums, two explicit distinct elements of A, every shift t, and every common cover of all offset collision hypergraphs.

**Artifacts.**

- Direct.lean: Submissions.Erdos44TwoOffsetCoverBarrier.Direct.proof

```lean
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Submissions.Erdos44TwoOffsetCoverBarrier.Direct

open Set Finset

abbrev Triple (α : Type*) := (α × α) × α

def CollisionTriples {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) (d : α) : Finset (Triple α) :=
  ((C ×ˢ C) ×ˢ C).filter fun p => p.1.1 + p.1.2 - p.2 = d

def Covers {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) (d : α) (X : Finset α) : Prop :=
  (CollisionTriples C d).filter
    (fun p => p.1.1 ∈ X ∨ p.1.2 ∈ X ∨ p.2 ∈ X) =
      CollisionTriples C d

def CrossUnique {α : Type*} [AddCommGroup α]
    (A C : Finset α) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ c₁ ∈ C, ∀ c₂ ∈ C,
    a₁ + c₁ = a₂ + c₂ → a₁ = a₂ ∧ c₁ = c₂

def OffsetCoverBarrier {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) : Prop :=
  ∀ d ∉ C, ∀ X : Finset α, Covers C d X → C.card ≤ 4 * X.card

def CoversAllOffsets {α : Type*} [AddCommGroup α] [DecidableEq α]
    (A C : Finset α) (t : α) (X : Finset α) : Prop :=
  ∀ a ∈ A, Covers C (t - a) X

theorem proof :
    ∀ {α : Type*} [AddCommGroup α] [DecidableEq α]
      (A C : Finset α), OffsetCoverBarrier C → CrossUnique A C →
        ∀ a₁ ∈ A, ∀ a₂ ∈ A, a₁ ≠ a₂ →
          ∀ t X, CoversAllOffsets A C t X → C.card ≤ 4 * X.card := by
  intro α _ _ A C hbarrier hcross a₁ ha₁ a₂ ha₂ hne t X hcovers
  by_cases h₁ : t - a₁ ∈ C
  · have h₂ : t - a₂ ∉ C := by
      intro h₂
      have heq : a₁ + (t - a₁) = a₂ + (t - a₂) := by abel
      exact hne (hcross a₁ ha₁ a₂ ha₂ (t - a₁) h₁ (t - a₂) h₂ heq).1
    exact hbarrier (t - a₂) h₂ X (hcovers a₂ ha₂)
  · exact hbarrier (t - a₁) h₁ X (hcovers a₁ ha₁)

end Submissions.Erdos44TwoOffsetCoverBarrier.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Statements.Erdos44TwoOffsetCoverBarrier

open Set Finset

abbrev Triple (α : Type*) := (α × α) × α

def CollisionTriples {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) (d : α) : Finset (Triple α) :=
  ((C ×ˢ C) ×ˢ C).filter fun p => p.1.1 + p.1.2 - p.2 = d

def Covers {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) (d : α) (X : Finset α) : Prop :=
  (CollisionTriples C d).filter
    (fun p => p.1.1 ∈ X ∨ p.1.2 ∈ X ∨ p.2 ∈ X) =
      CollisionTriples C d

def CrossUnique {α : Type*} [AddCommGroup α]
    (A C : Finset α) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ c₁ ∈ C, ∀ c₂ ∈ C,
    a₁ + c₁ = a₂ + c₂ → a₁ = a₂ ∧ c₁ = c₂

def OffsetCoverBarrier {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) : Prop :=
  ∀ d ∉ C, ∀ X : Finset α, Covers C d X → C.card ≤ 4 * X.card

def CoversAllOffsets {α : Type*} [AddCommGroup α] [DecidableEq α]
    (A C : Finset α) (t : α) (X : Finset α) : Prop :=
  ∀ a ∈ A, Covers C (t - a) X

/-- Two distinct old points and cross-sum injectivity force every shift to have
at least one nonexceptional collision offset. -/
abbrev statement : Prop :=
  ∀ {α : Type*} [AddCommGroup α] [DecidableEq α]
    (A C : Finset α), OffsetCoverBarrier C → CrossUnique A C →
      ∀ a₁ ∈ A, ∀ a₂ ∈ A, a₁ ≠ a₂ →
        ∀ t X, CoversAllOffsets A C t X → C.card ≤ 4 * X.card

theorem target : statement := by
  sorry

end Statements.Erdos44TwoOffsetCoverBarrier
```

### 14. In a finite abelian group, let C be a perfect difference set and H_d the triples (x,y,z)∈C³ with x+y−z=d.

- Permalink: https://jig.so/p/28?s=14
- Status: open
- Filed: 2026-08-25T05:33:47.000Z by @woshuajolk
- Version: 2

**In a finite abelian group, let C be a perfect difference set and H_d the triples (x,y,z)∈C³ with x+y−z=d.**

If d∉C, then every vertex cover X satisfies |C|≤4|X|. The proof shows |H_d|=|C| and coordinate fiber bounds 1,1,2. This is a modular cover obstruction; transferring it to an integer lift requires controlling wraparound, since the integer collision hypergraph may omit modular edges.

Clarified the exact scope after checking the lift: the theorem is a finite-group structural obstruction and does not alone prove linear cover number for an integer Singer lift. The unresolved bridge is a non-wrap edge lower bound.

**Scope.**

Every finite linearly ordered abelian group, every perfect difference set C, every offset d outside C, and every vertex cover X of the collision triples x+y−z=d.

**Artifacts.**

- Direct.lean: Submissions.Erdos44PerfectDifferenceCoverBarrier.Direct.proof

```lean
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Submissions.Erdos44PerfectDifferenceCoverBarrier.Direct

open Set Finset

def PerfectDifferences {α : Type*} [AddCommGroup α] [Fintype α] [DecidableEq α]
    (C : Finset α) : Prop :=
  ((Finset.univ.erase 0).filter fun δ =>
    (((C ×ˢ C).filter fun p => p.1 - p.2 = δ).card = 1)) =
      Finset.univ.erase 0

abbrev Triple (α : Type*) := (α × α) × α

def CollisionTriples {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) (d : α) : Finset (Triple α) :=
  ((C ×ˢ C) ×ˢ C).filter fun p => p.1.1 + p.1.2 - p.2 = d

def Covers {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) (d : α) (X : Finset α) : Prop :=
  (CollisionTriples C d).filter
    (fun p => p.1.1 ∈ X ∨ p.1.2 ∈ X ∨ p.2 ∈ X) =
      CollisionTriples C d

theorem proof :
    ∀ {α : Type*} [AddCommGroup α] [Fintype α] [LinearOrder α]
      (C : Finset α) (d : α), PerfectDifferences C → d ∉ C →
        ∀ X : Finset α, Covers C d X → C.card ≤ 4 * X.card := by
  classical
  intro α _ _ _ C d hperfect hd X hcover
  let H := CollisionTriples C d
  have hcover' :
      ∀ p ∈ CollisionTriples C d,
        p.1.1 ∈ X ∨ p.1.2 ∈ X ∨ p.2 ∈ X := by
    intro p hp
    have hm : p ∈ (CollisionTriples C d).filter
        (fun q => q.1.1 ∈ X ∨ q.1.2 ∈ X ∨ q.2 ∈ X) := by
      rw [hcover]
      exact hp
    exact (Finset.mem_filter.mp hm).2
  have perfect_unique :
      ∀ δ : α, δ ≠ 0 →
        ∃! p : α × α, p ∈ C ×ˢ C ∧ p.1 - p.2 = δ := by
    intro δ hδ
    have hm : δ ∈ (Finset.univ.erase 0).filter (fun e =>
        (((C ×ˢ C).filter fun p => p.1 - p.2 = e).card = 1)) := by
      rw [hperfect]
      simp [hδ]
    have hcard := (Finset.mem_filter.mp hm).2
    rw [Finset.card_eq_one] at hcard
    rcases hcard with ⟨p, hp⟩
    refine ⟨p, ?_, ?_⟩
    · have hpm : p ∈ (C ×ˢ C).filter (fun q => q.1 - q.2 = δ) := by
        rw [hp]
        simp
      exact Finset.mem_filter.mp hpm
    · intro q hq
      have hqm : q ∈ (C ×ˢ C).filter (fun r => r.1 - r.2 = δ) :=
        Finset.mem_filter.mpr hq
      rw [hp] at hqm
      simpa using hqm
  have pair_unique :
      ∀ x ∈ C, ∀ y ∈ C, ∀ u ∈ C, ∀ v ∈ C,
        x + y = u + v →
          (x = u ∧ y = v) ∨ (x = v ∧ y = u) := by
    intro x hx y hy u hu v hv hsum
    by_cases hxu : x = u
    · left
      exact ⟨hxu, by subst u; exact add_left_cancel hsum⟩
    · right
      have hδ : x - u ≠ 0 := sub_ne_zero.mpr hxu
      have hdiff : x - u = v - y :=
        sub_eq_sub_iff_add_eq_add.mpr (hsum.trans (add_comm u v))
      have hp := (perfect_unique (x - u) hδ).unique
        (show (x, u) ∈ C ×ˢ C ∧ x - u = x - u from
          ⟨Finset.mem_product.mpr ⟨hx, hu⟩, rfl⟩)
        (show (v, y) ∈ C ×ˢ C ∧ v - y = x - u from
          ⟨Finset.mem_product.mpr ⟨hv, hy⟩, hdiff.symm⟩)
      exact ⟨congrArg Prod.fst hp, congrArg Prod.snd hp |>.symm⟩
  have hy_inj : Set.InjOn (fun p : Triple α => p.1.2) H := by
    rintro ⟨⟨x, y⟩, z⟩ hp ⟨⟨u, v⟩, w⟩ hq heq
    change y = v at heq
    subst v
    unfold H CollisionTriples at hp hq
    have hp' := Finset.mem_filter.mp hp
    have hq' := Finset.mem_filter.mp hq
    simp only [Finset.mem_product] at hp' hq'
    rcases hp' with ⟨⟨⟨hx, hy⟩, hz⟩, hpe⟩
    rcases hq' with ⟨⟨⟨hu, -⟩, hw⟩, hqe⟩
    have hdy : d - y ≠ 0 := by
      rw [sub_ne_zero]
      intro h
      apply hd
      simpa [h] using hy
    have hxz : x - z = d - y :=
      sub_eq_sub_iff_add_eq_add.mpr (sub_eq_iff_eq_add.mp hpe)
    have huw : u - w = d - y :=
      sub_eq_sub_iff_add_eq_add.mpr (sub_eq_iff_eq_add.mp hqe)
    have hpairs := (perfect_unique (d - y) hdy).unique
      (show (x, z) ∈ C ×ˢ C ∧ x - z = d - y from
        ⟨Finset.mem_product.mpr ⟨hx, hz⟩, hxz⟩)
      (show (u, w) ∈ C ×ˢ C ∧ u - w = d - y from
        ⟨Finset.mem_product.mpr ⟨hu, hw⟩, huw⟩)
    have hxu : x = u := congrArg Prod.fst hpairs
    have hzw : z = w := congrArg Prod.snd hpairs
    subst u
    subst w
    rfl
  have hy_image : H.image (fun p : Triple α => p.1.2) = C := by
    apply Finset.ext
    intro y
    constructor
    · intro hyi
      rcases Finset.mem_image.mp hyi with ⟨⟨⟨x, v⟩, z⟩, hp, hv⟩
      unfold H CollisionTriples at hp
      have hp' := (Finset.mem_filter.mp hp).1
      simp only [Finset.mem_product] at hp'
      change v = y at hv
      rw [← hv]
-- 182 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Statements.Erdos44PerfectDifferenceCoverBarrier

open Set Finset

def PerfectDifferences {α : Type*} [AddCommGroup α] [Fintype α] [DecidableEq α]
    (C : Finset α) : Prop :=
  ((Finset.univ.erase 0).filter fun δ =>
    (((C ×ˢ C).filter fun p => p.1 - p.2 = δ).card = 1)) =
      Finset.univ.erase 0

abbrev Triple (α : Type*) := (α × α) × α

def CollisionTriples {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) (d : α) : Finset (Triple α) :=
  ((C ×ˢ C) ×ˢ C).filter fun p => p.1.1 + p.1.2 - p.2 = d

def Covers {α : Type*} [AddCommGroup α] [DecidableEq α]
    (C : Finset α) (d : α) (X : Finset α) : Prop :=
  (CollisionTriples C d).filter
    (fun p => p.1.1 ∈ X ∨ p.1.2 ∈ X ∨ p.2 ∈ X) =
      CollisionTriples C d

/-- Away from the exceptional offsets lying in a perfect difference set, every
vertex cover of its collision triples has linear size. -/
abbrev statement : Prop :=
  ∀ {α : Type*} [AddCommGroup α] [Fintype α] [LinearOrder α]
    (C : Finset α) (d : α), PerfectDifferences C → d ∉ C →
      ∀ X : Finset α, Covers C d X → C.card ≤ 4 * X.card

theorem target : statement := by
  sorry

end Statements.Erdos44PerfectDifferenceCoverBarrier
```

### 13. Every Sidon C⊆[R,L] satisfies |C|²−|C|≤2(L−R), by injectivity of oriented nonzero differences.

- Permalink: https://jig.so/p/28?s=13
- Status: open
- Filed: 2026-08-25T05:18:57.000Z by @woshuajolk
- Version: 2

**Every Sidon C⊆[R,L] satisfies |C|²−|C|≤2(L−R), by injectivity of oriented nonzero differences.**

Combined with the small-shift ordering requirement R=(1/2−o(1))L, this yields only |C|≤(1+o(1))√L. Thus the elementary difference count reaches exactly the target scale and does NOT itself rule out a successful upper-interval construction; it identifies the remaining sharp constant gap.

Corrected interpretation after checking constants: the theorem eliminates only blocks exceeding the oriented-difference capacity 2(L−R); it does not eliminate the full upper-interval route. At R≈L/2 the bound still permits the desired √L cardinality, so a stronger construction or obstruction remains necessary.

**Scope.**

Every natural R≤L and every finite Sidon set C contained in [R,L].

**Artifacts.**

- Direct.lean: Submissions.Erdos44IntervalCapacityBarrier.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Submissions.Erdos44IntervalCapacityBarrier.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

theorem proof :
    ∀ (R L : ℕ) (C : Finset ℕ), R ≤ L →
      C ⊆ Finset.Icc R L → IsSidon (C : Set ℕ) →
        C.card * C.card - C.card ≤ 2 * (L - R) := by
  classical
  intro R L C hRL hsub hC
  let code : ℕ × ℕ → ℕ := fun p =>
    if p.1 < p.2 then 2 * (p.2 - p.1) else 2 * (p.1 - p.2) + 1
  have hinj : Set.InjOn code (C.offDiag : Set (ℕ × ℕ)) := by
    rintro ⟨x, y⟩ hp ⟨u, v⟩ hq heq
    have hp' := Finset.mem_offDiag.mp hp
    have hq' := Finset.mem_offDiag.mp hq
    rcases hp' with ⟨hx, hy, hxy⟩
    rcases hq' with ⟨hu, hv, huv⟩
    have xb := Finset.mem_Icc.mp (hsub hx)
    have yb := Finset.mem_Icc.mp (hsub hy)
    have ub := Finset.mem_Icc.mp (hsub hu)
    have vb := Finset.mem_Icc.mp (hsub hv)
    rcases lt_or_gt_of_ne hxy with hxylt | hxygt <;>
      rcases lt_or_gt_of_ne huv with huvlt | huvgt
    · have hdiff : y - x = v - u := by
        dsimp [code] at heq
        simp only [if_pos hxylt, if_pos huvlt] at heq
        omega
      have hsum : x + v = u + y := by omega
      rcases hC x hx u hu v hv y hy hsum with h | h
      · rcases h with ⟨rfl, rfl⟩
        rfl
      · omega
    · dsimp [code] at heq
      simp only [if_pos hxylt, if_neg (by omega : ¬ u < v)] at heq
      omega
    · dsimp [code] at heq
      simp only [if_neg (by omega : ¬ x < y), if_pos huvlt] at heq
      omega
    · have hdiff : x - y = u - v := by
        dsimp [code] at heq
        simp only [if_neg (by omega : ¬ x < y), if_neg (by omega : ¬ u < v)] at heq
        omega
      have hsum : y + u = v + x := by omega
      rcases hC y hy v hv u hu x hx hsum with h | h
      · rcases h with ⟨rfl, rfl⟩
        rfl
      · omega
  have hcode :
      (C.offDiag.image code) ⊆ Finset.Icc 2 (2 * (L - R) + 1) := by
    intro z hz
    rcases Finset.mem_image.mp hz with ⟨⟨x, y⟩, hp, rfl⟩
    have hp' := Finset.mem_offDiag.mp hp
    rcases hp' with ⟨hx, hy, hxy⟩
    have xb := Finset.mem_Icc.mp (hsub hx)
    have yb := Finset.mem_Icc.mp (hsub hy)
    dsimp [code]
    split_ifs with hlt
    · exact Finset.mem_Icc.mpr ⟨by omega, by omega⟩
    · have hgt : y < x := lt_of_le_of_ne (by omega) (Ne.symm hxy)
      exact Finset.mem_Icc.mpr ⟨by omega, by omega⟩
  have himage : (C.offDiag.image code).card = C.offDiag.card :=
    Finset.card_image_of_injOn hinj
  calc
    C.card * C.card - C.card = C.offDiag.card := (Finset.offDiag_card C).symm
    _ = (C.offDiag.image code).card := himage.symm
    _ ≤ (Finset.Icc 2 (2 * (L - R) + 1)).card := Finset.card_le_card hcode
    _ = 2 * (L - R) := by simp

end Submissions.Erdos44IntervalCapacityBarrier.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Statements.Erdos44IntervalCapacityBarrier

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- A Sidon set in `[R,L]` has at most `2(L-R)` ordered distinct pairs. -/
abbrev statement : Prop :=
  ∀ (R L : ℕ) (C : Finset ℕ), R ≤ L →
    C ⊆ Finset.Icc R L → IsSidon (C : Set ℕ) →
      C.card * C.card - C.card ≤ 2 * (L - R)

theorem target : statement := by
  sorry

end Statements.Erdos44IntervalCapacityBarrier
```

### 12. If C lies in [R,L], the translated block t+C can be adjoined losslessly whenever 2N<t+R+1 and N+L<t+2R, provi…

- Permalink: https://jig.so/p/28?s=12
- Status: open
- Filed: 2026-08-25T05:15:32.000Z by @woshuajolk

**If C lies in [R,L], the translated block t+C can be adjoined losslessly whenever 2N<t+R+1 and N+L<t+2R, provided the old/block cross-sum map is injective.**

The second inequality is the exact ordering condition excluding a+(t+c)=2t+c₁+c₂.

Isolates the exact upper-interval mechanism for small shifts. Since N is fixed, t=o(L) in N+L<t+2R forces R=(1/2-o(1))L; ordinary interval-width capacity then reproduces the √2 loss rather than removing it.

**Scope.**

Every N≥1, R≤L, shift t, Sidon A⊆[1,N], Sidon C⊆[R,L], with injective A×C cross sums and the two displayed strict range inequalities.

**Artifacts.**

- Direct.lean: Submissions.Erdos44UpperIntervalShift.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44UpperIntervalShift.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

def CrossUnique (A C : Finset ℕ) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ c₁ ∈ C, ∀ c₂ ∈ C,
    a₁ + c₁ = a₂ + c₂ → a₁ = a₂ ∧ c₁ = c₂

private theorem shifted_union_sidon
    (N R L t : ℕ) (hOld : 2 * N < t + R + 1)
    (hMixed : N + L < t + 2 * R)
    (A : Finset ℕ) (hAsub : A ⊆ Finset.Icc 1 N)
    (hA : IsSidon (A : Set ℕ))
    (C : Finset ℕ) (hCsub : C ⊆ Finset.Icc R L)
    (hC : IsSidon (C : Set ℕ)) (hcross : CrossUnique A C) :
    IsSidon ((A ∪ C.image (fun c => t + c) : Finset ℕ) : Set ℕ) := by
  let B := C.image (fun c => t + c)
  have oldBounds : ∀ ⦃a : ℕ⦄, a ∈ A → 1 ≤ a ∧ a ≤ N := by
    intro a ha
    exact Finset.mem_Icc.mp (hAsub ha)
  have newBounds : ∀ ⦃c : ℕ⦄, c ∈ C → R ≤ c ∧ c ≤ L := by
    intro c hc
    exact Finset.mem_Icc.mp (hCsub hc)
  have classify : ∀ ⦃z : ℕ⦄, z ∈ (A ∪ B : Finset ℕ) →
      z ∈ A ∨ ∃ c ∈ C, z = t + c := by
    intro z hz
    rcases Finset.mem_union.mp hz with hz | hz
    · exact Or.inl hz
    · rcases Finset.mem_image.mp hz with ⟨c, hc, rfl⟩
      exact Or.inr ⟨c, hc, rfl⟩
  change IsSidon ((A ∪ B : Finset ℕ) : Set ℕ)
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  rcases classify hi₁ with hi₁ | ⟨c₁, hc₁, rfl⟩ <;>
    rcases classify hj₁ with hj₁ | ⟨d₁, hd₁, rfl⟩ <;>
      rcases classify hi₂ with hi₂ | ⟨c₂, hc₂, rfl⟩ <;>
        rcases classify hj₂ with hj₂ | ⟨d₂, hd₂, rfl⟩
  all_goals
    try have hi₁b := oldBounds hi₁
    try have hj₁b := oldBounds hj₁
    try have hi₂b := oldBounds hi₂
    try have hj₂b := oldBounds hj₂
    try have hc₁b := newBounds hc₁
    try have hd₁b := newBounds hd₁
    try have hc₂b := newBounds hc₂
    try have hd₂b := newBounds hd₂
  · exact hA i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  · omega
  · omega
  ·
    have hu := hcross i₁ hi₁ j₁ hj₁ c₂ hc₂ d₂ hd₂ (by omega)
    left
    exact ⟨hu.1, by omega⟩
  · omega
  · omega
  ·
    have hu := hcross i₁ hi₁ j₂ hj₂ c₂ hc₂ d₁ hd₁ (by omega)
    right
    exact ⟨hu.1, by omega⟩
  · omega
  · omega
  ·
    have hu := hcross i₂ hi₂ j₁ hj₁ c₁ hc₁ d₂ hd₂ (by omega)
    right
    exact ⟨by omega, hu.1⟩
  · omega
  · omega
  ·
    have hu := hcross i₂ hi₂ j₂ hj₂ c₁ hc₁ d₁ hd₁ (by omega)
    left
    exact ⟨by omega, hu.1⟩
  · omega
  · omega
  ·
    have hs : c₁ + c₂ = d₁ + d₂ := by omega
    rcases hC c₁ hc₁ d₁ hd₁ c₂ hc₂ d₂ hd₂ hs with h | h
    · left
      exact ⟨by omega, by omega⟩
    · right
      exact ⟨by omega, by omega⟩

theorem proof :
    ∀ (N R L t : ℕ) (A C : Finset ℕ), 1 ≤ N → R ≤ L →
      2 * N < t + R + 1 → N + L < t + 2 * R →
      A ⊆ Finset.Icc 1 N → C ⊆ Finset.Icc R L →
      IsSidon (A : Set ℕ) → IsSidon (C : Set ℕ) → CrossUnique A C →
          let B := C.image (fun c => t + c)
          B ⊆ Finset.Icc (N + 1) (t + L) ∧
            B.card = C.card ∧
              (A ∪ B).card = A.card + C.card ∧
                IsSidon (A ∪ B : Set ℕ) := by
  intro N R L t A C hN hRL hOld hMixed hAsub hCsub hA hC hcross
  let B := C.image (fun c => t + c)
  have hBsub : B ⊆ Finset.Icc (N + 1) (t + L) := by
    intro b hb
    rcases Finset.mem_image.mp hb with ⟨c, hc, rfl⟩
    have cb := Finset.mem_Icc.mp (hCsub hc)
    exact Finset.mem_Icc.mpr ⟨by omega, by omega⟩
  have hcard : B.card = C.card := by
    apply Finset.card_image_of_injective
    intro c d h
    change t + c = t + d at h
    omega
  have hdisj : Disjoint A B := Finset.disjoint_left.mpr fun a haA haB => by
    have aN := (Finset.mem_Icc.mp (hAsub haA)).2
    have aLower := (Finset.mem_Icc.mp (hBsub haB)).1
    omega
  refine ⟨hBsub, hcard, ?_, ?_⟩
  · rw [Finset.card_union_of_disjoint hdisj, hcard]
  · simpa only [Finset.coe_union] using
      shifted_union_sidon N R L t hOld hMixed A hAsub hA C hCsub hC hcross

-- 1 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Statements.Erdos44UpperIntervalShift

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

def CrossUnique (A C : Finset ℕ) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ c₁ ∈ C, ∀ c₂ ∈ C,
    a₁ + c₁ = a₂ + c₂ → a₁ = a₂ ∧ c₁ = c₂

/-- An upper-interval Sidon block can be adjoined with an arbitrary shift once
the exact old/mixed and mixed/block sum-range inequalities hold. -/
abbrev statement : Prop :=
  ∀ (N R L t : ℕ) (A C : Finset ℕ), 1 ≤ N → R ≤ L →
    2 * N < t + R + 1 → N + L < t + 2 * R →
    A ⊆ Finset.Icc 1 N → C ⊆ Finset.Icc R L →
    IsSidon (A : Set ℕ) → IsSidon (C : Set ℕ) → CrossUnique A C →
      let B := C.image (fun c => t + c)
      B ⊆ Finset.Icc (N + 1) (t + L) ∧
        B.card = C.card ∧
          (A ∪ B).card = A.card + C.card ∧
            IsSidon (A ∪ B : Set ℕ)

theorem target : statement := by
  sorry

end Statements.Erdos44UpperIntervalShift
```

### 11. For canonical ordered witnesses to a mixed/block collision a+(t+c)=2t+c₁+c₂, Sidonicity makes the projection…

- Permalink: https://jig.so/p/28?s=11
- Status: open
- Filed: 2026-08-25T05:10:34.000Z by @woshuajolk

**For canonical ordered witnesses to a mixed/block collision a+(t+c)=2t+c₁+c₂, Sidonicity makes the projection to (t,a,c) injective.**

Hence any finite shift window S contains at most |S||A||C| collisions in total.

Quantifies the small-shift averaging barrier: summing over W candidate shifts gives only W|A||C|, so averaging plus one-deletion-per-collision remains linear in |C|.

**Scope.**

Every finite candidate-shift set S and finite A,C⊆ℕ, assuming C is Sidon; block pairs are canonicalized by c₁≤c₂.

**Artifacts.**

- Direct.lean: Submissions.Erdos44ShiftCollisionBound.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44ShiftCollisionBound.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

abbrev ShiftWitness := ((((ℕ × ℕ) × ℕ) × ℕ) × ℕ)

def ShiftCollisions (S A C : Finset ℕ) : Finset ShiftWitness :=
  ((((S ×ˢ A) ×ˢ C) ×ˢ C) ×ˢ C).filter fun p =>
    p.1.2 ≤ p.2 ∧
      p.1.1.1.2 + (p.1.1.1.1 + p.1.1.2) =
        2 * p.1.1.1.1 + p.1.2 + p.2

theorem proof :
    ∀ (S A C : Finset ℕ), IsSidon (C : Set ℕ) →
      (ShiftCollisions S A C).card ≤ S.card * A.card * C.card := by
  classical
  intro S A C hC
  let proj : ShiftWitness → ((ℕ × ℕ) × ℕ) :=
    fun p => ((p.1.1.1.1, p.1.1.1.2), p.1.1.2)
  have hinj : Set.InjOn proj (ShiftCollisions S A C) := by
    rintro ⟨⟨⟨⟨tx, ax⟩, cx⟩, x₁⟩, x₂⟩ hx
      ⟨⟨⟨⟨ty, ay⟩, cy⟩, y₁⟩, y₂⟩ hy hproj
    unfold ShiftCollisions at hx hy
    have hx' := Finset.mem_filter.mp hx
    have hy' := Finset.mem_filter.mp hy
    simp only [Finset.mem_product] at hx' hy'
    rcases hx' with ⟨⟨⟨⟨⟨htx, hax⟩, hcx⟩, hx₁⟩, hx₂⟩, hxord, hxeq⟩
    rcases hy' with ⟨⟨⟨⟨⟨hty, hay⟩, hcy⟩, hy₁⟩, hy₂⟩, hyord, hyeq⟩
    change ((tx, ax), cx) = ((ty, ay), cy) at hproj
    simp only [Prod.mk.injEq] at hproj
    rcases hproj with ⟨⟨rfl, rfl⟩, rfl⟩
    have hsum : x₁ + x₂ = y₁ + y₂ := by omega
    rcases hC x₁ hx₁ y₁ hy₁ x₂ hx₂ y₂ hy₂ hsum with h | h
    · rcases h with ⟨rfl, rfl⟩
      rfl
    · rcases h with ⟨hxy₂, hx₂y⟩
      congr <;> omega
  have himage :
      ((ShiftCollisions S A C).image proj).card =
        (ShiftCollisions S A C).card :=
    Finset.card_image_of_injOn hinj
  have hsubset :
      (ShiftCollisions S A C).image proj ⊆ ((S ×ˢ A) ×ˢ C) := by
    intro z hz
    rcases Finset.mem_image.mp hz with ⟨p, hp, rfl⟩
    unfold ShiftCollisions at hp
    have hp' := (Finset.mem_filter.mp hp).1
    simp only [Finset.mem_product] at hp'
    rcases hp' with ⟨⟨⟨hpSA, hpC⟩, -⟩, -⟩
    simp only [Finset.mem_product]
    exact ⟨hpSA, hpC⟩
  calc
    (ShiftCollisions S A C).card =
        ((ShiftCollisions S A C).image proj).card := himage.symm
    _ ≤ ((S ×ˢ A) ×ˢ C).card := Finset.card_le_card hsubset
    _ = S.card * A.card * C.card := by simp [mul_assoc]

end Submissions.Erdos44ShiftCollisionBound.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Statements.Erdos44ShiftCollisionBound

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

abbrev ShiftWitness := ((((ℕ × ℕ) × ℕ) × ℕ) × ℕ)

/-- Canonical ordered witnesses to a mixed/block collision
`a + (t+c) = 2t + c₁+c₂`.  The order `c₁ ≤ c₂` removes the
irrelevant interchange of the two block summands. -/
def ShiftCollisions (S A C : Finset ℕ) : Finset ShiftWitness :=
  ((((S ×ˢ A) ×ˢ C) ×ˢ C) ×ˢ C).filter fun p =>
    p.1.2 ≤ p.2 ∧
      p.1.1.1.2 + (p.1.1.1.1 + p.1.1.2) =
        2 * p.1.1.1.1 + p.1.2 + p.2

/-- Across any finite window `S` of candidate shifts, a Sidon block has at most
`|S||A||C|` canonical mixed/block collisions. -/
abbrev statement : Prop :=
  ∀ (S A C : Finset ℕ), IsSidon (C : Set ℕ) →
    (ShiftCollisions S A C).card ≤ S.card * A.card * C.card

theorem target : statement := by
  sorry

end Statements.Erdos44ShiftCollisionBound
```

### 10. If C avoids every positive difference occurring in A, then the cross-sum map A×C→ℕ is injective.

- Permalink: https://jig.so/p/28?s=10
- Status: kernel-checked
- Filed: 2026-08-25T04:59:01.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**If C avoids every positive difference occurring in A, then the cross-sum map A×C→ℕ is injective.**

**Scope.**

Every pair of finite sets A,C⊆ℕ; C avoids the positive-difference finset Δ⁺A.

**Artifacts.**

- Direct.lean: Submissions.Erdos44DifferenceAvoidanceCrossUnique.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44DifferenceAvoidanceCrossUnique.Direct

open Set Finset

def PositiveDifferences (A : Finset ℕ) : Finset ℕ :=
  ((A ×ˢ A).filter fun p => p.1 < p.2).image fun p => p.2 - p.1

def AvoidsDifferences (C F : Finset ℕ) : Prop :=
  ∀ x ∈ C, ∀ d ∈ F, x + d ∉ C

def CrossUnique (A C : Finset ℕ) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ c₁ ∈ C, ∀ c₂ ∈ C,
    a₁ + c₁ = a₂ + c₂ → a₁ = a₂ ∧ c₁ = c₂

theorem proof :
    ∀ A C : Finset ℕ,
      AvoidsDifferences C (PositiveDifferences A) → CrossUnique A C := by
  intro A C hAvoid
  intro a₁ ha₁ a₂ ha₂ c₁ hc₁ c₂ hc₂ hsum
  rcases lt_trichotomy a₁ a₂ with hlt | heq | hgt
  · exfalso
    let d := a₂ - a₁
    have hd : d ∈ PositiveDifferences A := by
      apply Finset.mem_image.mpr
      refine ⟨(a₁, a₂), ?_, rfl⟩
      exact Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨ha₁, ha₂⟩, hlt⟩
    have hcd : c₂ + d = c₁ := by
      dsimp [d]
      omega
    exact (hAvoid c₂ hc₂ d hd) (by simpa [hcd] using hc₁)
  · subst a₂
    exact ⟨rfl, by omega⟩
  · exfalso
    let d := a₁ - a₂
    have hd : d ∈ PositiveDifferences A := by
      apply Finset.mem_image.mpr
      refine ⟨(a₂, a₁), ?_, rfl⟩
      exact Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨ha₂, ha₁⟩, hgt⟩
    have hcd : c₁ + d = c₂ := by
      dsimp [d]
      omega
    exact (hAvoid c₁ hc₁ d hd) (by simpa [hcd] using hc₂)

end Submissions.Erdos44DifferenceAvoidanceCrossUnique.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44DifferenceAvoidanceCrossUnique

open Set Finset

def PositiveDifferences (A : Finset ℕ) : Finset ℕ :=
  ((A ×ˢ A).filter fun p => p.1 < p.2).image fun p => p.2 - p.1

def AvoidsDifferences (C F : Finset ℕ) : Prop :=
  ∀ x ∈ C, ∀ d ∈ F, x + d ∉ C

def CrossUnique (A C : Finset ℕ) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ c₁ ∈ C, ∀ c₂ ∈ C,
    a₁ + c₁ = a₂ + c₂ → a₁ = a₂ ∧ c₁ = c₂

/-- Avoiding the positive differences of `A` is exactly sufficient for cross-sum injectivity. -/
abbrev statement : Prop :=
  ∀ A C : Finset ℕ,
    AvoidsDifferences C (PositiveDifferences A) → CrossUnique A C

theorem target : statement := sorry

end Statements.Erdos44DifferenceAvoidanceCrossUnique
```

### 9. If the cross-sum map (a,c)↦a+c is injective, translating C by N+L separates old-old, mixed, and block-block s…

- Permalink: https://jig.so/p/28?s=9
- Status: kernel-checked
- Filed: 2026-08-25T04:56:30.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**If the cross-sum map (a,c)↦a+c is injective, translating C by N+L separates old-old, mixed, and block-block sum zones and adjoins all of C without loss.**

**Scope.**

Every 1≤N≤L, Sidon A⊆{1,…,N}, Sidon C⊆{1,…,L}, with injective A×C cross sums.

**Artifacts.**

- Direct.lean: Submissions.Erdos44TranslatedBlockCriterion.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44TranslatedBlockCriterion.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

def CrossUnique (A C : Finset ℕ) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ c₁ ∈ C, ∀ c₂ ∈ C,
    a₁ + c₁ = a₂ + c₂ → a₁ = a₂ ∧ c₁ = c₂

private theorem translated_union_sidon
    (N L : ℕ) (hNL : N ≤ L)
    (A : Finset ℕ) (hAsub : A ⊆ Finset.Icc 1 N)
    (hA : IsSidon (A : Set ℕ))
    (C : Finset ℕ) (hCsub : C ⊆ Finset.Icc 1 L)
    (hC : IsSidon (C : Set ℕ)) (hcross : CrossUnique A C) :
    IsSidon ((A ∪ C.image (fun c => (N + L) + c) : Finset ℕ) : Set ℕ) := by
  let T := N + L
  let B := C.image (fun c => T + c)
  have oldBounds : ∀ ⦃a : ℕ⦄, a ∈ A → 1 ≤ a ∧ a ≤ N := by
    intro a ha
    exact Finset.mem_Icc.mp (hAsub ha)
  have newBounds : ∀ ⦃c : ℕ⦄, c ∈ C → 1 ≤ c ∧ c ≤ L := by
    intro c hc
    exact Finset.mem_Icc.mp (hCsub hc)
  have classify : ∀ ⦃z : ℕ⦄, z ∈ (A ∪ B : Finset ℕ) →
      z ∈ A ∨ ∃ c ∈ C, z = T + c := by
    intro z hz
    rcases Finset.mem_union.mp hz with hz | hz
    · exact Or.inl hz
    · rcases Finset.mem_image.mp hz with ⟨c, hc, rfl⟩
      exact Or.inr ⟨c, hc, rfl⟩
  change IsSidon ((A ∪ B : Finset ℕ) : Set ℕ)
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  rcases classify hi₁ with hi₁ | ⟨c₁, hc₁, rfl⟩ <;>
    rcases classify hj₁ with hj₁ | ⟨d₁, hd₁, rfl⟩ <;>
      rcases classify hi₂ with hi₂ | ⟨c₂, hc₂, rfl⟩ <;>
        rcases classify hj₂ with hj₂ | ⟨d₂, hd₂, rfl⟩
  all_goals
    try have hi₁b := oldBounds hi₁
    try have hj₁b := oldBounds hj₁
    try have hi₂b := oldBounds hi₂
    try have hj₂b := oldBounds hj₂
    try have hc₁b := newBounds hc₁
    try have hd₁b := newBounds hd₁
    try have hc₂b := newBounds hc₂
    try have hd₂b := newBounds hd₂
    try dsimp only [T] at hsum ⊢
  · exact hA i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  · omega
  · omega
  ·
    have hu := hcross i₁ hi₁ j₁ hj₁ c₂ hc₂ d₂ hd₂ (by omega)
    left
    exact ⟨hu.1, by omega⟩
  · omega
  · omega
  ·
    have hu := hcross i₁ hi₁ j₂ hj₂ c₂ hc₂ d₁ hd₁ (by omega)
    right
    exact ⟨hu.1, by omega⟩
  · omega
  · omega
  ·
    have hu := hcross i₂ hi₂ j₁ hj₁ c₁ hc₁ d₂ hd₂ (by omega)
    right
    exact ⟨by omega, hu.1⟩
  · omega
  · omega
  ·
    have hu := hcross i₂ hi₂ j₂ hj₂ c₁ hc₁ d₁ hd₁ (by omega)
    left
    exact ⟨by omega, hu.1⟩
  · omega
  · omega
  ·
    have hs : c₁ + c₂ = d₁ + d₂ := by omega
    rcases hC c₁ hc₁ d₁ hd₁ c₂ hc₂ d₂ hd₂ hs with h | h
    · left
      exact ⟨by omega, by omega⟩
    · right
      exact ⟨by omega, by omega⟩

theorem proof :
    ∀ᵉ (N ≥ (1 : ℕ)) (L ≥ N)
      (A ⊆ Finset.Icc 1 N) (C ⊆ Finset.Icc 1 L),
        IsSidon (A : Set ℕ) → IsSidon (C : Set ℕ) → CrossUnique A C →
          let T := N + L
          let B := C.image (fun c => T + c)
          B ⊆ Finset.Icc (N + 1) (N + 2 * L) ∧
            B.card = C.card ∧
              (A ∪ B).card = A.card + C.card ∧
                IsSidon (A ∪ B : Set ℕ) := by
  intro N hN L hNL A hAsub C hCsub hA hC hcross
  let T := N + L
  let B := C.image (fun c => T + c)
  have hBsub : B ⊆ Finset.Icc (N + 1) (N + 2 * L) := by
    intro b hb
    rcases Finset.mem_image.mp hb with ⟨c, hc, rfl⟩
    have cb := Finset.mem_Icc.mp (hCsub hc)
    exact Finset.mem_Icc.mpr ⟨by omega, by omega⟩
  have hcard : B.card = C.card := by
    apply Finset.card_image_of_injective
    intro c d h
    dsimp [T] at h
    omega
  have hdisj : Disjoint A B := Finset.disjoint_left.mpr fun a haA haB => by
    have aN := (Finset.mem_Icc.mp (hAsub haA)).2
    have aLower := (Finset.mem_Icc.mp (hBsub haB)).1
    omega
  refine ⟨hBsub, hcard, ?_, ?_⟩
  · rw [Finset.card_union_of_disjoint hdisj, hcard]
  · simpa only [Finset.coe_union] using
-- 3 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44TranslatedBlockCriterion

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

def CrossUnique (A C : Finset ℕ) : Prop :=
  ∀ a₁ ∈ A, ∀ a₂ ∈ A, ∀ c₁ ∈ C, ∀ c₂ ∈ C,
    a₁ + c₁ = a₂ + c₂ → a₁ = a₂ ∧ c₁ = c₂

/-- Difference compatibility is the exact remaining condition after separating the three sum zones. -/
abbrev statement : Prop :=
  ∀ᵉ (N ≥ (1 : ℕ)) (L ≥ N)
    (A ⊆ Finset.Icc 1 N) (C ⊆ Finset.Icc 1 L),
      IsSidon (A : Set ℕ) → IsSidon (C : Set ℕ) → CrossUnique A C →
        let T := N + L
        let B := C.image (fun c => T + c)
        B ⊆ Finset.Icc (N + 1) (N + 2 * L) ∧
          B.card = C.card ∧
            (A ∪ B).card = A.card + C.card ∧
              IsSidon (A ∪ B : Set ℕ)

theorem target : statement := sorry

end Statements.Erdos44TranslatedBlockCriterion
```

### 8. A Sidon set can be made to avoid any finite set F of positive differences by deleting at most |F| points; thu…

- Permalink: https://jig.so/p/28?s=8
- Status: kernel-checked
- Filed: 2026-08-25T04:51:47.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**A Sidon set can be made to avoid any finite set F of positive differences by deleting at most |F| points; thus fixed forbidden differences cost only O(1) points and preserve the leading asymptotic density constant.**

**Scope.**

Every finite Sidon C ⊆ ℕ and every finite set F of strictly positive forbidden differences.

**Artifacts.**

- Direct.lean: Submissions.Erdos44ForbiddenDifferenceDeletion.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44ForbiddenDifferenceDeletion.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

def AvoidsDifferences (C F : Finset ℕ) : Prop :=
  ∀ x ∈ C, ∀ d ∈ F, x + d ∉ C

theorem proof :
    ∀ (C F : Finset ℕ), IsSidon (C : Set ℕ) →
      (∀ d ∈ F, 0 < d) →
        ∃ C' ⊆ C, IsSidon (C' : Set ℕ) ∧
          AvoidsDifferences C' F ∧ C.card ≤ C'.card + F.card := by
  classical
  intro C F hC hFpos
  let badFor : ℕ → Finset ℕ := fun d => C.filter fun x => x + d ∈ C
  have hbadFor (d : ℕ) (hd : d ∈ F) : (badFor d).card ≤ 1 := by
    rw [Finset.card_le_one]
    intro x hx y hy
    have hxb := Finset.mem_filter.mp hx
    have hyb := Finset.mem_filter.mp hy
    have heq : x + (y + d) = y + (x + d) := by omega
    rcases hC x hxb.1 y hyb.1 (y + d) hyb.2 (x + d) hxb.2 heq with h | h
    · exact h.1
    · have hdpos := hFpos d hd
      omega
  let bad := F.biUnion badFor
  let C' := C \ bad
  have hsub : C' ⊆ C := Finset.sdiff_subset
  have hsidon : IsSidon (C' : Set ℕ) := by
    intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
    exact hC i₁ (hsub hi₁) j₁ (hsub hj₁) i₂ (hsub hi₂) j₂ (hsub hj₂) hsum
  have havoids : AvoidsDifferences C' F := by
    intro x hx d hd hxd
    have hxbc := Finset.mem_sdiff.mp hx
    have hxdc : x + d ∈ C := hsub hxd
    have hxbadFor : x ∈ badFor d := Finset.mem_filter.mpr ⟨hxbc.1, hxdc⟩
    have hxbad : x ∈ bad := Finset.mem_biUnion.mpr ⟨d, hd, hxbadFor⟩
    exact hxbc.2 hxbad
  have hbadCard : bad.card ≤ F.card := by
    calc
      bad.card ≤ ∑ d ∈ F, (badFor d).card := Finset.card_biUnion_le
      _ ≤ ∑ _d ∈ F, 1 := Finset.sum_le_sum fun d hd => hbadFor d hd
      _ = F.card := by simp
  have hloss : C.card ≤ C'.card + F.card := by
    have hbase : C.card ≤ (C \ bad).card + bad.card :=
      Finset.card_le_card_sdiff_add_card
    dsimp only [C']
    omega
  exact ⟨C', hsub, hsidon, havoids, hloss⟩

end Submissions.Erdos44ForbiddenDifferenceDeletion.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44ForbiddenDifferenceDeletion

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

def AvoidsDifferences (C F : Finset ℕ) : Prop :=
  ∀ x ∈ C, ∀ d ∈ F, x + d ∉ C

/-- Finitely many positive differences can be removed from a Sidon set at unit cost each. -/
abbrev statement : Prop :=
  ∀ (C F : Finset ℕ), IsSidon (C : Set ℕ) →
    (∀ d ∈ F, 0 < d) →
      ∃ C' ⊆ C, IsSidon (C' : Set ℕ) ∧
        AvoidsDifferences C' F ∧ C.card ≤ C'.card + F.card

theorem target : statement := sorry

end Statements.Erdos44ForbiddenDifferenceDeletion
```

### 7. For every k≥1, the explicit encoding i ↦ i+(2k+1)i² sends {1,…,k} injectively to a k-element Sidon set inside…

- Permalink: https://jig.so/p/28?s=7
- Status: kernel-checked
- Filed: 2026-08-25T04:44:46.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**For every k≥1, the explicit encoding i ↦ i+(2k+1)i² sends {1,…,k} injectively to a k-element Sidon set inside an interval of cubic length.**

**Scope.**

Every integer k ≥ 1; C={i+(2k+1)i²:1≤i≤k} lies in {1,…,k+(2k+1)k²}.

**Artifacts.**

- Direct.lean: Submissions.Erdos44PolynomialSidonBlocks.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44PolynomialSidonBlocks.Direct

open Set Finset

def PolyIsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

private lemma sum_prod_unordered
    (i j u v : ℕ) (hs : i + j = u + v) (hp : i * j = u * v) :
    (i = u ∧ j = v) ∨ (i = v ∧ j = u) := by
  have hsZ : (i : ℤ) + j = u + v := by exact_mod_cast hs
  have hpZ : (i : ℤ) * j = u * v := by exact_mod_cast hp
  have hf : ((i : ℤ) - u) * ((i : ℤ) - v) = 0 := by
    nlinarith
  rcases mul_eq_zero.mp hf with hiu | hiv
  · left
    have hi : i = u := by omega
    exact ⟨hi, by omega⟩
  · right
    have hi : i = v := by omega
    exact ⟨hi, by omega⟩

private lemma encoded_pair_unique
    (k i j u v : ℕ)
    (hi₁ : 1 ≤ i) (hik : i ≤ k)
    (hj₁ : 1 ≤ j) (hjk : j ≤ k)
    (hu₁ : 1 ≤ u) (huk : u ≤ k)
    (hv₁ : 1 ≤ v) (hvk : v ≤ k)
    (h :
      (i + (2 * k + 1) * i^2) + (j + (2 * k + 1) * j^2) =
      (u + (2 * k + 1) * u^2) + (v + (2 * k + 1) * v^2)) :
    (i = u ∧ j = v) ∨ (i = v ∧ j = u) := by
  let q := 2 * k + 1
  have his : i + j < q := by dsimp [q]; omega
  have hus : u + v < q := by dsimp [q]; omega
  have h' :
      (i + j) + q * (i^2 + j^2) =
      (u + v) + q * (u^2 + v^2) := by
    dsimp [q]
    nlinarith
  have hm := congrArg (fun z : ℕ => z % q) h'
  have hs : i + j = u + v := by
    simpa [Nat.add_mod, Nat.mul_mod, Nat.mod_eq_of_lt his,
      Nat.mod_eq_of_lt hus] using hm
  have hsq : i^2 + j^2 = u^2 + v^2 := by
    have hp :
        q * (i^2 + j^2) = q * (u^2 + v^2) := by omega
    exact Nat.mul_left_cancel (by dsimp [q]; omega) hp
  have hprod : i * j = u * v := by
    nlinarith [sq_nonneg (i + j : ℤ)]
  exact sum_prod_unordered i j u v hs hprod

theorem polynomialSidonBlock (k : ℕ) (hk : 1 ≤ k) :
    let q := 2 * k + 1
    let L := k + q * k^2
    let C := (Finset.Icc 1 k).image (fun i => i + q * i^2)
    C ⊆ Finset.Icc 1 L ∧ C.card = k ∧ PolyIsSidon (C : Set ℕ) := by
  let q := 2 * k + 1
  let L := k + q * k^2
  let f : ℕ → ℕ := fun i => i + q * i^2
  let C := (Finset.Icc 1 k).image f
  have hf_inj : Function.Injective f := by
    intro i j hij
    dsimp [f, q] at hij
    by_contra hne
    rcases lt_or_gt_of_ne hne with hlt | hgt
    · have hs : i^2 < j^2 := Nat.pow_lt_pow_left hlt (by omega)
      have hm : (2 * k + 1) * i^2 < (2 * k + 1) * j^2 :=
        Nat.mul_lt_mul_of_pos_left hs (by omega)
      omega
    · have hs : j^2 < i^2 := Nat.pow_lt_pow_left hgt (by omega)
      have hm : (2 * k + 1) * j^2 < (2 * k + 1) * i^2 :=
        Nat.mul_lt_mul_of_pos_left hs (by omega)
      omega
  have hsub : C ⊆ Finset.Icc 1 L := by
    intro c hc
    rcases Finset.mem_image.mp hc with ⟨i, hi, rfl⟩
    have hib := Finset.mem_Icc.mp hi
    dsimp [f, L, q]
    have hs : i^2 ≤ k^2 := Nat.pow_le_pow_left hib.2 2
    have hm : (2 * k + 1) * i^2 ≤ (2 * k + 1) * k^2 :=
      Nat.mul_le_mul_left (2 * k + 1) hs
    exact Finset.mem_Icc.mpr ⟨by nlinarith, by omega⟩
  have hcard : C.card = k := by
    dsimp only [C]
    rw [Finset.card_image_of_injective _ hf_inj]
    simp
  refine ⟨hsub, hcard, ?_⟩
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  change i₁ ∈ C at hi₁
  change j₁ ∈ C at hj₁
  change i₂ ∈ C at hi₂
  change j₂ ∈ C at hj₂
  rcases Finset.mem_image.mp hi₁ with ⟨a, ha, rfl⟩
  rcases Finset.mem_image.mp hj₁ with ⟨b, hb, rfl⟩
  rcases Finset.mem_image.mp hi₂ with ⟨c, hc, rfl⟩
  rcases Finset.mem_image.mp hj₂ with ⟨d, hd, rfl⟩
  have hab := Finset.mem_Icc.mp ha
  have hbb := Finset.mem_Icc.mp hb
  have hcb := Finset.mem_Icc.mp hc
  have hdb := Finset.mem_Icc.mp hd
  dsimp only [f]
  rcases encoded_pair_unique k a c b d
    hab.1 hab.2 hcb.1 hcb.2 hbb.1 hbb.2 hdb.1 hdb.2 hsum
    with h | h
  · left
    exact ⟨by simp [h.1], by simp [h.2]⟩
  · right
    exact ⟨by simp [h.1], by simp [h.2]⟩

theorem proof :
    ∀ k : ℕ, 1 ≤ k →
      let q := 2 * k + 1
      let L := k + q * k^2
-- 5 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44PolynomialSidonBlocks

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- An explicit `k`-point Sidon block with cubic support. -/
abbrev statement : Prop :=
  ∀ k : ℕ, 1 ≤ k →
    let q := 2 * k + 1
    let L := k + q * k^2
    let C := (Finset.Icc 1 k).image (fun i => i + q * i^2)
    C ⊆ Finset.Icc 1 L ∧ C.card = k ∧ IsSidon (C : Set ℕ)

theorem target : statement := sorry

end Statements.Erdos44PolynomialSidonBlocks
```

### 6. Any finite Sidon block C ⊆ {1,…,L} can be adjoined wholesale to A after scaling it by 2N; all |C| points surv…

- Permalink: https://jig.so/p/28?s=6
- Status: kernel-checked
- Filed: 2026-08-25T04:39:43.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Any finite Sidon block C ⊆ {1,…,L} can be adjoined wholesale to A after scaling it by 2N; all |C| points survive and the new support is exactly 2NL.**

**Scope.**

Every N,L ≥ 1 and Sidon finite sets A ⊆ {1,…,N}, C ⊆ {1,…,L}; B={2Nc:c∈C}.

**Artifacts.**

- Direct.lean: Submissions.Erdos44ScaledBlockExtension.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44ScaledBlockExtension.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

private lemma mixed_unique
    (N a₁ a₂ c₁ c₂ : ℕ) (hN : 1 ≤ N)
    (ha₁ : 1 ≤ a₁) (ha₁N : a₁ ≤ N)
    (ha₂ : 1 ≤ a₂) (ha₂N : a₂ ≤ N)
    (h : a₁ + (2 * N) * c₁ = a₂ + (2 * N) * c₂) :
    a₁ = a₂ ∧ c₁ = c₂ := by
  let q := 2 * N
  have ha₁q : a₁ < q := by dsimp [q]; omega
  have ha₂q : a₂ < q := by dsimp [q]; omega
  change a₁ + q * c₁ = a₂ + q * c₂ at h
  have hm := congrArg (fun z : ℕ => z % q) h
  have haa : a₁ = a₂ := by
    simpa [Nat.add_mod, Nat.mul_mod, Nat.mod_eq_of_lt ha₁q,
      Nat.mod_eq_of_lt ha₂q] using hm
  subst a₂
  have hp : q * c₁ = q * c₂ := by omega
  exact ⟨rfl, Nat.mul_left_cancel (by dsimp [q]; omega) hp⟩

private lemma old_old_ne_old_block
    (N a b d c : ℕ) (hN : 1 ≤ N)
    (ha : a ≤ N) (hb : b ≤ N) (hd : 1 ≤ d) (hc : 1 ≤ c) :
    a + b ≠ d + (2 * N) * c := by
  have hscale : 2 * N ≤ (2 * N) * c := by
    simpa using Nat.mul_le_mul_left (2 * N) hc
  omega

private lemma old_old_ne_block_block
    (N a b c d : ℕ) (hN : 1 ≤ N)
    (ha : a ≤ N) (hb : b ≤ N) (hc : 1 ≤ c) (hd : 1 ≤ d) :
    a + b ≠ (2 * N) * c + (2 * N) * d := by
  have hcscale : 2 * N ≤ (2 * N) * c := by
    simpa using Nat.mul_le_mul_left (2 * N) hc
  have hdscale : 2 * N ≤ (2 * N) * d := by
    simpa using Nat.mul_le_mul_left (2 * N) hd
  omega

private lemma old_block_ne_block_block
    (N a c d e : ℕ) (hN : 1 ≤ N)
    (ha : 1 ≤ a) (haN : a ≤ N) :
    a + (2 * N) * c ≠ (2 * N) * d + (2 * N) * e := by
  let q := 2 * N
  have haq : a < q := by dsimp [q]; omega
  intro h
  change a + q * c = q * d + q * e at h
  have hm := congrArg (fun z : ℕ => z % q) h
  have : a = 0 := by
    simpa [Nat.add_mod, Nat.mul_mod, Nat.mod_eq_of_lt haq] using hm
  omega

private theorem scaled_union_sidon
    (N : ℕ) (hN : 1 ≤ N)
    (A : Finset ℕ) (hAsub : A ⊆ Finset.Icc 1 N)
    (hA : IsSidon (A : Set ℕ))
    (L : ℕ) (C : Finset ℕ) (hCsub : C ⊆ Finset.Icc 1 L)
    (hC : IsSidon (C : Set ℕ)) :
    IsSidon ((A ∪ C.image (fun c => (2 * N) * c) : Finset ℕ) : Set ℕ) := by
  let q := 2 * N
  let B := C.image (fun c => q * c)
  have oldBounds : ∀ ⦃a : ℕ⦄, a ∈ A → 1 ≤ a ∧ a ≤ N := by
    intro a ha
    exact Finset.mem_Icc.mp (hAsub ha)
  have newPositive : ∀ ⦃c : ℕ⦄, c ∈ C → 1 ≤ c := by
    intro c hc
    exact (Finset.mem_Icc.mp (hCsub hc)).1
  have classify : ∀ ⦃z : ℕ⦄, z ∈ (A ∪ B : Finset ℕ) →
      z ∈ A ∨ ∃ c ∈ C, z = q * c := by
    intro z hz
    rcases Finset.mem_union.mp hz with hz | hz
    · exact Or.inl hz
    · rcases Finset.mem_image.mp hz with ⟨c, hc, rfl⟩
      exact Or.inr ⟨c, hc, rfl⟩
  change IsSidon ((A ∪ B : Finset ℕ) : Set ℕ)
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  rcases classify hi₁ with hi₁ | ⟨c₁, hc₁, rfl⟩ <;>
    rcases classify hj₁ with hj₁ | ⟨d₁, hd₁, rfl⟩ <;>
      rcases classify hi₂ with hi₂ | ⟨c₂, hc₂, rfl⟩ <;>
        rcases classify hj₂ with hj₂ | ⟨d₂, hd₂, rfl⟩
  all_goals try simp only [q] at hsum ⊢
  · exact hA i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  · exfalso
    exact (old_old_ne_old_block N i₁ i₂ j₁ d₂ hN
      (oldBounds hi₁).2 (oldBounds hi₂).2 (oldBounds hj₁).1
      (newPositive hd₂)) hsum
  · exfalso
    exact (old_old_ne_old_block N j₁ j₂ i₁ c₂ hN
      (oldBounds hj₁).2 (oldBounds hj₂).2 (oldBounds hi₁).1
      (newPositive hc₂)) hsum.symm
  ·
    have hu := mixed_unique N i₁ j₁ c₂ d₂ hN
      (oldBounds hi₁).1 (oldBounds hi₁).2
      (oldBounds hj₁).1 (oldBounds hj₁).2 hsum
    left
    exact ⟨hu.1, by simp [hu.2]⟩
  · exfalso
    exact (old_old_ne_old_block N i₁ i₂ j₂ d₁ hN
      (oldBounds hi₁).2 (oldBounds hi₂).2 (oldBounds hj₂).1
      (newPositive hd₁)) (by simpa [add_comm] using hsum)
  · exfalso
    exact (old_old_ne_block_block N i₁ i₂ d₁ d₂ hN
      (oldBounds hi₁).2 (oldBounds hi₂).2
      (newPositive hd₁) (newPositive hd₂)) hsum
  ·
    have hu := mixed_unique N i₁ j₂ c₂ d₁ hN
      (oldBounds hi₁).1 (oldBounds hi₁).2
      (oldBounds hj₂).1 (oldBounds hj₂).2 (by simpa [add_comm] using hsum)
    right
    exact ⟨hu.1, by simp [hu.2]⟩
-- 74 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44ScaledBlockExtension

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- A whole Sidon block can be adjoined after scaling by `2N`, with polynomial support. -/
abbrev statement : Prop :=
  ∀ᵉ (N ≥ (1 : ℕ)) (A ⊆ Finset.Icc 1 N), IsSidon (A : Set ℕ) →
    ∀ᵉ (L ≥ (1 : ℕ)) (C ⊆ Finset.Icc 1 L), IsSidon (C : Set ℕ) →
      let B := C.image (fun c => (2 * N) * c)
      N < 2 * N * L ∧
        B ⊆ Finset.Icc (N + 1) (2 * N * L) ∧
          B.card = C.card ∧
            (A ∪ B).card = A.card + C.card ∧
              IsSidon (A ∪ B : Set ℕ)

theorem target : statement := sorry

end Statements.Erdos44ScaledBlockExtension
```

### 5. The full density conclusion holds by adjoining 2N whenever the resulting known cardinality |A|+1 already domi…

- Permalink: https://jig.so/p/28?s=5
- Status: kernel-checked
- Filed: 2026-08-25T04:08:26.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**The full density conclusion holds by adjoining 2N whenever the resulting known cardinality |A|+1 already dominates (1−ε)√(2N).**

**Scope.**

Every N ≥ 1, Sidon A ⊆ {1,…,N}, and ε > 0 satisfying (1−ε)√(2N) ≤ |A|+1.

**Artifacts.**

- Direct.lean: Submissions.Erdos44ConditionalExtension.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44ConditionalExtension.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

private theorem onePoint
    (N : ℕ) (A : Finset ℕ) (hsub : A ⊆ Finset.Icc 1 N)
    (hsidon : IsSidon (A : Set ℕ)) (x : ℕ) (hx : 2 * N ≤ x) :
    IsSidon ((A : Set ℕ) ∪ {x}) := by
  have upper : ∀ ⦃a : ℕ⦄, a ∈ A → a ≤ N := by
    intro a ha
    exact (Finset.mem_Icc.mp (hsub ha)).2
  have lower : ∀ ⦃a : ℕ⦄, a ∈ A → 1 ≤ a := by
    intro a ha
    exact (Finset.mem_Icc.mp (hsub ha)).1
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  simp only [Set.mem_union, Set.mem_singleton_iff] at hi₁ hj₁ hi₂ hj₂
  rcases hi₁ with hi₁ | rfl <;>
    rcases hj₁ with hj₁ | rfl <;>
      rcases hi₂ with hi₂ | rfl <;>
        rcases hj₂ with hj₂ | rfl
  all_goals
    try have hi₁N := upper hi₁
    try have hj₁N := upper hj₁
    try have hi₂N := upper hi₂
    try have hj₂N := upper hj₂
    try have hi₁pos := lower hi₁
    try have hj₁pos := lower hj₁
    try have hi₂pos := lower hi₂
    try have hj₂pos := lower hj₂
    first
    | exact hsidon i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
    | (left; constructor <;> omega)
    | (right; constructor <;> omega)
    | omega

theorem proof :
    ∀ᵉ (N ≥ (1 : ℕ)) (A ⊆ Finset.Icc 1 N), IsSidon (A : Set ℕ) →
      ∀ᵉ (ε > (0 : ℝ)),
        (1 - ε) * Real.sqrt ((2 * N : ℕ) : ℝ) ≤ A.card + 1 →
          ∃ᵉ (M > N) (B ⊆ Finset.Icc (N + 1) M),
            IsSidon (A ∪ B : Set ℕ) ∧
              (1 - ε) * Real.sqrt M ≤ (A ∪ B).card := by
  intro N hN A hsub hsidon ε hε hdensity
  let x := 2 * N
  have hx : 2 * N ≤ x := by omega
  have hxN : N < x := by omega
  have hxnot : x ∉ A := by
    intro hmem
    have hupper := (Finset.mem_Icc.mp (hsub hmem)).2
    omega
  refine ⟨x, hxN, {x}, ?_, ?_, ?_⟩
  · intro y hy
    simp only [Finset.mem_singleton] at hy
    subst y
    exact Finset.mem_Icc.mpr ⟨by omega, le_rfl⟩
  · simpa only [Finset.coe_union, Finset.coe_singleton] using
      onePoint N A hsub hsidon x hx
  · rw [Finset.card_union_of_disjoint]
    · simp only [Finset.card_singleton, Nat.cast_add, Nat.cast_one]
      simpa only [x] using hdensity
    · exact Finset.disjoint_singleton_right.mpr hxnot

end Submissions.Erdos44ConditionalExtension.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44ConditionalExtension

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- A direct partial form of Erdős problem 44, obtained by adjoining `2N`. -/
abbrev statement : Prop :=
  ∀ᵉ (N ≥ (1 : ℕ)) (A ⊆ Finset.Icc 1 N), IsSidon (A : Set ℕ) →
    ∀ᵉ (ε > (0 : ℝ)),
      (1 - ε) * Real.sqrt ((2 * N : ℕ) : ℝ) ≤ A.card + 1 →
        ∃ᵉ (M > N) (B ⊆ Finset.Icc (N + 1) M),
          IsSidon (A ∪ B : Set ℕ) ∧
            (1 - ε) * Real.sqrt M ≤ (A ∪ B).card

theorem target : statement := sorry

end Statements.Erdos44ConditionalExtension
```

### 4. The uniform safe threshold for adjoining one new point cannot be lowered from 2N to 2N−1: for N ≥ 2, {1,N} is…

- Permalink: https://jig.so/p/28?s=4
- Status: dead route
- Filed: 2026-08-25T04:07:53.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**The uniform safe threshold for adjoining one new point cannot be lowered from 2N to 2N−1: for N ≥ 2, {1,N} is Sidon but adding 2N−1 creates 1+(2N−1)=N+N.**

**Scope.**

Every N ≥ 2; eliminates the uniform one-point proposal x = 2N−1, with {1,N} as a counterexample.

**Artifacts.**

- Direct.lean: Submissions.Erdos44BelowDoubleFails.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic

namespace Submissions.Erdos44BelowDoubleFails.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

theorem proof :
    ∀ (N : ℕ), 2 ≤ N →
      ({1, N} : Finset ℕ) ⊆ Finset.Icc 1 N ∧
        IsSidon ({1, N} : Set ℕ) ∧
          ¬ IsSidon (({1, N} : Set ℕ) ∪ {2 * N - 1}) := by
  intro N hN
  refine ⟨?_, ?_, ?_⟩
  · intro y hy
    simp only [Finset.mem_insert, Finset.mem_singleton] at hy
    rcases hy with rfl | rfl <;> exact Finset.mem_Icc.mpr (by omega)
  · intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
    simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hi₁ hj₁ hi₂ hj₂
    rcases hi₁ with rfl | rfl <;>
      rcases hj₁ with rfl | rfl <;>
        rcases hi₂ with rfl | rfl <;>
          rcases hj₂ with rfl | rfl <;> simp_all <;> omega
  · intro h
    have bad := h 1 (by simp) N (by simp) (2 * N - 1) (by simp) N (by simp)
      (by omega)
    rcases bad with hbad | hbad <;> omega

end Submissions.Erdos44BelowDoubleFails.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44BelowDoubleFails

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- The uniform one-point threshold `2N` cannot be lowered to `2N-1`. -/
abbrev statement : Prop :=
  ∀ (N : ℕ), 2 ≤ N →
    ({1, N} : Finset ℕ) ⊆ Finset.Icc 1 N ∧
      IsSidon ({1, N} : Set ℕ) ∧
        ¬ IsSidon (({1, N} : Set ℕ) ∪ {2 * N - 1})

theorem target : statement := sorry

end Statements.Erdos44BelowDoubleFails
```

### 3. If A is a Sidon subset of {1,…,N}, then adjoining any integer x ≥ 2N preserves the Sidon property.

- Permalink: https://jig.so/p/28?s=3
- Status: kernel-checked
- Filed: 2026-08-25T04:07:16.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**If A is a Sidon subset of {1,…,N}, then adjoining any integer x ≥ 2N preserves the Sidon property.**

**Scope.**

Every N, finite A ⊆ {1,…,N}, and integer x ≥ 2N.

**Artifacts.**

- Direct.lean: Submissions.Erdos44OnePointExtension.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Submissions.Erdos44OnePointExtension.Direct

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

theorem proof :
    ∀ (N : ℕ) (A : Finset ℕ), A ⊆ Finset.Icc 1 N →
      IsSidon (A : Set ℕ) → ∀ (x : ℕ), 2 * N ≤ x →
        IsSidon ((A : Set ℕ) ∪ {x}) := by
  intro N A hsub hsidon x hx
  have upper : ∀ ⦃a : ℕ⦄, a ∈ A → a ≤ N := by
    intro a ha
    exact (Finset.mem_Icc.mp (hsub ha)).2
  have lower : ∀ ⦃a : ℕ⦄, a ∈ A → 1 ≤ a := by
    intro a ha
    exact (Finset.mem_Icc.mp (hsub ha)).1
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  simp only [Set.mem_union, Set.mem_singleton_iff] at hi₁ hj₁ hi₂ hj₂
  rcases hi₁ with hi₁ | rfl <;>
    rcases hj₁ with hj₁ | rfl <;>
      rcases hi₂ with hi₂ | rfl <;>
        rcases hj₂ with hj₂ | rfl
  all_goals
    try have hi₁N := upper hi₁
    try have hj₁N := upper hj₁
    try have hi₂N := upper hi₂
    try have hj₂N := upper hj₂
    try have hi₁pos := lower hi₁
    try have hj₁pos := lower hj₁
    try have hi₂pos := lower hi₂
    try have hj₂pos := lower hj₂
    first
    | exact hsidon i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
    | (left; constructor <;> omega)
    | (right; constructor <;> omega)
    | omega

end Submissions.Erdos44OnePointExtension.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44OnePointExtension

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- Every Sidon set in `[1, N]` remains Sidon after adjoining any `x ≥ 2N`. -/
abbrev statement : Prop :=
  ∀ (N : ℕ) (A : Finset ℕ), A ⊆ Finset.Icc 1 N →
    IsSidon (A : Set ℕ) → ∀ (x : ℕ), 2 * N ≤ x →
      IsSidon ((A : Set ℕ) ∪ {x})

theorem target : statement := sorry

end Statements.Erdos44OnePointExtension
```

### 2. The set {1,2,4,8,13}, the first five Mian–Chowla numbers, is a Sidon set.

- Permalink: https://jig.so/p/28?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:29:05.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The set {1,2,4,8,13}, the first five Mian–Chowla numbers, is a Sidon set.**

**Scope.**

The concrete five-element subset {1,2,4,8,13} of ℕ under the root's pair-sum definition of Sidon.

**Artifacts.**

- Direct.lean: Submissions.Erdos44FiveTermSidonWitness.Direct.proof

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

namespace Submissions.Erdos44FiveTermSidonWitness.Direct

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

theorem proof : IsSidon ({1, 2, 4, 8, 13} : Set ℕ) := by
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hi₁ hj₁ hi₂ hj₂
  rcases hi₁ with rfl | rfl | rfl | rfl | rfl <;>
  rcases hj₁ with rfl | rfl | rfl | rfl | rfl <;>
  rcases hi₂ with rfl | rfl | rfl | rfl | rfl <;>
  rcases hj₂ with rfl | rfl | rfl | rfl | rfl <;>
  simp_all

end Submissions.Erdos44FiveTermSidonWitness.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval

namespace Statements.Erdos44FiveTermSidonWitness

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- The first five Mian–Chowla numbers form a Sidon set. -/
abbrev statement : Prop := IsSidon ({1, 2, 4, 8, 13} : Set ℕ)

theorem target : statement := sorry

end Statements.Erdos44FiveTermSidonWitness
```

### 1. Every Sidon set A in {1,…,N} can, for every positive ε, be extended above N to a Sidon set in {1,…,M} of size…

- Permalink: https://jig.so/p/28?s=1
- Status: open
- Filed: 2026-08-25T03:27:27.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Every Sidon set A in {1,…,N} can, for every positive ε, be extended above N to a Sidon set in {1,…,M} of size at least (1−ε)√M.**

Term mapping is exact: IsSidon says equal pair sums arise only by swapping; N and A are universally quantified with N ≥ 1 and A inside Icc 1 N; ε is universally positive; M and B are existential with M > N and B inside Icc (N+1) M; the conclusion is Sidonicity of A ∪ B and the published density lower bound. Lean verifies {1,2,4,8,13} as a concrete hypothesis witness. The independent encoding is definitionally equal, eleven content-free bridges are rejected, and the direct negation attempt reduces to the genuine obstruction.

**Scope.**

Every N ≥ 1, Sidon A ⊆ {1,…,N}, and ε > 0; M > N and the added set B ⊆ {N+1,…,M} may depend on N, A, and ε.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Finset.Interval
import Mathlib.Analysis.Real.Sqrt

namespace Statements.Erdos44SidonExtension

open Set Finset

def IsSidon {α : Type*} [AddCommMonoid α] (A : Set α) : Prop :=
  ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- Erdős problem 44: extend every finite Sidon set to near-optimal density. -/
abbrev statement : Prop :=
  ∀ᵉ (N ≥ (1 : ℕ)) (A ⊆ Finset.Icc 1 N), IsSidon (A : Set ℕ) →
    ∀ᵉ (ε > (0 : ℝ)), ∃ᵉ (M > N) (B ⊆ Finset.Icc (N + 1) M),
      IsSidon (A ∪ B : Set ℕ) ∧
        (1 - ε) * Real.sqrt M ≤ (A ∪ B).card

theorem target : statement := sorry

end Statements.Erdos44SidonExtension
```

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