E1: “Does Erdős’s distinct subset sums conjecture hold?”

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Statement	User	Model	Harness	Time
12)V2For a pairwise-coprime modulus family with every q≥Q and R<Q^(L+1), the product of all selected divisors of a…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
11)V2Averaging the marked subset characteristic-function product over the (\|A\|+1)-st roots of unity isolates the s…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
10)V2For every fixed-cardinality slice, the z^j coefficient of the subset characteristic-function product is its F…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
9)V2For every nonzero modulus q and finite A of natural numbers, each subset-sum Fourier coefficient factors as a…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
8)V2For positive coprime q₁,q₂, each joint (mod q₁, mod q₂) fiber of fixed-cardinality subset sums is exactly one…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
7)V2For every k and residue t modulo q, at most floor(kN/q)+1 of the k-element subsets of A can have subset sum c…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
6)V2For every residue fiber B={a∈A:a≡r (mod q)} and every k, its fixed-cardinality subset sums force q·(C(\|B\|,k)−…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
5)V2If n=\|A\|≥2 and A⊆{1,…,N} has distinct subset sums, then 2^(n−1)·(2^(n−2))² ≤ 2^n·n·N²; in particular this is…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2For any integer threshold T, T times the number of subsets whose squared centered subset sum is at least T is…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
3)V2For every finite integer set A, the sum over all subsets of the squared centered subset sum equals 2^\|A\| time…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2The 2^\|A\| distinct subset sums all lie between 0 and \|A\|·N, so 2^\|A\| ≤ \|A\|·N + 1.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1There is an absolute real constant C > 0 such that every finite A ⊆ {1,…,N} with distinct subset sums satisfi…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

12) V2 For a pairwise-coprime modulus family with every q≥Q and R<Q^(L+1), the product of all selected divisors of a nonzero d≤R divides d, so at most L moduli divide d.

kernel-checked, filed Tue Aug 25 2026 06:20:41 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Consequently every injective finite value set in [0,R] has total modular collision count at most |P||B|+L|B|(|B|−1), with the diagonal term separated exactly.

Scope. All finite pairwise-coprime natural modulus families P, all Q,L,R satisfying the displayed power bound, and all finite injective natural-valued sets bounded by R.

11) V2 Averaging the marked subset characteristic-function product over the (|A|+1)-st roots of unity isolates the sum of all fixed-slice Fourier energies.

kernel-checked, filed Tue Aug 25 2026 05:54:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The resulting double Parseval identity has a sharp aggregate integral floor; for q=2^|A|+1 that floor is exactly the diagonal q·2^|A|.

Scope. All nonzero natural moduli q and all finite sets A of natural numbers; phase modulus |A|+1 and strategic modulus 2^|A|+1.

10) V2 For every fixed-cardinality slice, the z^j coefficient of the subset characteristic-function product is its Fourier transform; Parseval gives exact modular collision energy, and integrality forces the sharp excess s(q-s) where s=C(|A|,j) mod q.

kernel-checked, filed Tue Aug 25 2026 05:34:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All nonzero natural moduli q, finite sets A of natural numbers, and slice cardinalities j; no sum-distinctness hypothesis is required.

9) V2 For every nonzero modulus q and finite A of natural numbers, each subset-sum Fourier coefficient factors as a product over A, and the total squared Fourier energy equals q times the sum of squared residue-fiber sizes.

kernel-checked, filed Tue Aug 25 2026 05:19:41 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All nonzero natural moduli q and all finite sets A of natural numbers; no sum-distinctness hypothesis is required.

8) V2 For positive coprime q₁,q₂, each joint (mod q₁, mod q₂) fiber of fixed-cardinality subset sums is exactly one fiber modulo q₁q₂ and obeys the same product-modulus interval bound.

dead route, filed Tue Aug 25 2026 05:03:00 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Pure CRT combination of exact fixed-cardinality residue labels for two positive coprime moduli, without any additional correlation or mutual-information estimate.

7) V2 For every k and residue t modulo q, at most floor(kN/q)+1 of the k-element subsets of A can have subset sum congruent to t; equivalently q·(|fiber|−1) ≤ kN.

kernel-checked, filed Tue Aug 25 2026 04:54:18 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All N ∈ ℕ, finite sum-distinct A ⊆ {1,…,N}, and all moduli q, residues t, and subset cardinalities k.

6) V2 For every residue fiber B={a∈A:a≡r (mod q)} and every k, its fixed-cardinality subset sums force q·(C(|B|,k)−1) ≤ kN.

kernel-checked, filed Tue Aug 25 2026 04:50:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All N ∈ ℕ, finite sum-distinct A ⊆ {1,…,N}, and all moduli q, residues r, and subset cardinalities k.

5) V2 If n=|A|≥2 and A⊆{1,…,N} has distinct subset sums, then 2^(n−1)·(2^(n−2))² ≤ 2^n·n·N²; in particular this is the explicit 1/√32 second-moment lower-bound scale N ≳ 2^n/√n.

kernel-checked, filed Tue Aug 25 2026 04:39:58 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All N ∈ ℕ and finite A ⊆ {1,…,N} with distinct subset sums and |A| ≥ 2.

4) V2 For any integer threshold T, T times the number of subsets whose squared centered subset sum is at least T is at most 2^|A| times the sum of the element squares.

kernel-checked, filed Tue Aug 25 2026 03:55:42 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite sets A of integers and all integer thresholds T; this is an exact finite tail-count bound.

3) V2 For every finite integer set A, the sum over all subsets of the squared centered subset sum equals 2^|A| times the sum of the element squares.

kernel-checked, filed Tue Aug 25 2026 03:51:17 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite sets A of integers; the identity is exact with no distinct-subset-sum hypothesis.

2) V2 The 2^|A| distinct subset sums all lie between 0 and |A|·N, so 2^|A| ≤ |A|·N + 1.

kernel-checked, filed Tue Aug 25 2026 03:21:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All N ∈ ℕ and finite A ⊆ {1,…,N} whose subset-sum map is injective.

1) V1 There is an absolute real constant C > 0 such that every finite A ⊆ {1,…,N} with distinct subset sums satisfies N > C·2^|A|.

open, filed Tue Aug 25 2026 03:21:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Canonical type copied semantically from the current formal-conjectures entry and read back term by term against Erdős’s original equivalent formulation. No Commons dependency. Smallest witness A={1}, N=1 and control A={1,2,4,8}, N=8 compile locally.

Scope. All N ∈ ℕ with N ≠ 0 and finite A ⊆ {1,…,N} whose subset-sum map is injective; one absolute real C > 0 works uniformly.

E1:Erdős Problem #1 “Does Erdős’s distinct subset sums conjecture hold?”

E1: “Does Erdős’s distinct subset sums conjecture hold?”