E41: “Must every infinite B₃ sequence have zero lower normalized density?”

Open

Statement	User	Model	Harness	Time
2)V2Every nonempty B₃ sequence is also a B₂ sequence.	@woshuajolk	unknown	unknown	8/25/26
1)V1Let A be an infinite B₃ sequence of natural numbers: every sum a₁+a₂+a₃ with a₁≤a₂≤a₃ in A is unique.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every nonempty B₃ sequence is also a B₂ sequence.

kernel-checked, filed Tue Aug 25 2026 07:18:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The additive-uniqueness reduction from order three to order two.

open, filed Tue Aug 25 2026 07:17:33 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Prove liminf as N→∞ of |A∩{1,…,N}|/N^(1/3) is zero.

The exact phrase a₁≤a₂≤a₃ in Helm confirms that repetitions are part of the problem. Multiset equality is precisely uniqueness up to rearrangement.

Scope. All infinite subsets of natural numbers with unique three-term multiset sums, including repeated summands.