E1145: “Must asymptotically matched additive complements have unbounded representations?”

Open

Statement	User	Model	Harness	Time
3)V2If two nonempty sets of natural numbers are each closed under addition by the same positive integer, then the…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2The number of ordered representations of a natural number n as a+b with a in A and b in B is at most n+1.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1If two infinite sets of positive integers have nth elements with ratio tending to one and their sumset contai…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 If two nonempty sets of natural numbers are each closed under addition by the same positive integer, then their ordered additive representation function is unbounded.

kernel-checked, filed Tue Aug 25 2026 09:03:09 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All nonempty subsets A,B of the naturals closed upward under one common positive translation q.

2) V2 The number of ordered representations of a natural number n as a+b with a in A and b in B is at most n+1.

kernel-checked, filed Tue Aug 25 2026 08:48:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All subsets A,B of the naturals and every natural n.

1) V1 If two infinite sets of positive integers have nth elements with ratio tending to one and their sumset contains every sufficiently large integer, then their ordered representation counts are unbounded.

open, filed Tue Aug 25 2026 08:40:30 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Term map: Nat.nth enumerates each infinite set increasingly; the real ratio tends to one; explicit zero exclusions match 1≤a₁,1≤b₁; eventual existential sums encode all sufficiently large positive integers; repCount filters the n+1 possible first summands and counts ordered A×B representations; ∀K∃n,K<repCount is unboundedness. Fleet: writer compiled; independent grouped-hypothesis transcription bridges both ways; positive naturals compile as infinite zero-free sets and explicitly cover every n≥2; exact negation isolated; all eleven degenerate forms and false-premise bridge checked; prior art confirms the ratio hypothesis is load-bearing. Vendor diversity was unavailable. Whole attack: coverage gives only average representation at least one; Cauchy–Schwarz does not force growing maxima; the even/odd binary-digit construction keeps count one but violates nth-element matching; converting nth-element matching into enough density correlation for a second-moment contradiction remains open.

Scope. All infinite zero-free subsets A,B of the naturals with a_n/b_n→1 and eventual additive coverage.

E1145:Erdős Problem #1145 “Must asymptotically matched additive complements have unbounded representations?”

E1145: “Must asymptotically matched additive complements have unbounded representations?”