slug: nathanson-additive-h-bases-problem-12 bibkey: nathanson2026hbases doi: null url: https://arxiv.org/abs/2605.26425v3 triage: theorem motivation_gids:

• D5/S3/Arith/NathansonAdditiveHBasisRefutation.result

Refutation of Nathanson’s additive h-basis strict inequality

Problem

Problem 12(2) of arXiv:2605.26425v3 asks whether, for integers h >= 2 and k >= 2, every k-element integer set A with a negative element satisfies ell_h(A) < n-flat_h(k) whenever ell_h(A) is defined.

Motivation

The preregistered issue #8540 records the literal quantifiers, the source statement, and the candidate counterexample before formalization. The source prints A = {-1, 1, 2} as a reflection example, while Problem 12(2) asks whether the strict inequality holds universally.

Gap

The checked source and repository searches found no earlier formal settlement of this exact Problem 12(2) statement. This dossier does not claim exhaustive literature coverage or publication priority.

Route

At h = 2, k = 3, and A = {-1, 1, 2}, the double sumset is {-2, 0, 1, 2, 3, 4}, so the covered initial segment reaches 4. The nonnegative three-set maximum is also 4: {0, 1, 2} reaches 4, while a three-element set whose double sumset covers 0 through 5 must contain 0 and 1; representing 3 then forces the third element to be 2 or 3, and neither case represents 5.

Falsifier

A proof of the literal universal strict inequality in Problem 12(2) would contradict the frozen theorem D5/S3/Arith/NathansonAdditiveHBasisRefutation.result.

Evidence

• Frozen result: D5/S3/Arith/NathansonAdditiveHBasisRefutation.result.
• The result has type Not claim and is kernel-checked in the pinned Lean toolchain.
• The witness bound nonnegativeMaximum 2 3 <= 4 is proved by the explicit case analysis in the result proof, not by a finite search shortcut.

Triage

theorem; resolution refuted for the literal strict statement only. No claim is made about Problem 12(1), other parameter values, minimality, or a corrected inequality.

ASSUMED-UNVERIFIED

The literature search is bounded to the cited arXiv source, repository surfaces, and the preregistered checks. Exhaustive coverage and publication priority remain unverified.