The additive-square-free spectrum below 139

Q: What does this research mean for the field?

Up to affine changes of the alphabet, all four-element sets of real numbers with a maximum additive-square-free word length less than 139 consist of the known endpoint-sum family and seven isolated classes. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Enkai Zhang

doi:10.5281/zenodo.22700368

September 11, 2026Open Access

The additive-square-free spectrum below 139

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Key Points

To classify all four-element sets of real numbers whose maximum length of an additive-square-free word is strictly less than 139.
Conducted exhaustive computational searches to identify and verify isolated parameter sets.
Constructed 77 explicit words to analyze and cover all remaining real parameter configurations up to affine transformations.
Classified all four-element sets with maximum word length below 139 into the known endpoint-sum family and seven isolated classes up to affine changes.
Established a sharp cutoff of length 139 for the set {0, 2, 3, 7}.
Demonstrated a sharp cutoff of length 167 within the parameter family {0, 1, x, x+2} for x > 1.

Implication

Combinatorial analysis classifies four-element real sets avoiding equal adjacent block sums, highlighting sharp cutoffs in additive-square-free word lengths.

Abstract

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Cite This Study

Enkai Zhang (2026) studied this question.

synapsesocial.com/papers/6aa7b2f70926e14a848b184c https://doi.org/10.5281/zenodo.22700368

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The additive-square-free spectrum below 139

Key Points

Implication

Abstract

Discussion

Cite This Study

The additive-square-free spectrum below 139

Key Points

Implication

Abstract

Discussion

Cite This Study

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