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bibkey: shin2026iterated authors: Henry Shin year: 2026 title: “Iterated-sumset spectra: The complete exponent law and its rank geometry” doi: 10.48550/arXiv.2609.01690 claim: The question following Corollary 10.5 asks whether every four-point sumset cardinality is realizable at the minimum diameter of a maximal-cardinality model. strata_touched:

  • D5/S0/Certificates/ShinFourPointCompression license: citation-only triage: anchor

Four-point sumset compression

Verified locator

DOI: https://doi.org/10.48550/arXiv.2609.01690

The pinned source is https://arxiv.org/html/2609.01690v1. Immediately before Corollary 10.5 it defines nu(h,4) as the least D for which all cardinalities of h-fold sumsets of four distinct integers occur in [0,D]. Corollary 10.5, equation (193), proves lower and upper bounds. The unnumbered paragraph after its proof, before Proposition 10.6, asks whether nu(h,4)=D_h^max for every h>=2. Here D_h^max=choose(h+2,2)+1 for odd h and choose(h+2,2) otherwise. The operation hA allows repetition and uses exactly h summands. This is an explicit question, without attributing an affirmative conjecture to Shin.

The source HTML read on 2026-09-13 has SHA-256 453e39bc6fc2773e50582eed117b46986f75341010fea1abc1db3a18bd506d60. The official abstract/version page https://arxiv.org/abs/2609.01690 listed only v1 in the same day’s check. A fresh public fetch of the pinned HTML returned HTTP 200. Only bibliographic information and the short question are paraphrased here; no source-paper text or code is redistributed.

Mathematical scope

The affirmative claim quantifies over every natural h>=2 and every finite four-element subset A of the integers. It asks for a four-element integer set B in the closed interval [0,D_h^max] with |hB|=|hA|. No additive-type, gcd, primitivity, reflection, or ordering requirement is imposed on A or B. The proposed negative instance is h=11, A={0,7,17,80}: |11A|=347, and that cardinality is absent from all four-element subsets of [0,79]. The Lean result consumes the full finite exclusion after translation normalization. It makes no assertion that nu(11,4)=80.

At repository base e6ad855ec601b2195efc5abdf307ab282b31fd4d, searches of D5, Library, Problems and Blueprint for 2609.01690, 2609.08915, iterated.?sumset, and sumset found no matching compression result. The sumset mention in GreedyThreeSumfreeTwoParameter concerns a different three-sumfree construction. Pinned mathlib revision db584cd6d46c92f209a44c0f1c829460d327499d has no matching compression theorem. Its repeated-pointwise-addition, finite ordering, translation-cardinality, binary-index, and bitwise lemmas are used directly in the proof.

The 2026-09-13 GitHub issue/PR searches for either paper ID within trureturing returned zero. The global 2609.08915 issue search found one arXiv feed issue. Formal Conjectures was checked at a2f4a1bb12a28e04a969da78feefac7d1ce49565; the exact 2609.01690 code query returned zero, and a fresh worker GitHub query returned total_count=0 with incomplete_results=false. Public arXiv queries sumset spectra and sumset compression, ordered newest first with size 50, included the Shin paper as a positive control and found no newer matching solution. These observations do not establish exhaustive publication priority.

Zhang’s https://arxiv.org/html/2609.08915v1#S12.Thmtheorem5, Question 12.5, independently calls the same label-level question open and states that its weak-type theorem does not answer it. The source is catalogued as zhang2026sharp. Current source status, source-to-Lean fidelity, and absence of prior solutions are literature/review obligations, not consequences of a kernel axiom check. Independent review, canonical admission, freezing and delivery are separate from the local proof.