slug: shin-four-point-sumset-compression
bibkey: shin2026iterated
doi: 10.48550/arXiv.2609.01690
triage: theorem
motivation_gids:
  - D5/S0/Certificates/FiniteExhaustion

Shin’s four-point sumset compression question

Problem

For every natural h>=2, put D_h=choose(h+2,2)+1 when h is odd, and D_h=choose(h+2,2) otherwise. Must every four-element integer set A have a four-element integer set B in [0,D_h] with |hB|=|hA|? Here hA is the set of sums of exactly h elements of A with repetitions allowed. This is the question after Corollary 10.5 and equation (193) in arXiv:2609.01690v1, equivalently nu(h,4)=D_h for all h>=2. It is restated as open in arXiv:2609.08915v1, Question 12.5.

Motivation

FiniteExhaustion provides the existing setting for complete finite certificate arguments. This source question has an unbounded universal statement, but one cardinality absent from a bounded interval suffices to refute its affirmative answer.

Gap

The question preserves only cardinality. A failure to preserve a complete additive relation type would not answer it. The obstruction must therefore exclude every four-element set in the proposed interval, including sets with nontrivial common divisors and unnormalized translates.

Route

Use h=11 and A={0,7,17,80}, whose repeated sumset has 347 elements. Order an arbitrary B in [0,79], subtract its least element, and obtain 0<a<b<c<=79 without changing the repeated-sum cardinality. A bitset recurrence starts at {0} and takes the union with its shifts by a,b,c eleven times. Its kernel-verified interpretation is the actual sumset. Its cardinality excludes 347 for all 79,079 normalized triples. The designated Lean result is Not claim for the full universal proposition.

Falsifier

Any normalized triple 0<a<b<c<=79 with count 347 invalidates the finite obstruction. A different witness cardinality, an incomplete normalization, or interpreting scalar images as repeated sums also invalidates it. The kernel proofs address each of these mathematical possibilities.

Evidence

The source definition and question are pinned in Library/Certificates/shin2026iterated.md; the independent restatement is in Library/Certificates/zhang2026sharp.md. The intended full result is D5/S0/Certificates/ShinFourPointCompression.result, negating its claim. The finite exclusion was kernel checked with only propext, Classical.choice and Quot.sound. Definitions, count soundness, normalization and the full negation are assembled in that single module.

Triage

theorem: a concrete finite obstruction with a full source-negation target. Neither a stronger equality nu(11,4)=80 nor positive cases at other h are part of the claimed result.

ASSUMED-UNVERIFIED

The bounded public searches found no prior matching solution; they do not prove first-publication priority or exhaust all literature. The kernel cannot prove correspondence with natural-language source text. Independent fidelity review, the canonical delivery gates, freeze/resolution binding, and merged delivery remain required before counting a solved problem.