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The Gaussian moat problem

Theorem 1.1: for every real $D$ there is a finite $B_D$ such that every connected component of the graph on Gaussian primes with edges of length at most $D$ has at most $B_D$ vertices, independent of the starting prime and including primes on the axes; so every walk through distinct Gaussian primes with steps at most $D$ has at most $B_D$ terms. The proof finds, for each $D$ , a finite set of split primes whose joint sieve leaves a periodic set with no infinite bounded-step walk, using geometric sampling and entropy estimates. $B_D$ is nonexplicit, so no numerical moat width is given for any particular $D$ .

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 26, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Gaussian moat problem · Gaussian moat problem

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The Gaussian moat problem

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No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorems 1.1 and 1.2 of the TeX source; the proof was not refereed. Lean-checked on the Comparator challenge GaussianMoat, listed in the release's formalization catalogue (OAI.GaussianMoat.fullMain, OAI/NumberTheory/GaussianMoat/Main.lean). Its statement has two parts: for every real $D$ there is no injective sequence of irreducible Gaussian integers with successive distances at most $D$ ; and for every real $D$ one natural number bounds the size of every connected component of the graph joining distinct Gaussian primes at distance at most $D$ , and the length of every injective bounded-step walk. That is the headline claim, axis primes and associates included. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.