Openai 2026 bounded step walks gaussian primes

../

proposition_2_1: The periodicity reduction of the claimed Gaussian moat resolution: a finite sieve set with no infinite bounded-step self-avoiding walk gives an explicit bound on every component of the distance-D Gaussian-prime graph.

theorem_1_1: The uniform component bound claimed as the resolution of the Gaussian moat problem (Problem 952): for every finite real D a finite, nonexplicit B_D bounds every component of the graph joining Gaussian primes at distance at most D.

theorem_1_2: The finite sieve obstruction behind the claimed Gaussian moat resolution: for every D at least 1, a finite set of split primes depending only on D leaves no infinite self-avoiding D-step walk avoiding zero modulo each factor.

OpenAI, Bounded-Step Walks on Gaussian Primes, OpenAI Math Release preprint, September 26, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_bounded_step_walks_gaussian_primes.pdf, and the release's TeX bundle sits in the same release folder.

@misc{OAI:Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026,
  author = {{OpenAI}},
  title = {{Bounded-Step Walks on Gaussian Primes}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf}{OAI:Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026}},
  year = {2026}
}

Attestation, as the source states it. The release's root README says its manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all of them have Lean formalizations, and that "Some of the unformalized results could have issues". The manuscript's own README adds only the title, the author line "OpenAI", the date September 26, 2026 and the citation block above; it carries no statement about human assistance. The PDF names "OpenAI" as author and prints no affiliation, acknowledgment or funding line. These are the source's own provenance statements, recorded here as historical attestations and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

Formalization, as the release lists it. The release's catalogue (lean/formalization.yaml) names this manuscript and the declaration OAI.GaussianMoat.fullMain in OAI/NumberTheory/GaussianMoat/Main.lean, with the comparator configuration ComparatorChallenges/GaussianMoat.json (permitted axioms propext, Quot.sound, Classical.choice). The release's own Lean page says the formalized result is the negative answer to the infinite-walk question for every real step bound $D$ together with one finite bound, depending only on $D$, on the size of every component of the bounded-step graph and on the length of every injective bounded-step walk, with axis primes and all associates included and no explicit function of $D$. The comparator statement file it names, ComparatorChallenges/GaussianMoat.lean, defines the graph on irreducible Gaussian integers with an edge when the Euclidean distance of the two points in $\mathbb C$ is at most $D$, states MainEndpoint (no injective sequence of irreducible Gaussian integers has all consecutive distances at most $D$, for every real $D$) and UniformEndpoint (for every real $D$ some natural $B$ bounds the cardinality of every component and the length of every injective finite walk with steps at most $D$), and poses fullMain : MainEndpoint ∧ UniformEndpoint with a sorry placeholder, the solution module supplying the proof through FiniteSieveEndpoint, the Lean counterpart of Theorem 1.2, and endpoints_of_finiteSieve, the counterpart of Proposition 2.1. All of this was read statically from the release's catalogue. The comparator bounds the Euclidean distance by a real $D$, whereas the formal-conjectures statement the problem page records bounds the integer norm of each step; the two conditions agree in content, and no bridging statement was checked here. The corpus's verification built the declaration OAI.GaussianMoat.fullMain and checked its axioms (propext, Classical.choice and Quot.sound only). That verification covers the whole question: the first conjunct, MainEndpoint, says that for every real $D$ no infinite sequence of distinct Gaussian primes has every step of length at most $D$, so the answer to the problem's question is no. The record is kept on the claim page of Problem 952, not on this card.

Companions: the release groups this manuscript alone, under the title "Uniformly bounded components of Gaussian-prime graphs"; it lists no companion, alternate proof or consequence paper.

Read status: claims checked for Theorem 1.1, Theorem 1.2 and Proposition 2.1, read clause by clause in the TeX source (main.tex lines 53--125 and periodicity.tex lines 8--51; PDF pp. 1--3) on 2026-10-07, together with the statements of Lemmas 3.1--3.3, 4.1--4.3, 5.1, 6.1, 8.1, Theorem 5.2 and Propositions 6.2 and 7.1 (preliminaries.tex, geometry.tex, coverage-transfer.tex, schedule.tex, information.tex); the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

• Section 1, The Gaussian moat problem (main.tex lines 53--178; pp. 1--3). A Gaussian prime is an irreducible element of $\mathbb Z[i]$; $a+bi$ is identified with $(a,b)\in\mathbb Z^2$, $N(a+bi)=a^2+b^2$ and $|z|=\sqrt{N(z)}$. For a finite real $D$, $G_D$ is the graph on all Gaussian primes with an edge between distinct $z,w$ when $|z-w|\le D$; the moat problem is whether, for some $D$, the graph $G_D$ has an infinite path that visits no vertex twice. Theorem 1.1 (Uniform component bound, p. 1): "For every finite real $D$ there is a finite $B_D$ such that every connected component of $G_D$ has at most $B_D$ vertices", the bound not depending on the starting prime, so no sequence of distinct Gaussian primes with all steps of length at most $D$ has more than $B_D$ terms. The manuscript says the bound is nonexplicit, since scales are chosen "sufficiently large" after $D$ is fixed, and that $B_D=1$ works for $D<1$. History as the manuscript gives it: Gethner and Stark trace the question to Gordon at the 1962 ICM (Gethner--Stark, p. 289); Erdős credits Gordon and Motzkin and the Pasadena meeting of November 1963 (Erdős 1977, p. 69). Prior work it names: prime-free disks centered on any line through two Gaussian integers and arbitrarily isolated real Gaussian primes (Gethner, Wagon and Wick, Theorems 4.1 and 4.4, the latter also credited to Vardi); the computational bound of Tsuchimura on the distance reachable from the origin with steps at most $6$, described as a result about one component; the periodic obstructions of Gethner and Stark for step bounds $\sqrt2$ and $2$ and their proposal to sieve by small Gaussian primes (pp. 290--292); and Vardi's periodic coprimality graphs and deduction of a uniform bound on component sizes from the nonexistence of an infinite walk (Section 6, Proposition 6.2). Subsection 1.1 defines, for a finite set $\mathcal P$ of rational primes $p\equiv1\pmod 4$ with chosen conjugate factors $\pi_p,\overline{\pi_p}$, the set $\mathcal A(\mathcal P)$ of Gaussian integers divisible by neither factor over any $p\in\mathcal P$, periodic under $Q\mathbb Z^2$ with $Q=\prod_{p\in\mathcal P}p$, and states Theorem 1.2 (Finite sieve obstruction, p. 2): "For every $D\ge1$ there is a finite set $\mathcal P_D$ of rational primes congruent to $1$ modulo $4$ such that $\mathcal A(\mathcal P_D)$ contains no infinite sequence of distinct points with successive distances at most $D$. The set $\mathcal P_D$ depends only on $D$." The rest of the section outlines the route (below) and calls the final telescoping step "methodologically related" to the entropy-decrement argument of Tao (Section 3 of the cited paper), while stating: "Here the signed residue batches, coverage transfer, and costs forced by zero avoidance are established in full within this paper." (p. 3).
• Section 2, Periodicity and the uniform bound (periodicity.tex; p. 3). Proposition 2.1 (From a periodic obstruction to a uniform bound): if $\mathcal A(\mathcal P)$ has no infinite self-avoiding walk with steps at most $D$, then with $Q$ as above, $K_D$ the number of Gaussian integers of modulus at most $D$, and $E$ the set of associates of the selected factors, every component of $G_D$ has at most $\max\{Q^2,\ |E|+(K_D-1)|E|Q^2\}$ vertices. The section then deduces Theorem 1.1 for $D\ge1$ from Theorem 1.2 and Proposition 2.1, and notes that the weaker infinite-walk conclusion follows by discarding a finite initial stretch of the walk that holds all the sieved primes.
• Section 3, Arithmetic, walks, and entropy (preliminaries.tex; pp. 4--6). Conventions ($A\ll_D B$, natural logarithms); the arithmetic of split primes: unique factorization in $\mathbb Z[i]$, stated without citation; the two nonassociate factors of norm $p$ over $p\equiv1\pmod4$ and their residue fields of size $p$, cited to Conrad's expository notes on $\mathbb Z[i]$; and consequences the text states as its own, uncited (divisibility by both factors means $p$ divides both coordinates; a nonzero multiple of $\pi$ has length at least $\sqrt p$; multiplication by $\alpha$ has determinant $N(\alpha)$); the batches $\mathcal B_T$ of primes $p\equiv1\pmod4$ in $[T,2T]$, their count $k_T$ and mean logarithm $L_T^*$, with $k_T=(1/2+o(1))T/\log T$ from the prime number theorem in progressions for the fixed modulus $4$ (Selberg 1950, equations (1.1)--(1.2)); entropy, conditional mutual information, total variation, relative entropy and Pinsker's inequality (Cover and Thomas, Chapter 2 and Lemma 11.6.1), with a short derivation of Pinsker from the log-sum inequality; the sampling convention that the walk is a fixed deterministic sequence and only times are random; Lemma 3.1 (displacement entropy: $H(z_{T_1}-z_{T_0})\le 2\log(n+1)+O_D(1)$ when $|T_1-T_0|\le n$); Lemma 3.2 (continuity of conditional entropy in total variation); Lemma 3.3 (a short-list entropy bound adapted from the proof of Fano's inequality, Cover and Thomas, Theorem 2.10.1).
• Section 4, Entropy enrichment from a forward segment (geometry.tex; pp. 6--13). Lemma 4.1 (Many differences): a segment of $n$ steps with diameter $R$ and width $W$ has $|E-E|\ge c_D RW$ with $cn\le RW\le D^2n^2$, proved by a degree argument for a map from a parameter torus to $\mathbb R^2/\Lambda$ (Hatcher, Section 3.3). Lemma 4.2 (A randomly signed product avoids a thin rectangle): for $r$ distinct primes $p\equiv1\pmod4$ in $[T,2T]$, $T\ge5$, with product $P$ and independent fair sign choices, and an origin-centered rectangle with half-lengths $R_1\ge W_1\ge1$ and $R_1W_1\le P^{1-d}$, where $0<d<1/4$ and $d^3r\ge C_0$, the probability that the product of the chosen factors has a nonzero multiple in the rectangle is at most $Ce^{-cd^3r}$ ($c,C,C_0$ absolute); the proof uses determinant divisibility, Hoeffding's inequality (Theorem 2, equation (2.6)) and an edge-isoperimetric inequality on the Boolean cube (proved in the text, with Harper 1964 cited). Subsection 4.3 defines the averaged residue entropy $f_Z(s)$ over uniformly chosen signed subsets of size $s$, notes its concavity, and defines the forward transition: from a time $t$, choose a uniform difference of the segment $E_t$ of $n=\lfloor e^{gU}\rfloor$ steps, then one endpoint. Lemma 4.3 (Entropy enrichment): in the stated asymptotic regime the normalized entropy after the transition is at least half of one plus the normalized entropy before it, up to $O_D(g)$.
• Section 5, Coverage from shared continuation data (coverage-transfer.tex; pp. 13--16). Lemma 5.1 (One shared conditioning cost) in the setting of homomorphisms from a lattice to finite abelian groups of orders in $[T,2T]$. The definition of $(\tau,\beta)$ coverage: fewer than $p^{1-\beta}$ residues have probability below $\tau/p$. Theorem 5.2 (One-step coverage transfer): under listed numerical hypotheses on $\tau,\delta,\beta,\beta',\eta,\eta',n,p,T$, coverage of the conditional terminal laws at the next checkpoint, high joint entropy and a cheap shared data vector give coverage of the mixed terminal law with a better exceptional-set exponent; the proof uses a multiplicative lower tail for binomial counts (Lugosi's notes, Exercise 8, and Boucheron, Lugosi and Massart, with a derivation given) and Lemma 3.3.
• Section 6, A common sampling schedule (schedule.tex; pp. 17--23). Fixes $\beta_0=1/200$, $e_0=\beta_0/50$, then $l_0$, $g_0$, $c_0$, then $B>2$, then the asymptotic parameter $M$; scale windows $X_m=100^m$ for $\lceil M/2\rceil\le m\le M$; batches $T=B^j$ with $X_m\le\log T\le1.05X_m$; three bands per window (Table 1: top with $g_0$, middle with $g_1=e^{-M^3}$, bottom with $g_2=e^{-100M^{20}}$), transitions of Lemma 4.3 applied at decreasing scales, then a uniform offset in $\{0,\dots,N-1\}$, $N=\lceil T_+^{10}\rceil$, giving the common terminal time $t_*$. Lemma 6.1 (Future shifts and smoothing): forward shifts from a checkpoint before scale $V$ are at most $S_V$ with $\log(1+S_V)\le2gV$, and translates of the law of $t_*$ by at most $T_+$ are within $T_+^{-9}$ in total variation. Subsections 6.2--6.3 iterate Lemma 4.3 to get joint entropy of order $T$ per batch (equation (6.4)), make all smaller alphabets cheap by the choice of $B$ (equation (6.5)), define checkpoints $A_0,\dots,A_J$ with $\beta_i=2^{-i}\beta_0$ and $r_{i+1}=r_i-2\beta_i$, and obtain an almost uniform one-coordinate marginal at $t_*$. Proposition 6.2 (Terminal residue coverage): from every exact start at $A_i$, all but a fraction $\eta_i$ of the signed primes have $(\tau_i,\beta_i)$ coverage at $t_*$, by downward induction on $i$ using Theorem 5.2; at $A_0$ this is mass at least $1/(4p)$ outside fewer than $p^{199/200}$ residues for $99\%$ of the signed primes.
• Section 7, Zero avoidance costs information (information.tex lines 1--290; pp. 23--27). Assumes the walk lies in $\mathcal A(\mathcal P_M)$, $\mathcal P_M$ the union of the selected batches; defines increment words $W_L(t)$ and dyadic lengths $L_T$ with $\frac12T^{2/5}<L_T\le T^{2/5}$. Proposition 7.1 (Information cost of one batch): for some $c_1>0$ depending only on the schedule constants, for all sufficiently large $M$ the conditional mutual information per increment between the batch's selected residue vector at $z_{t_*}$ and the next $L_T$ increments, given the smaller batches' residues and averaged over the auxiliary prime choices, is at least $c_1/\log T$; the proof tests candidate residues against repeated packages of displacement and word, using coverage, self-avoidance (two hits within a word would give a nonzero difference of length below $\sqrt p$) and the smoothing of $t_*$.
• Section 8, The common-law entropy budget (information.tex lines 292--417; pp. 27--28). Lemma 8.1 (A common-law entropy telescope): the sum over batches of the per-increment information is at most $\log K_D+o(1)$, by nesting the residue vectors, splitting longer dyadic words into blocks and using Lemmas 3.2 and 6.1. Proof of Theorem 1.2: the lower bounds $c_1/\log T$ summed over the batches grow like a constant times the number of windows, of order $M$, contradicting Lemma 8.1 for large $M$; $\mathcal P_D$ is $\mathcal P_M$ for one such $M$.
• References (references.tex; pp. 28--29): fourteen items, listed under Dependencies on the result pages. The Conrad notes are marked "accessed", one day after the manuscript's date.

Nothing in the manuscript is described as numerical, computer-assisted or conditional; the only unproved inputs are the cited standard results, and the bound $B_D$ is stated to be nonexplicit. The release folder holds the PDF, its README and a build folder with the TeX source, and nothing else.

Bears on

• Problem 952: claimed resolution of the whole question, negatively. Theorem 1.1 asserts that for every finite real $D$ no sequence of distinct Gaussian primes with successive distances at most $D$ has more than $B_D$ terms, which denies the infinite sequence the problem asks for at every constant $C$ and with any starting point, and it asserts the stronger uniform bound on component sizes. The claim is unverified here: no proof step was checked, and the page's status rests on acceptance evidence, not on this card.
• vardi_1998_prime_percolation: Theorem 1.1 is a claimed proof of the content of the Conjectures 1.1 and 1.2 that card records (no infinite component of Gaussian primes with steps at most $k$, and a bounded largest component, for every $k$); the manuscript does not name those conjectures and attributes the question to Gordon, so the identification is this corpus's reading of the two statements. Proposition 2.1 is the manuscript's own version of the periodicity reduction it cites to that paper's Section 6, Proposition 6.2. Nothing on either card is independently verified.