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Statement

Setting (pp. 283--284). Section 6 considers walks along the Gaussian integers relatively prime to a fixed integer $N$, viewed modulo $N$ in the square $[0,N-1]\times[0,N-1]$, and assumes $N$ even; the setting is stated in full on the Proposition 6.1 page. The proposition names no step size; the surrounding text applies it to walks of a fixed step $k$.

Proposition 6.2 (p. 284, quoted). "If there is no walk to infinity along Gaussian integers relatively prime to $N$, then there is an upper bound on the largest connected component, so Conjecture 1.2 holds."

The bound is for the Gaussian integers coprime to $N$; the passage to Conjecture 1.2 for the same step uses that all but finitely many Gaussian primes are coprime to $N$, which the paper does not spell out. The paper adds (p. 284) that only a single $N$ need be found to prove Conjecture 1.2, and so Conjecture 1.1.

Source. Ilan Vardi, Prime percolation, Experimental Mathematics 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373: Section 6, pp. 283--284. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The paper gives no proof.

Proof pointer

None in the paper beyond the remark (p. 284) that the fundamental square tiles the plane under the translations by $(N,0)$ and $(0,N)$, after which Propositions 6.1--6.3 are called clear.

Dependencies

None.

Bears on

• #952: the reduction from a single periodic obstruction to a bound on every component of Gaussian primes with the given step, hence to the negative answer for that step; it supplies no obstruction itself. The 2026 OpenAI manuscript cites this proposition for the periodicity principle and proves its own explicit version, its Proposition 2.1.