The Rule 30 Prize Problems

Rule 30 Prize, Problem 1 (non-periodicity). The center column of Rule 30 is not eventually periodic: there is no positive period p and threshold N past which the column repeats with period p.

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Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

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@misc{cairn-rule30,
  title        = {The Rule 30 Prize Problems},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/rule30}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

centerColumn_not_eventually_periodic. Rule 30 Prize, Problem 1 (non-periodicity). The center column of Rule 30 is not eventually periodic: there is no positive period $p$ and threshold $N$ past which the column repeats with period $p$ .

centerColumn_frequency_half. Rule 30 Prize, Problem 2 (equal frequency). Each color occurs on average equally often in the center column: the set of times at which it is black has natural density $1/2$ . This is Wolfram's phrasing that the discrete limit of $Total [c [t]] / t$ as $t \to \infty$ is $1/2$ .

Rule 30 is the elementary cellular automaton with local update $c^{'} = l \oplus (c \lor r)$ , the Boolean rule numbered $30$ by Wolfram. Started from a single black cell on a bi-infinite row, its center column t ↦ (state t) 0 looks random — it was long Mathematica's default pseudorandom generator — yet nothing about that randomness is proven. In 2019 Wolfram offered the Rule 30 Prizes for three questions about it; we formalize the first two.

Problem 1. Is the center column non-periodic (never eventually periodic)?
Problem 2. Does each color occur on average equally often, i.e. does the running average of the values converge to $1/2$ ?

Problem 3 — whether computing the $n$ -th cell requires at least $O (n)$ work — is omitted. It is model-relative and has no canonical model-independent phrasing. All three are open.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Other.Rule30 (2 statements).

theorem centerColumn_not_eventually_periodic :
    ¬ ∃ p : ℕ, 0 < p ∧ ∃ N : ℕ, ∀ t : ℕ, N ≤ t → centerColumn (t + p) = centerColumn t

theorem centerColumn_frequency_half :
    {t : ℕ | centerColumn t}.HasDensity (1 / 2)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Announcing the Rule 30 Prizes, Stephen Wolfram, 2019.
Rule 30 Prizes.
Wikipedia: Rule 30.

Source and licence

Imported from Formal Conjectures (Other), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.