[#P2812] Positive cycle entropy for finite cyclic Rule 30

Problem. For \(n\ge1\), let \(F_n:\{0,1\}^n\to\{0,1\}^n\) be cyclic Rule 30, \((F_n(x))_i=x_{i-1}\mathbin{\mathsf{xor}}(x_i\mathbin{\mathsf{or}}x_{i+1})\), with indices modulo \(n\). Let \(M_n\) be the largest eventual period of an orbit of \(F_n\). Is \(\limsup_{n\to\infty} n^{-1}\log_2 M_n>0\)?

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Definitions and notation

1Problem setup

Definition 1. The eventual period of \(x\) is the least positive \(p\) for which \(F_n^{t+p}(x)=F_n^t(x)\) for some \(t\ge0\).

Remark 1. The limsup inequality is equivalent to the existence of constants \(c>0\) and infinitely many \(n\) with \(M_n\ge2^{cn}\).

Remark 2. The displayed Boolean formula fixes the Rule 30 convention and the orientation of the neighborhood.

2What counts as a solution

Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\).
A disproof may also establish \(\limsup n^{-1}\log_2M_n=0\); finite tables alone do not settle either side.

1Status

Saved packet · July 31, 2026

What counts as a solution

Packet-reported status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. Exact maximum-period tables exist for small widths, and the checked structural results for Rule 30 do not prove an exponential subsequence of finite-ring periods. Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\).^[1]

This research-packet status has not passed editorial review.

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Record	Kind	Assessment
UNKNOWN as of 2026-07-31. Exact maximum-period tables exist for small widths, and the checked structural results for Rule 30 do not prove an exponential subsequence of finite-ring periods. Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 UNKNOWN as of 2026-07-31. Exact maximum-period tables exist for small widths, and the checked structural results for Rule 30 do not prove an exponential subsequence of finite-ring periods. Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\).^[1] Relevance to this problem For rule 30 positive cycle entropy, pins the dated research frontier: UNKNOWN as of 2026-07-31. Exact maximum-period tables exist for small widths, and the checked structural results for Rule 30 do not prove an exponential subsequence of finite-ring periods. Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\). Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-07-31. Exact maximum-period tables exist for small widths, and the checked structural results for Rule 30 do not prove an exponential subsequence of finite-ring periods. A complete resolution must satisfy this condition: Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\). Open full record →
Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\).	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\).^[1] Relevance to this problem For Positive cycle entropy for finite cyclic Rule 30, record rule-30-positive-cycle-entropy-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\). Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that \(M_n\ge2^{cn}\) for some explicit \(c>0\) and infinitely many \(n\), or prove \(\log M_n=o(n)\). Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Finite Rule 30 tables contain large and highly irregular periods. A construction relating widths, a transfer description of cycles, or certified lower bounds on an infinite subsequence would turn those isolated computations into reusable dynamics.

Original intake status. UNKNOWN as of 2026-07-27. Exact maximum-period tables exist for small widths, and the checked structural results for Rule 30 do not prove an exponential subsequence of finite-ring periods.

2026-07-27 prior-art search checked finite-width Rule 30 maximum periods, Rule 30 expansivity, periodic configurations, and cellular-automaton cycle growth; no asymptotic lower bound matching the statement was found.
A single record cycle at one width gives no asymptotic conclusion. Useful progress should identify a construction that embeds or combines cycles across widths.
Left permutivity controls preimages in one direction, while long temporal cycles on a ring also require the wraparound constraints to close.

Recorded example 1. The exact maxima at widths 4, 7, 14, 16, and 17 are respectively 8, 63, 1428, 6016, and 10846.

Computational notes

Complete functional-graph enumeration for widths 1 through 18 gave \(M_n=1,1,1,8,5,1,63,40,171,15,154,102,832,1428,1455,6016,10846,2844\). Every transition and cycle length was computed with integer bit operations.

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The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemPositive cycle entropy for finite cyclic Rule 30

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

A half-edge transient bound for majority dynamics on the square torusdynamics
Strong block universality of Conway's Game of Lifedynamics
Maximum cycle length of cyclic Rule 30 on 24 cellscellular automata

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“Positive cycle entropy for finite cyclic Rule 30.” TheoremDB. P2812. Problem statement; statement text SHA-256 f059c4d338241e9f2d0be7db6c01fc57a24dd56868b1fa0afece0c969990eed1. https://theoremdb.org/statement/?ref=P2812

BibTeX

@misc{theoremdb-problem-f059c4d338241e9f2d0be7db6c01fc57a24dd56868b1fa0afece0c969990eed1,
  title = {{Positive cycle entropy for finite cyclic Rule 30}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 f059c4d338241e9f2d0be7db6c01fc57a24dd56868b1fa0afece0c969990eed1},
  url = {https://theoremdb.org/statement/?ref=P2812}
}

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This problem includes 2 records joined by 1 typed links, sourced from doi.org^[1], current as of July 31, 2026.

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1References

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Packet source. Status assessment source. See dataset.references[0] for the exact external source and locator. ↗scholarly publication · reference sourceReference review pending: publication-style citation.Source use: citation only.Source used to assess the problem's recorded status.Also cited at Editorial research route recorded 2026-07-31.Source named by the research packet.
Rapid left expansivity, a commonality between Wolfram's Rule 30 and powers of p/q. OEIS entry A334497, checked 2026-08-01. Entry definition, data, comments, and linked references relevant to Positive cycle entropy for finite cyclic Rule 30; checked 2026-08-01. ↗reference database · reference source · checked 2026-07-31Source use: citation only.Source used to assess the problem's recorded status.For Positive cycle entropy for finite cyclic Rule 30: UNKNOWN as of 2026-07-27. Exact maximum-period tables exist for small widths, and the checked structural results for Rule 30 do not prove an exponential subsequence of finite-ring periods.

Original CC0 asymptotic question extracted from exact finite-ring cycle data.

[#P2812] Positive cycle entropy for finite cyclic Rule 30

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