Pith Number

pith:PANEHHS6

pith:2026:PANEHHS6CTJRVATXHQF55YVOGP

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Group Theory of the Kolakoski Sequence

Noah MacAulay

Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths.

arxiv:2605.14234 v1 · 2026-05-14 · math.GR

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Record completeness

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

As an application we determine the number of maximal-length orbits of the automata given an arbitrary input sequence for odd n.

C2weakest assumption

That the transformation groups K^{p,q}_n are subgroups of (and likely equal to) the recursively defined group J_n^{p,q} whose limit is weakly regular branch.

C3one line summary

Transformation groups of run-length decoding automata for Kolakoski-like sequences are subgroups of binary tree automorphisms with recursive structure, allowing exact count of maximal-length orbits when the iteration depth is odd.

References

5 extracted · 5 resolved · 0 Pith anchors

[1] Rufus Oldenburger, Exponent trajectories in symbolic dynamics, Trans. Amer. Math. Soc., Vol. 46 (1939), pp. 453-466 1939

[2] Kolakoski, Self Generating Runs, Problem 5304, American Math 1965

[3] The Kolakoski sequence and related conjectures about orbits 2020

[4] Notes on the Kolakoski sequence 1994

[5] Branch groups 2003

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:10.711358Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

781a439e5e14d31a82773c0bdee2ae33fe12cfc911b76c92865926e4112c815b

Aliases

arxiv: 2605.14234 · arxiv_version: 2605.14234v1 · doi: 10.48550/arxiv.2605.14234 · pith_short_12: PANEHHS6CTJR · pith_short_16: PANEHHS6CTJRVATX · pith_short_8: PANEHHS6

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/PANEHHS6CTJRVATXHQF55YVOGP \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 781a439e5e14d31a82773c0bdee2ae33fe12cfc911b76c92865926e4112c815b

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "e7cec9a6e9b300c83f65ec0759b2cf287ddf9832a238f56bf055e041251e4784",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.GR",
    "submitted_at": "2026-05-14T00:59:30Z",
    "title_canon_sha256": "a35cc5342783cc7ccbaf14eee78eff3dc7fe3f7ec5a7d6847f3bb05a6714aade"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14234",
    "kind": "arxiv",
    "version": 1
  }
}