Pith Number

pith:V5PQ6DHX

pith:2026:V5PQ6DHXOWFEZIN5ORW6IKSXMH

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On a Constraint on Invariant Measures of Certain Cellular Automata

Matan Tal

Fixed positive indices determine uniform conditional probabilities on a coset at zero for invariant measures of bi-permutative cellular automata.

arxiv:2604.10124 v4 · 2026-04-11 · math.DS · cs.FL · math.PR

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Claims

C1strongest claim

Fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional probability at the zero index; in the finite group multiplication case this set is a coset of some subgroup. The constraint induces a factor with respect to the shift, and zero-entropy invariant measures on that factor correspond to positive-entropy measures on the original system.

C2weakest assumption

The conjecture that the class of RLP subshifts is much larger than bi-permutative cellular automata, while only one additional example is proved to belong to it; the paper relies on the prior observation in [6] without re-deriving it from scratch.

C3one line summary

A constraint linking fixed positive-index values to uniform conditional probabilities at the zero index is strengthened for bi-permutative cellular automata, with group-structure relations, zero-entropy factors, and a partial generalization to RLP subshifts.

Receipt and verification

First computed	2026-05-27T01:04:58.034888Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

af5f0f0cf7758a4ca1bd746de42a5761c3fb55198a0420f85b6ae981af5a23f2

Aliases

arxiv: 2604.10124 · arxiv_version: 2604.10124v4 · doi: 10.48550/arxiv.2604.10124 · pith_short_12: V5PQ6DHXOWFE · pith_short_16: V5PQ6DHXOWFEZIN5 · pith_short_8: V5PQ6DHX

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/V5PQ6DHXOWFEZIN5ORW6IKSXMH \
  | 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: af5f0f0cf7758a4ca1bd746de42a5761c3fb55198a0420f85b6ae981af5a23f2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "c613cf2f4264ed54e2e93e6cbc680b73eab1905cbcb7e3842aac6b51e1cfdc0b",
    "cross_cats_sorted": [
      "cs.FL",
      "math.PR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2026-04-11T09:38:43Z",
    "title_canon_sha256": "bb4b5f2ae4b36e26bfc919fcb976156300465a3e9371b81151eb035c76c8284a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.10124",
    "kind": "arxiv",
    "version": 4
  }
}