Pith Number

pith:M3NN6BNF

pith:2026:M3NN6BNFNBHGGFSFD7LCIJYLWL

not attested not anchored not stored refs pending

The martingale evolution of probability measures defined via the sum-of-digits functions

Dawid Tar{\l}owski

The Cusick conjecture on sum-of-digits densities follows from a general median-preserving property of martingales on binary trees.

arxiv:2605.08624 v2 · 2026-05-09 · math.PR · math.NT

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\usepackage{pith}
\pithnumber{M3NN6BNFNBHGGFSFD7LCIJYLWL}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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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

the Cusick conjecture is a special case of a more general claim about the asymmetric evolution of the binary trees associated to the martingale

C2weakest assumption

We will assume that the random walk starts from zero, and thus we will work with the family of measures P_t determined by the convolution μ_t=μ_1∗P_t. The reindexing of odd integers via partial order leads to the nonautonomous dynamics on pairs of measures.

C3one line summary

A martingale from binary tree stopping times describes the evolution of sum-of-digits difference measures, generalizing the Cusick conjecture to asymmetric tree growth.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-22T01:04:05.657060Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

66dadf05a5684e6316451fd624270bb2dd85510ee39dbfea935c8c903c51ae74

Aliases

arxiv: 2605.08624 · arxiv_version: 2605.08624v2 · doi: 10.48550/arxiv.2605.08624 · pith_short_12: M3NN6BNFNBHG · pith_short_16: M3NN6BNFNBHGGFSF · pith_short_8: M3NN6BNF

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/M3NN6BNFNBHGGFSFD7LCIJYLWL \
  | 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: 66dadf05a5684e6316451fd624270bb2dd85510ee39dbfea935c8c903c51ae74

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9fff65754db0d8300d80edb6ce1603d446ea9ae6c63504ff618b3c3c2fb04c8a",
    "cross_cats_sorted": [
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-09T02:41:38Z",
    "title_canon_sha256": "2e425c99d5ee0f03a137a1f97736d9bcab41fcd595d4cdf7eccaac04e51618f8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.08624",
    "kind": "arxiv",
    "version": 2
  }
}