Pith Number

pith:FOPLMBBP

pith:2012:FOPLMBBPTND4T7M2GPMKANSK7I

not attested not anchored not stored refs pending

Doob's optional sampling and maximal inequality for $G$-martingales

Krzysztof Paczka

arxiv:1205.6976 v3 · 2012-05-31 · math.PR

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{FOPLMBBPTND4T7M2GPMKANSK7I}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

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.

Receipt and verification

First computed	2026-05-18T03:39:52.940193Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

2b9eb6042f9b47c9fd9a33d8a0364afa3291ab15b4c1d2b0698b8eb0d17ffd37

Aliases

arxiv: 1205.6976 · arxiv_version: 1205.6976v3 · doi: 10.48550/arxiv.1205.6976 · pith_short_12: FOPLMBBPTND4 · pith_short_16: FOPLMBBPTND4T7M2 · pith_short_8: FOPLMBBP

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/FOPLMBBPTND4T7M2GPMKANSK7I \
  | 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: 2b9eb6042f9b47c9fd9a33d8a0364afa3291ab15b4c1d2b0698b8eb0d17ffd37

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "4143f414df0aae344d20a708ede14c66e3968ea643592008f9ed7e420add35a7",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2012-05-31T12:38:24Z",
    "title_canon_sha256": "eb87ff2c978fa8e20636397b20f8f2f31995adaf81b5a45c3e9bbcaf3e97e7dd"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1205.6976",
    "kind": "arxiv",
    "version": 3
  }
}