Pith Number

pith:GCFU5BFI

pith:2009:GCFU5BFITEEOOID7BXHU2ULTNS

not attested not anchored not stored refs pending

Context-free pairs of groups. II - cuts, tree sets, and random walks

Wolfgang Woess

arxiv:0911.0134 v1 · 2009-11-01 · math.GR · 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{GCFU5BFITEEOOID7BXHU2ULTNS}

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	2026-05-18T03:39:29.656815Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

308b4e84a89908e7207f0dcf4d51736c836b754e10e31febf3e97da35a23e21b

Aliases

arxiv: 0911.0134 · arxiv_version: 0911.0134v1 · doi: 10.48550/arxiv.0911.0134 · pith_short_12: GCFU5BFITEEO · pith_short_16: GCFU5BFITEEOOID7 · pith_short_8: GCFU5BFI

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/GCFU5BFITEEOOID7BXHU2ULTNS \
  | 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: 308b4e84a89908e7207f0dcf4d51736c836b754e10e31febf3e97da35a23e21b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "245c6e1e7eaa89fbc3fd5a65b6e8fa131eec9a659e45822a4165dedfb510ed21",
    "cross_cats_sorted": [
      "math.PR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GR",
    "submitted_at": "2009-11-01T06:55:49Z",
    "title_canon_sha256": "5e265550b8d34a324657d7ef80ac7329f5ab3ad6530af26e117d8416feb40592"
  },
  "schema_version": "1.0",
  "source": {
    "id": "0911.0134",
    "kind": "arxiv",
    "version": 1
  }
}