Pith Number

pith:YWA7VINP

pith:2000:YWA7VINPDAODXSZNEBBHUBSMXH

not attested not anchored not stored refs pending

The critical group of a directed graph

David G. Wagner

arxiv:math/0010241 v1 · 2000-10-25 · math.CO · math.RA

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{YWA7VINPDAODXSZNEBBHUBSMXH}

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.

Cited by

1 paper in Pith

Chip-Firing Games and Critical Groups

Receipt and verification

First computed	2026-07-04T14:34:53.121637Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c581faa1af181c3bcb2d20427a064cb9e0f854d37201c14f08f8bfeba0b46fae

Aliases

arxiv: math/0010241 · arxiv_version: math/0010241v1 · doi: 10.48550/arxiv.math/0010241 · pith_short_12: YWA7VINPDAOD · pith_short_16: YWA7VINPDAODXSZN · pith_short_8: YWA7VINP

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/YWA7VINPDAODXSZNEBBHUBSMXH \
  | 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: c581faa1af181c3bcb2d20427a064cb9e0f854d37201c14f08f8bfeba0b46fae

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "dbbe8fdeb97fc55b68464d03a1c7616f1428d3611bb4b9969b778a25e330b6fb",
    "cross_cats_sorted": [
      "math.RA"
    ],
    "license": "",
    "primary_cat": "math.CO",
    "submitted_at": "2000-10-25T19:53:45Z",
    "title_canon_sha256": "1989aaca6b879af2dc130319e929eb8d902dbbc1545652e03d5a77504100bcd9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "math/0010241",
    "kind": "arxiv",
    "version": 1
  }
}