Pith Number

pith:OX65742W

pith:2026:OX65742WCXUKL7JUH5QDPTV2AH

not attested not anchored not stored refs pending

The Twin-Width of Graphs of Bounded VC-Dimension

Sophie Spirkl, Taite LaGrange, Therese Biedl

arxiv:2606.21640 v1 · 2026-06-19 · math.CO · cs.DM

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{OX65742WCXUKL7JUH5QDPTV2AH}

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	2026-06-23T01:13:17.573624Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

75fddff35615e8a5fd343f6037ceba01c7638272fb0ccc2eb6057e327194e0fa

Aliases

arxiv: 2606.21640 · arxiv_version: 2606.21640v1 · doi: 10.48550/arxiv.2606.21640 · pith_short_12: OX65742WCXUK · pith_short_16: OX65742WCXUKL7JU · pith_short_8: OX65742W

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/OX65742WCXUKL7JUH5QDPTV2AH \
  | 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: 75fddff35615e8a5fd343f6037ceba01c7638272fb0ccc2eb6057e327194e0fa

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "95580b58263d3fc1d406ec14135422f19aa58868ba7bf448373aff851c197d91",
    "cross_cats_sorted": [
      "cs.DM"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-06-19T17:48:49Z",
    "title_canon_sha256": "72d307b9350202dbe69cc4bf9c295928ea9145d073b946d8bbc2661f5ed5101f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2606.21640",
    "kind": "arxiv",
    "version": 1
  }
}