Pith Number

pith:PP5KWKIK

pith:2009:PP5KWKIKXBXAIAPOD7CQNVNQLL

not attested not anchored not stored refs pending

Approximate groups and doubling metrics

Tom Sanders

arxiv:0912.0305 v2 · 2009-12-02 · math.CA · math.CO

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{PP5KWKIKXBXAIAPOD7CQNVNQLL}

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	2026-05-18T03:39:29.646922Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7bfaab290ab86e0401ee1fc506d5b05ae03a69234cd3343c160ab8642943db27

Aliases

arxiv: 0912.0305 · arxiv_version: 0912.0305v2 · doi: 10.48550/arxiv.0912.0305 · pith_short_12: PP5KWKIKXBXA · pith_short_16: PP5KWKIKXBXAIAPO · pith_short_8: PP5KWKIK

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/PP5KWKIKXBXAIAPOD7CQNVNQLL \
  | 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: 7bfaab290ab86e0401ee1fc506d5b05ae03a69234cd3343c160ab8642943db27

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "c69b7a6b939b350af201499bc6ddb6f917d85894efcd60a03993b464a21759fb",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CA",
    "submitted_at": "2009-12-02T00:12:16Z",
    "title_canon_sha256": "f11eaeef42adf9126e2e52590a4ee6be11bfafde014d77ccc1f21b13e10c3088"
  },
  "schema_version": "1.0",
  "source": {
    "id": "0912.0305",
    "kind": "arxiv",
    "version": 2
  }
}