Pith Number

pith:4KGWK3NJ

pith:2012:4KGWK3NJCEX2CL4NJBSCJULUYW

not attested not anchored not stored refs pending

Explicit growth and expansion for SL_2

Emmanuel Kowalski

arxiv:1201.1139 v3 · 2012-01-05 · math.NT · math.GR

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{4KGWK3NJCEX2CL4NJBSCJULUYW}

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	2026-05-18T03:52:11.998816Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e28d656da9112fa12f8d486424d174c5a2bc2af656a7f25c0fc2e998037594d9

Aliases

arxiv: 1201.1139 · arxiv_version: 1201.1139v3 · doi: 10.48550/arxiv.1201.1139 · pith_short_12: 4KGWK3NJCEX2 · pith_short_16: 4KGWK3NJCEX2CL4N · pith_short_8: 4KGWK3NJ

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/4KGWK3NJCEX2CL4NJBSCJULUYW \
  | 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: e28d656da9112fa12f8d486424d174c5a2bc2af656a7f25c0fc2e998037594d9

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9106136a7eaa815adb36469c6b08d1bb24aafe4cc4a71e2a0872c5209ea81761",
    "cross_cats_sorted": [
      "math.GR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2012-01-05T11:59:45Z",
    "title_canon_sha256": "5d0a76888cedf3cf4752365965aed3d09d38cd8d63e28ab9f43d2e92f12c87ef"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1201.1139",
    "kind": "arxiv",
    "version": 3
  }
}