Pith Number

pith:52OVBDMC

pith:2011:52OVBDMCE4RQASKDZTMQEIJHTY

not attested not anchored not stored refs pending

Random groups and Property (T): \.Zuk's theorem revisited

Marcin Kotowski, Michal Kotowski

arxiv:1106.2242 v2 · 2011-06-11 · 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{52OVBDMCE4RQASKDZTMQEIJHTY}

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	2026-05-18T03:16:50.017226Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ee9d508d822723004943ccd90221279e14d96b786d05e826a63bb58d6c511b8f

Aliases

arxiv: 1106.2242 · arxiv_version: 1106.2242v2 · doi: 10.48550/arxiv.1106.2242 · pith_short_12: 52OVBDMCE4RQ · pith_short_16: 52OVBDMCE4RQASKD · pith_short_8: 52OVBDMC

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/52OVBDMCE4RQASKDZTMQEIJHTY \
  | 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: ee9d508d822723004943ccd90221279e14d96b786d05e826a63bb58d6c511b8f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b470dc9101cf460fac76b4e5939f70c0e54c1a67da2e23064521cab6d07d08b4",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GR",
    "submitted_at": "2011-06-11T14:33:17Z",
    "title_canon_sha256": "798ca369c1ee6977a29afa1d537e1af3970298c01692b86d5095356ce3e2ccf7"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1106.2242",
    "kind": "arxiv",
    "version": 2
  }
}