Pith Number

pith:G232HUL5

pith:2013:G232HUL5KH6MPSKMWVO567RJF3

not attested not anchored not stored refs pending

Beyond Expansion IV: Traces of Thin Semigroups

Alex Kontorovich, Jean Bourgain

arxiv:1310.7190 v3 · 2013-10-27 · math.NT

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{G232HUL5KH6MPSKMWVO567RJF3}

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	2026-05-18T00:18:56.255064Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

36b7a3d17d51fcc7c94cb55ddf7e292ef63ba24d8a6f621319734ac19b7d9cd2

Aliases

arxiv: 1310.7190 · arxiv_version: 1310.7190v3 · doi: 10.48550/arxiv.1310.7190 · pith_short_12: G232HUL5KH6M · pith_short_16: G232HUL5KH6MPSKM · pith_short_8: G232HUL5

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/G232HUL5KH6MPSKMWVO567RJF3 \
  | 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: 36b7a3d17d51fcc7c94cb55ddf7e292ef63ba24d8a6f621319734ac19b7d9cd2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f4c4a305b1d10f8a4585f125fcdf7bbe40d1ce706409188b2c6f9543bc5326e1",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2013-10-27T12:43:51Z",
    "title_canon_sha256": "34603c322c78736bf1d25743c561d14a7fee2f60851862f4688268a5c079bd29"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1310.7190",
    "kind": "arxiv",
    "version": 3
  }
}