Pith Number

pith:YNDNEI7X

pith:2009:YNDNEI7XYQKEOIFL33ZRELUBKL

not attested not anchored not stored refs pending

Lattices with and lattices without spectral gap

Alexander Lubotzky, Bachir Bekka

arxiv:0909.1006 v2 · 2009-09-05 · math.DS · 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{YNDNEI7XYQKEOIFL33ZRELUBKL}

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	2026-05-18T02:28:05.716860Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c346d223f7c4144720abdef3122e8152c6fe71a41e07bc97f77943e3b907a79d

Aliases

arxiv: 0909.1006 · arxiv_version: 0909.1006v2 · doi: 10.48550/arxiv.0909.1006 · pith_short_12: YNDNEI7XYQKE · pith_short_16: YNDNEI7XYQKEOIFL · pith_short_8: YNDNEI7X

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/YNDNEI7XYQKEOIFL33ZRELUBKL \
  | 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: c346d223f7c4144720abdef3122e8152c6fe71a41e07bc97f77943e3b907a79d

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "770312db3004effd8d2226c98746a86c3a61a250246252ab2a5a22f2f2f2b2ad",
    "cross_cats_sorted": [
      "math.GR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2009-09-05T08:40:09Z",
    "title_canon_sha256": "5f04d156929bed3dcd568c7115dd72581408c3a8cfdb95e2dccc2720b6cb7575"
  },
  "schema_version": "1.0",
  "source": {
    "id": "0909.1006",
    "kind": "arxiv",
    "version": 2
  }
}