Pith Number

pith:5WISRY6W

pith:2025:5WISRY6WO36UUZIUOS4P2FVECW

not attested not anchored not stored refs pending

Random graphs, expanding families and the construction of noncompact hyperbolic surfaces with uniform spectral gaps

Bobo Hua, Qi Guo, Yang Shen

arxiv:2507.16794 v2 · 2025-07-22 · math.DG · math.CO · math.GT

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{5WISRY6WO36UUZIUOS4P2FVECW}

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	2026-07-05T11:44:09.867630Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ed9128e3d676fd4a651474b8fd16a415b9ed20a9e2f4d8f4390a2b67c3bd09bb

Aliases

arxiv: 2507.16794 · arxiv_version: 2507.16794v2 · doi: 10.48550/arxiv.2507.16794 · pith_short_12: 5WISRY6WO36U · pith_short_16: 5WISRY6WO36UUZIU · pith_short_8: 5WISRY6W

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/5WISRY6WO36UUZIUOS4P2FVECW \
  | 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: ed9128e3d676fd4a651474b8fd16a415b9ed20a9e2f4d8f4390a2b67c3bd09bb

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b03b7b0f93d158b64f5c135c4fb40c40699d9f2c0c70e1768bca20e459c6328c",
    "cross_cats_sorted": [
      "math.CO",
      "math.GT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2025-07-22T17:44:22Z",
    "title_canon_sha256": "9c3ca56b2381b7c558b03615ba085d8bb233950bb9ae5a184d8d99086e412c90"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2507.16794",
    "kind": "arxiv",
    "version": 2
  }
}