Pith Number

pith:PJTUJ6LE

pith:2026:PJTUJ6LEUQGQP22F5PEKITF6CR

not attested not anchored not stored refs pending

A phase transition for the hard sphere model on the hyperbolic plane

Lewis Bowen, Marcus Michelen, Will Perkins

arxiv:2607.23873 v1 · 2026-07-26 · math-ph · math.DS · math.MP · math.PR

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{PJTUJ6LEUQGQP22F5PEKITF6CR}

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	2026-07-28T01:23:15.525529Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7a6744f964a40d07eb45ebc8a44cbe14771a304d0a5ff5f0b2c9adf8352ef0b1

Aliases

arxiv: 2607.23873 · arxiv_version: 2607.23873v1 · doi: 10.48550/arxiv.2607.23873 · pith_short_12: PJTUJ6LEUQGQ · pith_short_16: PJTUJ6LEUQGQP22F · pith_short_8: PJTUJ6LE

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/PJTUJ6LEUQGQP22F5PEKITF6CR \
  | 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: 7a6744f964a40d07eb45ebc8a44cbe14771a304d0a5ff5f0b2c9adf8352ef0b1

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b1e6108fbac992442e2c0bd9fbe53a0aa190dec2c503d92b81831d9f8e8aad4b",
    "cross_cats_sorted": [
      "math.DS",
      "math.MP",
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math-ph",
    "submitted_at": "2026-07-26T22:28:10Z",
    "title_canon_sha256": "874c7a58f3f8743a659dd65731a49ca7b477782710f4f40f138cebd4ad4c6182"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2607.23873",
    "kind": "arxiv",
    "version": 1
  }
}