Pith Number

pith:X5XKJTL3

pith:2026:X5XKJTL3FICHP5EPTGO2VB7NDY

not attested not anchored not stored refs pending

A Response Calculus for Liouville Brownian Motion I: Simple Spectrum, Joint Eigenvalue Densities, and Ward Identities

Chunhao Cai

arxiv:2608.02459 v1 · 2026-08-03 · math.PR · math-ph · math.MP · math.SP

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{X5XKJTL3FICHP5EPTGO2VB7NDY}

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	2026-08-04T02:43:37.599575Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

bf6ea4cd7b2a0477f48f999daa87ed1e004dfce99c2697ae59f3114b87cf0e33

Aliases

arxiv: 2608.02459 · arxiv_version: 2608.02459v1 · doi: 10.48550/arxiv.2608.02459 · pith_short_12: X5XKJTL3FICH · pith_short_16: X5XKJTL3FICHP5EP · pith_short_8: X5XKJTL3

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/X5XKJTL3FICHP5EPTGO2VB7NDY \
  | 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: bf6ea4cd7b2a0477f48f999daa87ed1e004dfce99c2697ae59f3114b87cf0e33

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "32a8a7367cc3726aff1c757e6d1ec3d1a1ae8282ceea240e1fdea048c84f19bb",
    "cross_cats_sorted": [
      "math-ph",
      "math.MP",
      "math.SP"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-08-03T16:31:17Z",
    "title_canon_sha256": "3d18e53188e47602e2580e4e66163c1d86d9ee7a8c695cc283000d775682d1b1"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2608.02459",
    "kind": "arxiv",
    "version": 1
  }
}