Pith Number

pith:O4MJG4KF

pith:2023:O4MJG4KFPUF2O45B7SEEESBPDQ

not attested not anchored not stored refs pending

Gravitational Lensing of Spherically Symmetric Black Holes in Dark Matter Halos

Chen-Kai Qiao, Jun Tao, Yi-Gao Liu

arxiv:2312.15760 v4 · 2023-12-25 · gr-qc · astro-ph.CO · astro-ph.GA

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{O4MJG4KFPUF2O45B7SEEESBPDQ}

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.

Cited by

1 paper in Pith

Reference-renormalized curvature-primitive Gauss-Bonnet formalism for finite-distance weak gravitational lensing in static spherical spacetimes

Receipt and verification

First computed	2026-07-05T09:25:19.532491Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

77189371457d0ba773a1fc8842482f1c156541eff8dde78ed56dd6b2a2f794ed

Aliases

arxiv: 2312.15760 · arxiv_version: 2312.15760v4 · doi: 10.48550/arxiv.2312.15760 · pith_short_12: O4MJG4KFPUF2 · pith_short_16: O4MJG4KFPUF2O45B · pith_short_8: O4MJG4KF

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/O4MJG4KFPUF2O45B7SEEESBPDQ \
  | 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: 77189371457d0ba773a1fc8842482f1c156541eff8dde78ed56dd6b2a2f794ed

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "de44e25aa3d10dbd6511a1ad378120be01c206417d768b78b7d0a57f486d6e45",
    "cross_cats_sorted": [
      "astro-ph.CO",
      "astro-ph.GA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "gr-qc",
    "submitted_at": "2023-12-25T15:51:57Z",
    "title_canon_sha256": "fb53bab377414030dc467ebe687f5f5ac694b22b3d4b1a32f74f49606f8948ad"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2312.15760",
    "kind": "arxiv",
    "version": 4
  }
}