Pith Number

pith:VSKPG2W2

pith:2025:VSKPG2W2KAYYAXAOCHXUMB5R44

not attested not anchored not stored refs pending

Gravitational Lensing by Black Holes in Einstein-nonlinear Electrodynamic Theories with Multiple Photon Spheres

Benrong Mu, Peng Wang, Siyuan Hui

arxiv:2507.13048 v1 · 2025-07-17 · gr-qc

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{VSKPG2W2KAYYAXAOCHXUMB5R44}

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

2 papers in Pith

Boundary-only weak deflection angles from isothermal optical geometry

Topologically equivalent yet radiatively distinct orbits in EMRI system

Receipt and verification

First computed	2026-07-05T11:38:53.827619Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ac94f36ada5031805c0e11ef4607b1e72e39ea0327fa13d73c26e0fd4385c878

Aliases

arxiv: 2507.13048 · arxiv_version: 2507.13048v1 · doi: 10.48550/arxiv.2507.13048 · pith_short_12: VSKPG2W2KAYY · pith_short_16: VSKPG2W2KAYYAXAO · pith_short_8: VSKPG2W2

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/VSKPG2W2KAYYAXAOCHXUMB5R44 \
  | 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: ac94f36ada5031805c0e11ef4607b1e72e39ea0327fa13d73c26e0fd4385c878

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e078040fe5b1a13f6c5e1011f89814d04989cf6a5c7c5c8b5d628a937f6fefa6",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "gr-qc",
    "submitted_at": "2025-07-17T12:21:04Z",
    "title_canon_sha256": "82b6fd0fbe82e41a3e78e358e977f4efe87c4f6f770ee37c5834fca13ba72642"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2507.13048",
    "kind": "arxiv",
    "version": 1
  }
}