Pith Number

pith:4FTBIYHO

pith:2010:4FTBIYHO4FB6HSD3VULTJ7JX6H

not attested not anchored not stored refs pending

Perturbation theory of multi-plane lens effects in terms of mass ratios: Approximate expressions of lensed-image positions for two lens planes

Hideki Asada, Koji Izumi

arxiv:1012.0070 v3 · 2010-12-01 · astro-ph.CO · 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{4FTBIYHO4FB6HSD3VULTJ7JX6H}

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	2026-05-18T04:01:02.502030Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e1661460eee143e3c87bad1734fd37f1f5348b87e9a8f771fc292b4faaa11540

Aliases

arxiv: 1012.0070 · arxiv_version: 1012.0070v3 · doi: 10.48550/arxiv.1012.0070 · pith_short_12: 4FTBIYHO4FB6 · pith_short_16: 4FTBIYHO4FB6HSD3 · pith_short_8: 4FTBIYHO

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/4FTBIYHO4FB6HSD3VULTJ7JX6H \
  | 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: e1661460eee143e3c87bad1734fd37f1f5348b87e9a8f771fc292b4faaa11540

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b94e4713038d67444afb9349d567c886ef860c496e18814661fe006ca8ac1e13",
    "cross_cats_sorted": [
      "gr-qc"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "astro-ph.CO",
    "submitted_at": "2010-12-01T00:30:43Z",
    "title_canon_sha256": "8fabf87fcd64a3ad8be1482e95258850014f634ea64aa59f6c11202679aa413e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1012.0070",
    "kind": "arxiv",
    "version": 3
  }
}