Pith Number

pith:4RXKNIRJ

pith:2004:4RXKNIRJXTLYOU37AEHDPIR2LU

not attested not anchored not stored refs pending

Average magnification effect of clumping of matter

Richard Lieu, T.W.B. Kibble

arxiv:astro-ph/0412275 v2 · 2004-12-11 · astro-ph

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{4RXKNIRJXTLYOU37AEHDPIR2LU}

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

Light path averages in spacetimes with non-vanishing average spatial curvature

Receipt and verification

First computed	2026-07-04T16:45:24.024718Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e46ea6a229bcd787537f010e37a23a5d1d13e02b00b9e00af9aaec17c763d409

Aliases

arxiv: astro-ph/0412275 · arxiv_version: astro-ph/0412275v2 · doi: 10.48550/arxiv.astro-ph/0412275 · pith_short_12: 4RXKNIRJXTLY · pith_short_16: 4RXKNIRJXTLYOU37 · pith_short_8: 4RXKNIRJ

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/4RXKNIRJXTLYOU37AEHDPIR2LU \
  | 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: e46ea6a229bcd787537f010e37a23a5d1d13e02b00b9e00af9aaec17c763d409

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "fe5e21de52b33fcbdb9ea361f53dd0e96a9d1a97a3a1588247eef100302f52d7",
    "cross_cats_sorted": [],
    "license": "",
    "primary_cat": "astro-ph",
    "submitted_at": "2004-12-11T23:54:00Z",
    "title_canon_sha256": "c2038af049b8d3b8918c5dc9fdb9c5d392a865b13b2f622e9da863f540c3e7bd"
  },
  "schema_version": "1.0",
  "source": {
    "id": "astro-ph/0412275",
    "kind": "arxiv",
    "version": 2
  }
}