Pith Number

pith:KKOZX3SX

pith:2026:KKOZX3SX6TTP22AEU53QXMRUPL

not attested not anchored not stored refs pending

Gauge coupling beta functions and gauge field anomalous dimensions at four loops in the Standard Model

B.A.Kniehl, M.A.Bezuglov, V.N.Velizhanin

arxiv:2607.21586 v1 · 2026-07-23 · hep-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{KKOZX3SX6TTP22AEU53QXMRUPL}

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	2026-07-24T01:24:41.545661Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

529d9bee57f4e6fd6804a7770bb2347adfd49df5fe65eb0573c68415883d1069

Aliases

arxiv: 2607.21586 · arxiv_version: 2607.21586v1 · doi: 10.48550/arxiv.2607.21586 · pith_short_12: KKOZX3SX6TTP · pith_short_16: KKOZX3SX6TTP22AE · pith_short_8: KKOZX3SX

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/KKOZX3SX6TTP22AEU53QXMRUPL \
  | 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: 529d9bee57f4e6fd6804a7770bb2347adfd49df5fe65eb0573c68415883d1069

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9ba0858a371a5e764e21853e1aacb46f683f12888fed4e5d68317c01d29d627a",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "hep-ph",
    "submitted_at": "2026-07-23T17:57:40Z",
    "title_canon_sha256": "cc220c7391838bd1b341b773cdd6ef741afa2a2e5fabc831569b91e4c163ed55"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2607.21586",
    "kind": "arxiv",
    "version": 1
  }
}