Pith Number

pith:NP5QPYWF

pith:2026:NP5QPYWFCFAMJ3UGF6ZZPWUPZ7

not attested not anchored not stored refs pending

A First-Order Mean Field Control Analysis of Transformer Layers under Cross-Entropy Training

Cheng Huan, Hongwei Yuan

arxiv:2606.23235 v1 · 2026-06-22 · math.OC · math.DS · stat.ML

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{NP5QPYWFCFAMJ3UGF6ZZPWUPZ7}

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	2026-06-23T03:14:14.519940Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6bfb07e2c51140c4ee862fb397da8fcfece5491c3b0e03ef0a431c0103eddb7e

Aliases

arxiv: 2606.23235 · arxiv_version: 2606.23235v1 · doi: 10.48550/arxiv.2606.23235 · pith_short_12: NP5QPYWFCFAM · pith_short_16: NP5QPYWFCFAMJ3UG · pith_short_8: NP5QPYWF

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/NP5QPYWFCFAMJ3UGF6ZZPWUPZ7 \
  | 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: 6bfb07e2c51140c4ee862fb397da8fcfece5491c3b0e03ef0a431c0103eddb7e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "eaa4b69ed2eb455472c3cbabaf1493147aaff731fa9ed14c770e52476325b0c2",
    "cross_cats_sorted": [
      "math.DS",
      "stat.ML"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-06-22T12:21:37Z",
    "title_canon_sha256": "70c95f366bc444c3140906f2e26a3f213a302d3baf98223c1672b470f41f35c5"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2606.23235",
    "kind": "arxiv",
    "version": 1
  }
}