Pith Number

pith:OCNWHQG5

pith:2026:OCNWHQG5TSI74JWCSPYC4IN5GC

not attested not anchored not stored refs pending

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning

Nima Rezaei, Stephan Wojtowytsch

arxiv:2607.25905 v1 · 2026-07-28 · math.AP · 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{OCNWHQG5TSI74JWCSPYC4IN5GC}

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	2026-07-29T01:26:10.352346Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

709b63c0dd9c91fe26c293f02e21bd3087146ed0608816eecaf6c924248831ef

Aliases

arxiv: 2607.25905 · arxiv_version: 2607.25905v1 · doi: 10.48550/arxiv.2607.25905 · pith_short_12: OCNWHQG5TSI7 · pith_short_16: OCNWHQG5TSI74JWC · pith_short_8: OCNWHQG5

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/OCNWHQG5TSI74JWCSPYC4IN5GC \
  | 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: 709b63c0dd9c91fe26c293f02e21bd3087146ed0608816eecaf6c924248831ef

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "ff2211c6be8b5534114b3620dfd79b6d98009bb055c986c26449440a6309b1b2",
    "cross_cats_sorted": [
      "stat.ML"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-07-28T16:01:41Z",
    "title_canon_sha256": "424d223cffa81a27f4881a939eda5575bcba73f0066e39ae2d2b3a4af2e78a0f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2607.25905",
    "kind": "arxiv",
    "version": 1
  }
}