Pith Number

pith:KWPGVB7Q

pith:2026:KWPGVB7QKZCX6YMUIZXWFML7UX

not attested not anchored not stored refs pending

Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$

Dachun Yang, Rupert L. Frank, Yinqin Li

arxiv:2607.06415 v1 · 2026-07-07 · math.AP · math.CA · math.FA

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{KWPGVB7QKZCX6YMUIZXWFML7UX}

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	2026-07-08T01:19:24.733629Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

559e6a87f056457f6194466f62b17fa5d753fc5b8affb7367126c02969b48d17

Aliases

arxiv: 2607.06415 · arxiv_version: 2607.06415v1 · doi: 10.48550/arxiv.2607.06415 · pith_short_12: KWPGVB7QKZCX · pith_short_16: KWPGVB7QKZCX6YMU · pith_short_8: KWPGVB7Q

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/KWPGVB7QKZCX6YMUIZXWFML7UX \
  | 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: 559e6a87f056457f6194466f62b17fa5d753fc5b8affb7367126c02969b48d17

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "43f1273b5bda90c6d24dd9c290fc5cb6862b6dc356275373dd37d12876af1261",
    "cross_cats_sorted": [
      "math.CA",
      "math.FA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-07-07T15:45:48Z",
    "title_canon_sha256": "a58f5f348d8c15c50401dbd24e14b72322c4a94752cfeba365702936a4b15c3e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2607.06415",
    "kind": "arxiv",
    "version": 1
  }
}