Pith Number

pith:BT4O23ZH

pith:2026:BT4O23ZHSAJZBQDRZIDN45R7EU

not attested not anchored not stored refs pending

Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group

Wenjing Chen, Zexi Wang

arxiv:2608.09034 v1 · 2026-08-10 · math.AP

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{BT4O23ZHSAJZBQDRZIDN45R7EU}

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	2026-08-11T02:21:38.921040Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

0cf8ed6f27901390c071ca06de763f253daa0f8ce586cd2efc88db2c649047eb

Aliases

arxiv: 2608.09034 · arxiv_version: 2608.09034v1 · doi: 10.48550/arxiv.2608.09034 · pith_short_12: BT4O23ZHSAJZ · pith_short_16: BT4O23ZHSAJZBQDR · pith_short_8: BT4O23ZH

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/BT4O23ZHSAJZBQDRZIDN45R7EU \
  | 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: 0cf8ed6f27901390c071ca06de763f253daa0f8ce586cd2efc88db2c649047eb

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "bfbce52dca71e63ba68a66816f536aff2286d6cf12e03ab87ac6b3fc441897ab",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-08-10T02:36:40Z",
    "title_canon_sha256": "198bca0c3a9d10f60d0e00f905ba694fc0b0fd8d382e76167e756492d47bba5a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2608.09034",
    "kind": "arxiv",
    "version": 1
  }
}