Pith Number

pith:WQNTYZ6X

pith:2026:WQNTYZ6X4NJKMIWHWKNNASXE2Z

not attested not anchored not stored refs pending

Sharp and Endpoint Two-Weight Fractional Integral Estimates for Schr"odinger Operators with Inverse-Square Potentials

Haochen Liu, Hongyan Zhou, Qinghao Yu

arxiv:2607.09585 v1 · 2026-07-10 · math.CA · math-ph · math.AP · math.MP

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{WQNTYZ6X4NJKMIWHWKNNASXE2Z}

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	2026-07-13T01:20:21.923988Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b41b3c67d7e352a622c7b29ad04ae4d64b8b0efb64926ce9500d7492dd97b03f

Aliases

arxiv: 2607.09585 · arxiv_version: 2607.09585v1 · doi: 10.48550/arxiv.2607.09585 · pith_short_12: WQNTYZ6X4NJK · pith_short_16: WQNTYZ6X4NJKMIWH · pith_short_8: WQNTYZ6X

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/WQNTYZ6X4NJKMIWHWKNNASXE2Z \
  | 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: b41b3c67d7e352a622c7b29ad04ae4d64b8b0efb64926ce9500d7492dd97b03f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3c0f61f25fd88638f5c24832d600247f1917865be877f22d7ea83ce561d0ea52",
    "cross_cats_sorted": [
      "math-ph",
      "math.AP",
      "math.MP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CA",
    "submitted_at": "2026-07-10T16:35:43Z",
    "title_canon_sha256": "f78fbd47d915a3bd175b59a76032c16731a73ecf01f94b49696ea566a7d704cf"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2607.09585",
    "kind": "arxiv",
    "version": 1
  }
}