Pith Number

pith:ZO2E76P6

pith:2026:ZO2E76P6AX2YMVRA7VLF3VSIX5

not attested not anchored not stored refs pending

Probability, Curvature and Spectrum on Graphs

Tianhong Zhao

arxiv:2607.21639 v1 · 2026-07-21 · math-ph · math.MP · math.PR · math.SP · quant-ph

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{ZO2E76P6AX2YMVRA7VLF3VSIX5}

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	2026-07-27T00:20:12.386012Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

cbb44ff9fe05f5865620fd565dd648bf55e9de5f338babc60b9daaf205bc040a

Aliases

arxiv: 2607.21639 · arxiv_version: 2607.21639v1 · doi: 10.48550/arxiv.2607.21639 · pith_short_12: ZO2E76P6AX2Y · pith_short_16: ZO2E76P6AX2YMVRA · pith_short_8: ZO2E76P6

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/ZO2E76P6AX2YMVRA7VLF3VSIX5 \
  | 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: cbb44ff9fe05f5865620fd565dd648bf55e9de5f338babc60b9daaf205bc040a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "bac8a7a8ece321c84d1078007425437085a47e02504ee71a72ca1bd99dc5373f",
    "cross_cats_sorted": [
      "math.MP",
      "math.PR",
      "math.SP",
      "quant-ph"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math-ph",
    "submitted_at": "2026-07-21T13:54:06Z",
    "title_canon_sha256": "8a79270b0c1056a00ff6857aa74e6d12010a7d69530b908f234d70ffb9a7aee5"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2607.21639",
    "kind": "arxiv",
    "version": 1
  }
}