Pith Number

pith:QN6NZ5RF

pith:2026:QN6NZ5RFLRNEEWWMZUZM3A3S7X

not attested not anchored not stored refs pending

Hidden critical and Morse equivalence behind duality: Theory and Applications

Dong Zhang

arxiv:2606.27004 v1 · 2026-06-25 · math.FA · math.CO · math.MG · math.SP

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{QN6NZ5RFLRNEEWWMZUZM3A3S7X}

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	2026-06-26T01:16:06.458411Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

837cdcf6255c5a425acccd32cd8372fded8384aec75c346604accfee26e2cb07

Aliases

arxiv: 2606.27004 · arxiv_version: 2606.27004v1 · doi: 10.48550/arxiv.2606.27004 · pith_short_12: QN6NZ5RFLRNE · pith_short_16: QN6NZ5RFLRNEEWWM · pith_short_8: QN6NZ5RF

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/QN6NZ5RFLRNEEWWMZUZM3A3S7X \
  | 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: 837cdcf6255c5a425acccd32cd8372fded8384aec75c346604accfee26e2cb07

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "82e5bb95b15d96412d4564dce089643cdf1f8c3cdc0d51363930796692dfa931",
    "cross_cats_sorted": [
      "math.CO",
      "math.MG",
      "math.SP"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2026-06-25T13:20:14Z",
    "title_canon_sha256": "04aec0d6717b1326420d596e5cf069447f3986d7122ec894fab8a21753d7a0d6"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2606.27004",
    "kind": "arxiv",
    "version": 1
  }
}