Pith Number

pith:5DH7D2TF

pith:2018:5DH7D2TFBQCPSWE2GSMF7PGOEJ

not attested not anchored not stored refs pending

A Morse Lemma for degenerate critical points of solutions of nonlinear equations in $\R^2$

Massimo Grossi

arxiv:1806.08559 v1 · 2018-06-22 · 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{5DH7D2TFBQCPSWE2GSMF7PGOEJ}

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	2026-05-18T00:12:36.813782Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e8cff1ea650c04f9589a34985fbcce226717e563d12408864ac653ff2186b90c

Aliases

arxiv: 1806.08559 · arxiv_version: 1806.08559v1 · doi: 10.48550/arxiv.1806.08559 · pith_short_12: 5DH7D2TFBQCP · pith_short_16: 5DH7D2TFBQCPSWE2 · pith_short_8: 5DH7D2TF

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/5DH7D2TFBQCPSWE2GSMF7PGOEJ \
  | 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: e8cff1ea650c04f9589a34985fbcce226717e563d12408864ac653ff2186b90c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "c4fde47e6871b2ddaa1c680a0201dd42606e5141a57d296da1dfc2e78c90a26e",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2018-06-22T08:53:42Z",
    "title_canon_sha256": "9aaf4f45cd32b2801010e1c3f35ed8f6be0ac85604a0bd7c765adb865d2e07f8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1806.08559",
    "kind": "arxiv",
    "version": 1
  }
}