Pith Number

pith:NFMP5U6B

pith:2015:NFMP5U6BPOCMRQ3EFP6EKCIBCA

not attested not anchored not stored refs pending

A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group

Alexander Lazar

arxiv:1509.03380 v3 · 2015-09-11 · math.CO · math.AG

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{NFMP5U6BPOCMRQ3EFP6EKCIBCA}

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	2026-05-18T00:25:12.079871Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6958fed3c17b84c8c3642bfc450901100be4b77815d8661ed260d8f6eb9f98c7

Aliases

arxiv: 1509.03380 · arxiv_version: 1509.03380v3 · doi: 10.48550/arxiv.1509.03380 · pith_short_12: NFMP5U6BPOCM · pith_short_16: NFMP5U6BPOCMRQ3E · pith_short_8: NFMP5U6B

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/NFMP5U6BPOCMRQ3EFP6EKCIBCA \
  | 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: 6958fed3c17b84c8c3642bfc450901100be4b77815d8661ed260d8f6eb9f98c7

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8c320069b6659a51dec5fb0a622c1768a4b46bd3b4462ab79943520ab3d70589",
    "cross_cats_sorted": [
      "math.AG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2015-09-11T03:14:11Z",
    "title_canon_sha256": "7baaa4be6c28ccfb3ba6f5cb8df7f15deda52b026130dec2b60151603d3ffbb5"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1509.03380",
    "kind": "arxiv",
    "version": 3
  }
}