Pith Number

pith:6RA3EMST

pith:2026:6RA3EMSTGXBO4JI67R4ZZPQGR7

not attested not anchored not stored refs pending

Subdivided expanders and counterexamples to the Tree Product Conjecture

Andrea Munaro

arxiv:2608.04659 v1 · 2026-08-05 · math.CO

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{6RA3EMSTGXBO4JI67R4ZZPQGR7}

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	2026-08-06T01:46:06.991580Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f441b2325335c2ee251efc799cbe068fefb3b35547be5c3d289b28988b013882

Aliases

arxiv: 2608.04659 · arxiv_version: 2608.04659v1 · doi: 10.48550/arxiv.2608.04659 · pith_short_12: 6RA3EMSTGXBO · pith_short_16: 6RA3EMSTGXBO4JI6 · pith_short_8: 6RA3EMST

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/6RA3EMSTGXBO4JI67R4ZZPQGR7 \
  | 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: f441b2325335c2ee251efc799cbe068fefb3b35547be5c3d289b28988b013882

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0e58e35dd0f8d839180742890d2a921043bde780ac78e7362ac3bc7bc518c9f6",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-08-05T10:14:20Z",
    "title_canon_sha256": "e90e214eb25acea2c0c4e3a0b86981f8c31db4b3134feb700d09322ad84e379d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2608.04659",
    "kind": "arxiv",
    "version": 1
  }
}