Pith Number

pith:L3WFYWYZ

pith:2026:L3WFYWYZLX4NV2LGGNKPTGKYF5

not attested not anchored not stored refs pending

Disproof of the tree product conjecture via the Heisenberg group

Freddie Illingworth, Raphael Steiner, Sergey Norin

arxiv:2607.03041 v1 · 2026-07-03 · math.CO · math.GR · math.MG

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{L3WFYWYZLX4NV2LGGNKPTGKYF5}

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.

Cited by

1 paper in Pith

Subdivided expanders and counterexamples to the Tree Product Conjecture

Receipt and verification

First computed	2026-07-07T01:16:40.884627Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5eec5c5b195df8dae9663354f999582f40014990723c8fa23738c2894b00ca6d

Aliases

arxiv: 2607.03041 · arxiv_version: 2607.03041v1 · doi: 10.48550/arxiv.2607.03041 · pith_short_12: L3WFYWYZLX4N · pith_short_16: L3WFYWYZLX4NV2LG · pith_short_8: L3WFYWYZ

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/L3WFYWYZLX4NV2LGGNKPTGKYF5 \
  | 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: 5eec5c5b195df8dae9663354f999582f40014990723c8fa23738c2894b00ca6d

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6699d8ab728169a27db069b9405d1eba9cacbd25a6a7cd8e75d090be5030a1c5",
    "cross_cats_sorted": [
      "math.GR",
      "math.MG"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-07-03T07:31:59Z",
    "title_canon_sha256": "8c60a26f30bf338785d4c3b673039ebf1a38edd57687e3040bc504fb4827448a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2607.03041",
    "kind": "arxiv",
    "version": 1
  }
}