Pith Number

pith:LPS5TBRG

pith:2026:LPS5TBRGTIWU7ELFMZHFSM5HZC

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Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks

Duxin Chen, Jiawen Chen, Qi Shao, Wenwu Yu, Zhiqiang Ge

Upper and lower neighborhoods suffice to achieve the full expressive power of the Combinatorial Complex Weisfeiler-Lehman test.

arxiv:2605.00725 v2 · 2026-05-01 · cs.LG

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Claims

C1strongest claim

We further prove that upper and lower neighborhoods are sufficient among the four adjacent WL tests to reach the expressivity of the full CCWL framework across topological structures of combinatorial complexes.

C2weakest assumption

That the four neighborhood relations defined in the combinatorial complex framework are exhaustive and that the axiomatic extension of the WL test preserves all relevant topological distinctions without additional constraints on the complexes.

C3one line summary

The Combinatorial Complex Weisfeiler-Lehman test unifies expressive-power analysis for topological neural networks on combinatorial complexes, with a proof that upper and lower neighborhoods suffice for maximum expressivity.

Receipt and verification

First computed	2026-06-19T16:09:58.762305Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5be5d986269a2d4f9165664e5933a7c89cdd65b02d55235b711dae6484db6402

Aliases

arxiv: 2605.00725 · arxiv_version: 2605.00725v2 · doi: 10.48550/arxiv.2605.00725 · pith_short_12: LPS5TBRGTIWU · pith_short_16: LPS5TBRGTIWU7ELF · pith_short_8: LPS5TBRG

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/LPS5TBRGTIWU7ELFMZHFSM5HZC \
  | 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: 5be5d986269a2d4f9165664e5933a7c89cdd65b02d55235b711dae6484db6402

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "5d8255e42472a9c9b110513e6ef537bf6b6154aa31d6cac8fe1e317edde4ee00",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-01T15:19:40Z",
    "title_canon_sha256": "56cff3031d4c4a74d444e4542dfc16da40597f4f9e8516f95146106f44b69a3f"
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  "source": {
    "id": "2605.00725",
    "kind": "arxiv",
    "version": 2
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}