Pith Number

pith:T4HD7RGA

pith:2026:T4HD7RGAMDXLFP6YCG5VGHKUYY

not attested not anchored not stored refs pending

Grokking Finite-Dimensional Algebra

Guillaume Dumas, Guillaume Rabusseau, Pascal Jr Tikeng Notsawo

Neural networks grok algebra multiplication once they recover the bilinear product from the structure tensor.

arxiv:2602.19533 v2 · 2026-02-23 · cs.LG · cs.AI · math.RA

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\usepackage{pith}
\pithnumber{T4HD7RGAMDXLFP6YCG5VGHKUYY}

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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.

Claims

C1strongest claim

Learning multiplication in finite-dimensional algebras amounts to learning a bilinear product specified by the algebra's structure tensor, and grokking emerges naturally as models learn discrete representations for algebras over finite fields.

C2weakest assumption

That the experimental models are actually learning the algebra's multiplication via the structure tensor rather than some other shortcut that happens to correlate with the target operation.

C3one line summary

Neural networks learning multiplication in finite-dimensional algebras show grokking whose timing depends on algebraic properties like commutativity and the rank/sparsity of the structure tensor.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:15.997675Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

9f0e3fc4c060eeb2bfd811bb531d54c60b5fd151f776a6801ae661123109ddf4

Aliases

arxiv: 2602.19533 · arxiv_version: 2602.19533v2 · doi: 10.48550/arxiv.2602.19533 · pith_short_12: T4HD7RGAMDXL · pith_short_16: T4HD7RGAMDXLFP6Y · pith_short_8: T4HD7RGA

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/T4HD7RGAMDXLFP6YCG5VGHKUYY \
  | 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: 9f0e3fc4c060eeb2bfd811bb531d54c60b5fd151f776a6801ae661123109ddf4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "58560f6e7a9c2e2cca50ad7d25a1547a43029aaa41cc9c5723add94232ffe862",
    "cross_cats_sorted": [
      "cs.AI",
      "math.RA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-02-23T05:55:52Z",
    "title_canon_sha256": "feed14295d552fef19e547aaa5c0248c6852af8b588d8875faa02f59a6fabbe1"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2602.19533",
    "kind": "arxiv",
    "version": 2
  }
}