Pith Number

pith:K57ODBMB

pith:2024:K57ODBMBQMSKOWU6CMNU7L63PZ

not attested not anchored not stored refs pending

Emergence in non-neural models: grokking modular arithmetic via average gradient outer product

Adityanarayanan Radhakrishnan, Daniel Beaglehole, Libin Zhu, Mikhail Belkin, Neil Mallinar, Parthe Pandit

arxiv:2407.20199 v3 · 2024-07-29 · stat.ML · cs.LG

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Add to your LaTeX paper

\usepackage{pith}
\pithnumber{K57ODBMBQMSKOWU6CMNU7L63PZ}

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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.

Cited by

8 papers in Pith

K-Inverse-RFM: A Modified RFM that Bridges the Gap to Neural Networks for Data-Corrupted Mathematical Tasks

Law of Neural Interaction: Depth-Width Shape, Interaction Efficiency, and Generalization

Feature Learning in Linear-Width Two-Layer Networks: Two vs. One Step of Gradient Descent

The Geometric Structure of Models Learning Sparse Data

Receipt and verification

First computed	2026-07-05T11:33:59.163208Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

577ee185818324a75a9e131b4fafdb7e5c904ab97c949825bee554cf8f3af4ac

Aliases

arxiv: 2407.20199 · arxiv_version: 2407.20199v3 · doi: 10.48550/arxiv.2407.20199 · pith_short_12: K57ODBMBQMSK · pith_short_16: K57ODBMBQMSKOWU6 · pith_short_8: K57ODBMB

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/K57ODBMBQMSKOWU6CMNU7L63PZ \
  | 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: 577ee185818324a75a9e131b4fafdb7e5c904ab97c949825bee554cf8f3af4ac

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0b7ae370a1bacddeea6f1a0eeb59a894548a21e349bd2f4210b70f6bd08c14c4",
    "cross_cats_sorted": [
      "cs.LG"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "stat.ML",
    "submitted_at": "2024-07-29T17:28:58Z",
    "title_canon_sha256": "524c0fdde5fe1f1ad955548396fd79cf9ba8bb1f95de9e5258c73bfe2aa55f9b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2407.20199",
    "kind": "arxiv",
    "version": 3
  }
}