Pith Number

pith:ZZJFDQRQ

pith:2026:ZZJFDQRQRB6AU6CLDKFZM3BYOY

not attested not anchored not stored refs pending

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication

Huu Danh Nguyen (Stanford University)

arxiv:2606.17399 v1 · 2026-06-16 · cs.LG · cs.AI

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{ZZJFDQRQRB6AU6CLDKFZM3BYOY}

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	2026-06-19T16:10:11.880975Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ce5251c230887c0a784b1a8b966c38762cf31f47ca341a5e601bd7927f0f6b6c

Aliases

arxiv: 2606.17399 · arxiv_version: 2606.17399v1 · doi: 10.48550/arxiv.2606.17399 · pith_short_12: ZZJFDQRQRB6A · pith_short_16: ZZJFDQRQRB6AU6CL · pith_short_8: ZZJFDQRQ

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/ZZJFDQRQRB6AU6CLDKFZM3BYOY \
  | 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: ce5251c230887c0a784b1a8b966c38762cf31f47ca341a5e601bd7927f0f6b6c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3727543b4c13be5db3c1c912f7364063b153d4eb74b2a2bef49aa9eeb64e70d5",
    "cross_cats_sorted": [
      "cs.AI"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-06-16T01:16:26Z",
    "title_canon_sha256": "6a43b224ddbf16d99267fd5de4edf060fab9a67a14c9a051d3af24219acaafe9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2606.17399",
    "kind": "arxiv",
    "version": 1
  }
}