Pith Number

pith:XL4BQDF2

pith:2026:XL4BQDF2V5BVJ23PUX3YMIOY3I

not attested not anchored not stored refs resolved

Spectral Flattening Is All Muon Needs: How Orthogonalization Controls Learning Rate and Convergence

James Bailey, Minh-Phuc Truong, Tien-Phat Nguyen, Trung Le, Truong Nguyen, Tuc Nguyen

Muon orthogonalizes its momentum buffer to flatten the gradient spectrum, allowing stable learning rates scaled to the average singular value rather than the largest.

arxiv:2605.13079 v1 · 2026-05-13 · cs.LG · cs.AI

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\usepackage{pith}
\pithnumber{XL4BQDF2V5BVJ23PUX3YMIOY3I}

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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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

We prove that Muon's maximal stable step size scales with the average singular value of the gradient rather than the largest, which bottlenecks standard gradient descent.

C2weakest assumption

The improvement in effective convergence factor is shown under a Kronecker-factored curvature model for the loss landscape.

C3one line summary

Muon achieves faster convergence and larger stable learning rates by flattening the singular value spectrum of the momentum buffer through orthogonalization, scaling step size with average rather than maximum singular values.

References

15 extracted · 15 resolved · 3 Pith anchors

[1] Old Optimizer, New Norm: An Anthology · arXiv:2409.20325

[2] Muon optimizes under spectral norm constraints

[3] An exploration of non-euclidean gradient descent: Muon and its many variants.arXiv preprint arXiv:2510.09827

[4] arXiv preprint arXiv:2512.04299 , year =

[5] Effective quantization of muon optimizer states.arXiv preprint arXiv:2509.23106, 2025

Receipt and verification

First computed	2026-05-18T03:08:58.726066Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

baf8180cbaaf4354eb6fa5f78621d8da28d3455fd710fefd1cd6db22c9a8caae

Aliases

arxiv: 2605.13079 · arxiv_version: 2605.13079v1 · doi: 10.48550/arxiv.2605.13079 · pith_short_12: XL4BQDF2V5BV · pith_short_16: XL4BQDF2V5BVJ23P · pith_short_8: XL4BQDF2

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/XL4BQDF2V5BVJ23PUX3YMIOY3I \
  | 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: baf8180cbaaf4354eb6fa5f78621d8da28d3455fd710fefd1cd6db22c9a8caae

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "825f1b6a3b0309227215f80f71f1595fd868be0beea38e553c8fa38e294d3f87",
    "cross_cats_sorted": [
      "cs.AI"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-13T06:54:01Z",
    "title_canon_sha256": "20a75af27bd355c3ec2ba041878db76ece16dd934d3bf27eddedba55f2d63068"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13079",
    "kind": "arxiv",
    "version": 1
  }
}