Paper Citation Record · LEDGER

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

As of 23 August 2026, this Paper Citation Record lists 41 of 41 outbound references and 7 inbound Pith citation observations for arXiv:2512.04632.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2512.04632 v2

Coverage vector

measured 41 of 41 reference resolution

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measured 48 of 48 standing notices

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measured 7 of 7 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-16T00:38:28.970754Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: arxiv_reference, observed 2026-07-02T07:06:45.022170Z

Reference resolution

41 of 41 outbound references displayed

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Outbound references

Observation e4f63cf7-c8d4-4c14-8550-8d64c2e04a14 · outbound

This paper cites Dion: Distributed orthonormalized updates.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Dion: Distributed orthonormalized updates

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.532446Z

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Observation babde7d5-bc63-41fb-b477-b35921fe3165 · outbound

This paper cites The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm

Reference 2

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Observation 69cc5d65-e4bc-43a2-9b8c-a8d922109c50 · outbound

This paper cites Sorting out lipschitz function approximation.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Sorting out lipschitz function approximation

Reference 3

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Observation 3fdcaf19-4189-4876-8c09-113e4dc5b7b1 · outbound

This paper cites Deriving muon, 2025, 2025.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Deriving muon, 2025, 2025

Reference 4

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Observation 919cd2e0-b8ce-46d5-94ad-988198c87dd9 · outbound

This paper cites Old optimizer, new norm: An anthology.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Old optimizer, new norm: An anthology

Reference 5

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Observation c363d293-b98a-4fad-830a-f451fa270cc6 · outbound

This paper cites Modular duality in deep learning.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Modular duality in deep learning

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.548315Z

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Observation 029601f8-be9e-4cef-a5de-0576d119d093 · outbound

This paper cites An iterative algorithm for computing the best estimate of an orthogonal matrix.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates An iterative algorithm for computing the best estimate of an orthogonal matrix

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.550806Z

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Observation bf79dfdb-d1ab-4168-99c8-4ce89b875b54 · outbound

This paper cites Sur quelques Propri´et´es des D´eterminants Gauches, page 332–336.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Sur quelques Propri´et´es des D´eterminants Gauches, page 332–336

Reference 8

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Observation 464e2826-1dc2-4e3f-920b-06579504902d · outbound

This paper cites Layer sharding for large-scale training with muon, 2025, 2025.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Layer sharding for large-scale training with muon, 2025, 2025

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.556217Z

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Observation 8103f007-e03a-46d7-bce2-ad0958764e80 · outbound

This paper cites an unresolved cited work.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Unresolved cited work

Reference 10

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Observation 6f1662e9-311b-4054-8b97-faa5ce4cecb5 · outbound

This paper cites ¨uber die abgrenzung der eigen- werte einer matrix.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates ¨uber die abgrenzung der eigen- werte einer matrix

Reference 11

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Observation bb3e3072-a9fc-4383-a4c5-e9884c1748de · outbound

This paper cites Accelerating Newton-Schulz Iteration for Orthogonalization via Chebyshev-type Polynomials, 2025.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Accelerating Newton-Schulz Iteration for Orthogonalization via Chebyshev-type Polynomials, 2025

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.563618Z

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Observation 5d4bc41c-b692-4c15-91a6-81eea4126313 · outbound

This paper cites A recipe for improved certifiable robustness.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates A recipe for improved certifiable robustness

Reference 13

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Observation 2e5d1781-ba45-4cd7-bdb8-509d14c68c3d · outbound

This paper cites cifar10-airbench.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates cifar10-airbench

Reference 14

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Observation 8ff16b11-03b1-4f4f-b702-fdf893c031ff · outbound

This paper cites an unresolved cited work.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Unresolved cited work

Reference 15

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Observation 68c68328-36e2-4839-b3f5-cd92b2cb6059 · outbound

This paper cites Muon: An optimizer for hidden layers in neural networks, 2024, 2024.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Muon: An optimizer for hidden layers in neural networks, 2024, 2024

Reference 16

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Observation fe1cc965-bf8b-4c53-96ff-d8290b7a63b9 · outbound

This paper cites Yutian, Chen.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Yutian, Chen

Reference 17

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Observation 6174d491-fcec-443a-bdc2-a023814f3ba2 · outbound

This paper cites Kingma and Jimmy Ba.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Kingma and Jimmy Ba

Reference 18

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Observation 51a9a0dc-9613-4843-93cf-b48631470b43 · outbound

This paper cites Learning multiple layers of features from tiny images.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Learning multiple layers of features from tiny images

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.583300Z

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Observation 28ffd6f0-942b-4560-beaf-031503559d72 · outbound

This paper cites Heavy-tailed class imbalance and why adam outperforms gradient descent on language mod- els.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Heavy-tailed class imbalance and why adam outperforms gradient descent on language mod- els

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.585718Z

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Observation 9f11cc32-4b04-44d7-a1b1-d9b3e29381de · outbound

This paper cites Th´eorie analytique des probabilit´es.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Th´eorie analytique des probabilit´es

Reference 21

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Observation 02d09013-dc5f-4feb-8919-bb28db0eda8c · outbound

This paper cites Scalable optimization in the modular norm.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Scalable optimization in the modular norm

Reference 22

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Observation c2bb1134-140e-4db8-98a9-cb82011c8149 · outbound

This paper cites Flash-muon: An efficient implementation of muon optimizer, 2025.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Flash-muon: An efficient implementation of muon optimizer, 2025

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.594120Z

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Observation 77f8b84a-73f9-4ea2-bf6f-7b336a2505e0 · outbound

This paper cites Muon is Scalable for LLM Training.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Muon is Scalable for LLM Training

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.596778Z

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Observation e766ff3c-7b83-46f8-b76f-74712398ab09 · outbound

This paper cites Decoupled weight de- cay regularization.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Decoupled weight de- cay regularization

Reference 25

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This paper cites The fineweb datasets: Decanting the web for the finest text data at scale.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates The fineweb datasets: Decanting the web for the finest text data at scale

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-03T18:37:49.603478Z

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Observation c8866c18-069a-43a8-a9a5-815d47a7a090 · outbound

This paper cites Training Deep Learning Models with Norm-Constrained LMOs.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Training Deep Learning Models with Norm-Constrained LMOs

Reference 27

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Observation d5d988ee-b092-4229-a1f5-093a02f17fa1 · outbound

This paper cites Almost-orthogonal layers for efficient general-purpose lipschitz networks.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Almost-orthogonal layers for efficient general-purpose lipschitz networks

Reference 28

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Observation 9d4399c0-eca0-40b3-aa63-356191834095 · outbound

This paper cites Language models are unsuper- vised multitask learners.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Language models are unsuper- vised multitask learners

Reference 29

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Observation 107d7054-db86-4268-8265-90a09bc9ff55 · outbound

This paper cites Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs).

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs)

Reference 30

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Observation 55a7b33f-d76b-44ca-9ad5-5b8da9da975d · outbound

This paper cites Practical Efficiency of Muon for Pretraining.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Practical Efficiency of Muon for Pretraining

Reference 31

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Observation 42f7814c-a06f-47a4-b73a-64b5f9638b77 · outbound

This paper cites Beyond the ideal: Analyzing the inexact muon update.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Beyond the ideal: Analyzing the inexact muon update

Reference 32

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This paper cites AdaMuon: Adap- tive Muon Optimizer, 2025.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates AdaMuon: Adap- tive Muon Optimizer, 2025

Reference 33

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This paper cites A tail-index analysis of stochastic gradient noise in deep neural networks.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates A tail-index analysis of stochastic gradient noise in deep neural networks

Reference 34

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Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Skew orthogonal convolu- tions

Reference 35

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Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Tri- ton: an intermediate language and compiler for tiled neu- ral network computations

Reference 36

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Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates High-dimensional probability, 2009

Reference 37

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This paper cites Fantastic Pretraining Optimizers and Where to Find Them.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Fantastic Pretraining Optimizers and Where to Find Them

Reference 38

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This paper cites L ´evy distribution - Wikipedia, the free encyclope- dia.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates L ´evy distribution - Wikipedia, the free encyclope- dia

Reference 39

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Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates GLM-4.5: Agentic, Reasoning, and Coding (ARC) Foundation Models

Reference 40

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This paper cites Kimi K2: Open Agentic Intelligence.

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates Kimi K2: Open Agentic Intelligence

Reference 2025

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Pith citing papers

Observation 308cfef5-d800-4388-87f0-ad4e2c0b42f9 · inbound

MuonQ: Enhancing Low-Bit Muon Quantization via Directional Fidelity Optimization cites this paper.

MuonQ: Enhancing Low-Bit Muon Quantization via Directional Fidelity Optimization Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

Reference 3

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Observation 07a1919c-1023-4966-bcd6-fde5ff06f631 · inbound

MuonQ: Enhancing Low-Bit Muon Quantization via Directional Fidelity Optimization cites this paper.

MuonQ: Enhancing Low-Bit Muon Quantization via Directional Fidelity Optimization Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

Reference 2020

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Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers cites this paper.

Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

Reference 17

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Observation d73d0ad4-b19a-41a1-8885-e1b7cb689d17 · inbound

Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers cites this paper.

Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

Reference 17

Resolution: verified exact; arxiv_id, observed 2026-07-07T03:18:51.225765Z

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Observation 0f084e89-b3c0-4260-b663-5db5261248dc · inbound

Why Muon Outperforms Adam: A Curvature Perspective cites this paper.

Why Muon Outperforms Adam: A Curvature Perspective Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

Reference 127

Resolution: verified exact; arxiv_id, observed 2026-07-07T03:18:51.225765Z

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Observation 32602136-5368-45e8-81d1-9a4edd0a6678 · inbound

Muse: Representation Geometry of Muon Beyond Normalized Momentum cites this paper.

Muse: Representation Geometry of Muon Beyond Normalized Momentum Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

Reference 2009

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Dion3: Full-Stack Orthogonal Updates cites this paper.

Dion3: Full-Stack Orthogonal Updates Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

Reference 6

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