Source
Shumaylov et al. (2026), LIT-456 — read as NOTE-208.
The account
Take the exact local Taylor expansion of the loss along an update direction. Two quantities fall out with no appeal to any norm: how well the update aligns with the stochastic gradient, and how much descent the direction affords at second order. Every optimizer implicitly trades one against the other, and the trade only pays at a step size tuned to exploit it.
Evaluate that trade under exact line search in a random-feature model and the ranking everyone expects reverses: plain gradient descent extracts more loss reduction per step than spectral descent. The catch is that its optimal step size oscillates violently from step to step, so the schedule that would realize the advantage cannot be run. Spectral descent’s optimal step size is essentially constant throughout training, because it is proportional to a quantity — the inner product of the gradient with its own polar factor — that barely moves.
So the claim is about realizability. Muon is not better because it points in a geometrically privileged direction. It is better because its best learning rate is a number you can find once and keep, and a schedule you can actually implement beats an optimum you cannot.
What was actually shown
The controls are LIT-456’s, run on NanoGPT and WikiText-2 at 118M tokens, alongside its theorems placing them outside the LMO framework and its random-feature analysis of optimal step sizes.
Three controls, each of which could have come out the other way.
TruncatedSGD zeroes the largest singular values, testing the simple
story that spectral optimizers work by removing noisy directions. It improves
on SGD and does not reach Muon: the simple story is necessary and
insufficient.
Freon sweeps the Schatten family and keeps going past p = 1. If the
LMO account were right the optimum would sit at a value where a norm exists.
On GPT-2 it sits strictly in the quasi-norm regime, where no unitarily
invariant norm does — and Theorem 2.1 says an update there cannot be
steepest descent for any such norm.
Kaon is the one that settles it. Replace the singular values with noise
from a chaotic recurrence: no oracle, no norm, no coherent geometry at all.
It matches Muon, and retains a convergence rate (Theorem 2.5). An explanation
that cannot distinguish its object from a randomized control is not doing
explanatory work.
What this does not say
It does not say Muon is a bad optimizer, or that the record should stop recommending it. SOTA-121 and SOTA-165 are untouched: what is contested is the account, not the technique, which is the separation ADR-031 built this scheme for and ADR-034 restated. Read as an argument against using Muon, this document would be a misreading of both the paper and the scheme.
It does not refute THEORY-024’s derivation. That orthogonalisation is
the duality map for the spectral norm, and that muP and Shampoo are partial
approximations of one map, are mathematical statements which no experiment
reaches. What this reaches is the sentence a reader takes away from them —
that being the duality map is why it works. THEORY-024 moves to
Proposed on this evidence, not to Rejected.
It is not established, and the scale is why. NanoGPT and WikiText-2 at
118M tokens, three seeds, one architecture family. The practices this argues
about are evidenced from 90M to 1.6T parameters. A null result at small scale
against an effect demonstrated at large scale is a reason to look, not a
verdict — which is what Proposed says here and what promote_when asks for.
And the positive half is weaker than the negative half. The step-size account is exact only in the random-feature model, which the authors show fails as a quadratic approximation of a GPT-2 layer. Alignment and descent potential cannot be computed during training; the framework diagnoses after the fact. The destructive result is strong, the constructive one is a hypothesis.
Why this is filed separately from the rank account
THEORY-032 is also a non-geometric explanation of the same practice, from an unrelated group, and it says something different: that the advantage comes from the rank structure of the activations and gradients. Neither paper cites the other. Whether they are two descriptions of one mechanism is open, and collapsing them into one document would assert an answer nobody has.