Manifold Muon: Probes & Optimization

• Manifold Muon is a concept where muons act as carriers of geometric information across quantum gravity, matrix optimization, and accelerator physics.
• It integrates methods such as probing spacetime near black holes, employing spectral descent on Stiefel and oblique manifolds, and engineering low-emittance muon beams.
• Applications include testing noncommutative metrics, advancing optimization algorithms in machine learning, and enhancing design efficiency in muon collider technology.

Across several research programs, “Manifold Muon” is used, or can be understood, in multiple technical senses. In one literal sense, a muon moving near a microscopic black hole is treated as a probe of the spacetime manifold and of a possible noncommutative antisymmetric metric sector, with the Michel spectrum serving as the observable. In another, Muon-family optimizers treat matrix updates through Stiefel-, oblique-, or low-rank-manifold geometry, so that the update direction rather than the parameter itself is constrained or normalized by a manifold structure. In accelerator and beam-physics work, the phrase is used metaphorically for the deliberate engineering of a compact muon phase-space manifold, as in low-emittance sources and beam-compression schemes (0901.4308, Mehta et al., 29 Sep 2025, Alesini et al., 2019, Belosevic et al., 2019).

1. Terminological scope and principal usages

The phrase spans at least three distinct but structurally related domains: spacetime geometry, optimization geometry, and beam phase-space geometry. In each case, the muon is not merely a particle species but the object through which a manifold structure becomes observable or operational.

Domain	Manifold object	Muon’s role
Quantum gravity	Spacetime manifold in Fermi normal coordinates, with a possible antisymmetric noncommutative sector	Decay-spectrum probe near a microscopic Schwarzschild black hole
Optimization	Stiefel, oblique, and low-rank manifolds; spectral-norm trust regions	Matrix update or adapter direction constrained by matrix-sign or related projections
Accelerator and beam physics	Narrow 6D phase-space regions and cooling/compression manifolds	Beam whose production, compression, and transport are engineered to occupy a small manifold

This usage pattern suggests a unifying theme: the muon serves as a carrier of geometry. In the gravitational setting, geometry is the local structure of spacetime itself. In optimization, geometry is the constraint or trust-region structure of matrix space. In accelerator physics, geometry is the occupied region of phase space and the lattice-defined transport manifold (0901.4308, Mehta et al., 29 Sep 2025, Alesini et al., 2019, Belosevic et al., 2019, Zisman, 2011).

2. The muon as a probe of the spacetime manifold

In the most literal formulation, a single muon in circular orbit around a microscopic Schwarzschild black hole is used to probe the local spacetime manifold. The setup takes a black hole of mass $M = 3 \times 10^{-12}\,\mathrm{cm}$ in geometric units, with orbital radius $r_0 \sim 6M$, and describes the neighborhood of the worldline in Fermi normal coordinates $X^\mu=(T,X^i)$. The metric is expanded to quadratic order in $X^i$, and the muon obeys the covariant Dirac equation

$$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$

The observable is the decay process

$\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$

specifically the Michel spectrum $\hat{\Gamma}(x)=d\Gamma/dx$ (0901.4308).

The key theoretical move is a reinterpretation of Moffat’s nonsymmetric gravity in which the antisymmetric part of the metric,

$g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$

is identified with the noncommutative sector of spacetime. Noncommutativity is introduced through

$$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$

so that the antisymmetric operator-valued metric couples directly to coordinate commutators. Two ansätze are studied: a constant structure $\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$, and a coordinate-dependent Lie-type structure $r_0 \sim 6M$0 (0901.4308).

These structures modify the spin connection and hence the decay amplitude. A central structural result is that the noncommutative correction to the decay rate is linear in the noncommutative scale $r_0 \sim 6M$1 but only survives orbital averaging through curvature-squared terms; terms linear in curvature and linear in $r_0 \sim 6M$2 vanish upon averaging over one orbital period. In the Schwarzschild background, curvature alone suppresses the decay rate relative to flat space, reproducing the curvature-induced stabilization effect found by Singh and Mobed. With noncommutativity included, the high-$r_0 \sim 6M$3 Michel spectrum is pushed upward, so that the noncommutative contribution counteracts the stabilization. For the sign convention $r_0 \sim 6M$4 for $r_0 \sim 6M$5, the effect is upward for $r_0 \sim 6M$6 (0901.4308).

Two quantitative scales are singled out. For the constant ansatz, a visible effect is shown for $r_0 \sim 6M$7; for the coordinate-dependent ansatz, a comparable effect requires $r_0 \sim 6M$8. The same paper notes a tension with the flat-space noncommutative-QED bound $r_0 \sim 6M$9, and it is explicit that experimental feasibility at the LHC is unclear. The proposal is therefore presented as a theoretical probe of principle rather than an immediately realizable measurement (0901.4308).

3. Stiefel-manifold Muon: spectral steepest descent in matrix space

In optimization, Muon is a matrix-aware optimizer that treats each weight matrix $X^\mu=(T,X^i)$0 as a geometric object and updates it through the orthogonalized momentum. The basic update is

$X^\mu=(T,X^i)$1

with

$X^\mu=(T,X^i)$2

for $X^\mu=(T,X^i)$3. The update direction is thus the polar factor of the momentum and has spectral norm $X^\mu=(T,X^i)$4 (Mehta et al., 29 Sep 2025).

This construction is presented in two formally related ways. First, Muon is the solution of the spectral-norm steepest-descent problem

$X^\mu=(T,X^i)$5

whose solution is exactly $X^\mu=(T,X^i)$6. Second, for square matrices, the method is interpreted as natural gradient descent on the Stiefel manifold $X^\mu=(T,X^i)$7, with the matrix sign supplying the optimal orthogonal approximation of the momentum. The same paper also gives a nonconvex convergence guarantee under $X^\mu=(T,X^i)$8-smoothness and bounded variance, with an $X^\mu=(T,X^i)$9 rate in average squared gradient norm (Mehta et al., 29 Sep 2025).

The empirical claims in that study are explicitly tied to decoder-only transformers with $X^i$0M–$X^i$1M parameters. Across that range, Muon reaches a target loss with $X^i$2–$X^i$3 of the compute required by AdamW, typically attains $X^i$4–$X^i$5 lower final loss at fixed compute, and remains stable up to $X^i$6M tokens per batch. The paper also reports multiplicative gains when Muon is combined with Multi-Head Latent Attention and Mixture-of-Experts, including $X^i$7 memory reduction, $X^i$8 inference speedup, and $X^i$9–$$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$0 perplexity improvement for the joint MLA+MoE+Muon system (Mehta et al., 29 Sep 2025).

A complementary theoretical analysis studies Muon under nonconvex Hölder-smooth empirical risk minimization with heavy-tailed stochastic noise. There the orthogonalized search direction is explicitly identified with a projection onto the Stiefel manifold,

$$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$1

and the paper proves convergence to a stationary point under a bounded $$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$2-moment noise assumption. In that analysis, Muon is shown to converge faster than mini-batch SGD in gradient norm, because the manifold projection fixes the scale of the search direction while preserving descent under heavy-tailed noise (Iiduka, 16 Mar 2026).

4. Oblique, anisotropic, and low-rank generalizations

Later work moves from the Stiefel-only picture toward more structured manifold geometries. Muon+ keeps Muon’s momentum and orthogonalization,

$$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$3

and then applies column-wise, row-wise, or composed row/column $$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$4-normalization. This is interpreted as a move from a Stiefel-like polar factor to an oblique-manifold update direction. The reported gains are consistent across GPT-style models from $$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$5M to $$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$6M parameters and LLaMA-style models from $$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$7M to $$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$8B parameters, including compute-optimal and $$\left[i \gamma^\mu(X)\left(\partial_\mu + i\,\Gamma_\mu(X)\right)-m\right]\psi(X)=0.$$9 regimes. The paper’s ablations further argue that much of the improvement comes from the normalization step itself rather than from second-moment scaling (Zhang et al., 25 Feb 2026).

Mano replaces Muon’s heavy global spectral normalization with projection of the momentum onto the tangent space of a rotational Oblique manifold, followed by manifold normalization of the update but not of the parameter. In matrix form, with $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$0 selecting column-wise or row-wise normalization, the update is constructed from

$\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$1

The paper argues that this preserves curvature information that Muon flattens away, while remaining much cheaper than Newton–Schulz orthogonalization. On LLaMA and Qwen3 models, Mano is reported to outperform both AdamW and Muon while using less memory than AdamW and less computational complexity than Muon (Gu et al., 30 Jan 2026).

Mousse instead modifies the trust-region geometry. It uses Shampoo-style Kronecker-factored statistics $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$2 and $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$3 to whiten the gradient,

$\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$4

applies the matrix sign in the whitened space, and maps back: $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$5 This is formulated as spectral steepest descent under an anisotropic trust region rather than Muon’s isotropic spectral ball. The reported outcome is around $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$6 reduction in training steps relative to Muon, with negligible computational overhead, on LLMs ranging from $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$7M to $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$8M parameters (Zhang et al., 10 Mar 2026).

A distinct line extends the same geometric logic to low-rank adaptation. LoRA-Muon derives spectral steepest descent on the low-rank manifold

$\mu^- \to e^- + \bar{\nu}_e + \nu_\mu,$9

using tangent-space updates for LoRA factors and a split weight-decay rule chosen so that the product-level dynamics match dense Muon or Shampoo-family behavior up to higher-order corrections. In the reported TinyShakespeare experiments, a rank-$\hat{\Gamma}(x)=d\Gamma/dx$0 proxy recovers the dense best tested learning rate, and a rank-$\hat{\Gamma}(x)=d\Gamma/dx$1 LoRA-Muon run attains lower mean validation loss than the dense baseline in the seed-averaged sweep (Cesista et al., 11 Jun 2026).

At the framework level, “Manifold Constrained Steepest Descent” generalizes Muon-like norm-constrained linear minimization oracles to embedded manifolds. Its spectral Stiefel specialization, SPEL, computes the LMO from the Riemannian gradient and then projects back to the manifold with the matrix sign. The paper supplies deterministic and stochastic convergence guarantees and reports improved stability on PCA, orthogonality-constrained CNNs, and manifold-constrained LLM adapter tuning (Yang et al., 29 Jan 2026).

5. Phase-space manifold engineering in muon sources and accelerators

In accelerator physics the manifold language is explicitly phase-space oriented. The LEMMA concept, a positron-driven muon source for a muon collider, is described as engineering a very special low-dimensional manifold in 6D phase space. A $\hat{\Gamma}(x)=d\Gamma/dx$2 GeV positron beam stored in a low-emittance ring is sent onto thin targets, producing muon pairs through

$\hat{\Gamma}(x)=d\Gamma/dx$3

operated near threshold at $\hat{\Gamma}(x)=d\Gamma/dx$4. Because production is near threshold in the center of mass but highly boosted in the laboratory, the muons are highly collimated and born with small transverse and longitudinal emittances. The paper summarizes this with

$\hat{\Gamma}(x)=d\Gamma/dx$5

and for a $\hat{\Gamma}(x)=d\Gamma/dx$6 GeV positron beam gives $\hat{\Gamma}(x)=d\Gamma/dx$7. In that sense, the muon beam occupies a narrow tube in transverse phase space from the moment of production (Alesini et al., 2019).

The same logic drives the target-line design. A localized multi-slice target preserves the manifold better than a multiple-interaction-point lattice because long chromatic transport fans out the initially narrow muon manifold. Representative source parameters in the paper include a conversion efficiency of $\hat{\Gamma}(x)=d\Gamma/dx$8 for $\hat{\Gamma}(x)=d\Gamma/dx$9 of Be, and initial production emittances from $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$0 to $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$1 nm depending on optics. This is offered as an alternative to proton-driven schemes that begin with much larger phase-space volume and therefore require very aggressive cooling (Alesini et al., 2019).

The muCool program at PSI adopts a different phase-space strategy. It aims to reduce the phase space of a standard $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$2 beam by a factor of $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$3 with $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$4 efficiency by stopping the beam in cryogenic helium gas and using electric and magnetic fields together with gas-density gradients to compress the thermalized muons. In one transverse stage, the beam is compressed from roughly $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$5 mm to roughly $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$6 mm over a few microseconds in a $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$7 T field and a $g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$8–$g_{\mu\nu}=g_{(\mu\nu)}+g_{[\mu\nu]},$9 K density gradient. In the longitudinal stage, a $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$0 mm swarm is compressed to within $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$1 mm around $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$2 in about $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$3 ns. Here the manifold is the occupied phase-space volume of the compressed muon swarm prior to extraction into vacuum (Belosevic et al., 2019).

The broader muon-collider literature uses the same language at the level of the whole accelerator chain. A collider complex based on $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$4 production must reduce a large initial 6D phase space through ionization cooling, bunching, bunch merging, and final cooling. The paper on muon collider accelerators identifies a target normalized transverse emittance of about $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$5 and treats the design space of proton driver, target, capture, cooling, acceleration, and collider ring as a coupled high-dimensional manifold of feasible parameter choices (Zisman, 2011).

6. Muon colliders as a manifold probe of Higgs interactions

A further use of the phrase appears in collider phenomenology, where the muon becomes a manifold probe of Higgs interactions. One study examines a muon collider at $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$6 and $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$7 TeV, considering all processes involving direct production of electroweak bosons $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$8, $$[\hat{x}^\mu,\hat{x}^\nu]=i\,\theta^{\mu\nu}(\hat{x})\equiv i\,\hbar\,\mathcal{J}^{\mu\nu}(\hat{x}),$$9, and $\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$0 with up to five particles in the final state. The analysis is performed in both HEFT and SMEFT under the assumption that the dominant BSM effects originate from the muon Yukawa sector. The paper’s central conclusion is that a muon collider probes Higgs–muon interactions far beyond the LHC because it does not rely solely on the branching fraction $\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$1, and that the $\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$2 TeV option is markedly more powerful than $\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$3 TeV, with multi-Higgs production especially effective (Celada et al., 2023).

The EFT parameterization is organized by couplings $\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$4 in

$\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$5

and Higgs self-couplings $\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$6. In the general HEFT picture the $\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$7 are independent, whereas in SMEFT6 they are correlated; specifically,

$\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$8

At high energy, multi-Higgs channels acquire simple scaling laws. For example,

$\mathcal{J}^{\mu\nu}=iC_{(0)}^{\mu\nu}$9

so the $r_0 \sim 6M$00, $r_0 \sim 6M$01, and $r_0 \sim 6M$02 rates grow as positive powers of $r_0 \sim 6M$03. Numerically, the paper reports at $r_0 \sim 6M$04 TeV and $r_0 \sim 6M$05 ab$r_0 \sim 6M$06 approximately $r_0 \sim 6M$07 events for $r_0 \sim 6M$08, $r_0 \sim 6M$09 for $r_0 \sim 6M$10, $r_0 \sim 6M$11 for $r_0 \sim 6M$12, and $r_0 \sim 6M$13 for $r_0 \sim 6M$14 when the corresponding $r_0 \sim 6M$15 benchmark is used (Celada et al., 2023).

These rates translate into strong bounds. In the HEFT interpretation, the $r_0 \sim 6M$16 TeV machine reaches approximately

$r_0 \sim 6M$17

from multi-Higgs channels alone at $r_0 \sim 6M$18 CL. In SMEFT6, the best single constraint is reported from $r_0 \sim 6M$19, giving

$r_0 \sim 6M$20

This makes the muon collider not only a measurement device for the ordinary muon Yukawa but a direct test of the entire tower of $r_0 \sim 6M$21 interactions and of the difference between linear SMEFT correlations and general HEFT behavior (Celada et al., 2023).