Euclidean steepest ascent is bounded by ‖g‖. Over the unit ‖·‖₂ ball, no direction beats the gradient: ⟨g,d⟩ ≤ ‖g‖. Pure Cauchy–Schwarz — the geometry behind plain SGD.

…and the normalized gradient attains it. g/‖g‖ is a unit vector with ⟨g, g/‖g‖⟩ = ‖g‖, so the SGD direction g/‖g‖ is the steepest-ascent maximizer and the dual norm is ‖g‖₂.

‖·‖∞-steepest ascent is bounded by Σ|gᵢ|. Over the box |dᵢ| ≤ 1, the pairing Σ gᵢ dᵢ ≤ Σ|gᵢ| = ‖g‖₁. The geometry behind sign and Adam-style coordinate updates.

…and sign(g) attains it. The box-corner dᵢ = ±1 = sign(gᵢ) is feasible and gives Σ gᵢ·sign(gᵢ) = Σ|gᵢ|, so the sign update is the ‖·‖∞-steepest direction, dual norm ‖g‖₁.

The polar factor attains the nuclear norm — the achievability half. Given an SVD G = U Σ Vᵀ (U,V orthogonal, Σ = diagonal s), the polar factor U Vᵀ pairs with G to give the nuclear norm Σσᵢ: ⟨G, UVᵀ⟩_F = Σ sᵢ. (The upper half — Σσᵢ is the max of ⟨G,·⟩ over the operator-norm ball, von Neumann's trace inequality — is muon_polar_is_max.)

No feasible direction beats the polar factor — von Neumann's trace inequality. Over the operator-norm unit ball, the gradient pairing is bounded by the nuclear norm: ⟨G, D⟩_F ≤ Σσᵢ for every contraction D. Here ‖D‖op ≤ 1 is spelled elementarily as the Euclidean contraction (D x)·(D x) ≤ x·x (*ᵥ = mulVec, ⬝ᵥ = dotProduct), which avoids a matrix operator-norm instance while saying exactly that.

Proof: with G = U Σ Vᵀ, cyclic trace gives ⟨G,D⟩_F = Σᵢ sᵢ Mᵢᵢ for M = Uᵀ D V, and each diagonal entry Mᵢᵢ = uᵢ · (D vᵢ) is bounded by 1 — Cauchy–Schwarz (‖uᵢ‖ = 1) then the contraction (‖D vᵢ‖ ≤ ‖vᵢ‖ = 1), i.e. Mᵢᵢ² ≤ (uᵢ·uᵢ)((Dvᵢ)·(Dvᵢ)) ≤ 1. Since sᵢ ≥ 0, Σ sᵢ Mᵢᵢ ≤ Σ sᵢ: a per-singular-vector Cauchy–Schwarz, summed against Σ.

The polar factor UVᵀ is the steepest-ascent direction under the operator norm. For an SVD G = U Σ Vᵀ (s ≥ 0), the polar factor UVᵀ is the argmax of ⟨G,·⟩_F over the operator-norm unit ball, with optimal value the dual (nuclear) norm Σσᵢ:

• feasible — UVᵀ is an isometry (‖UVᵀ x‖ = ‖x‖), hence a contraction, so it lies in the ball;
• attains — ⟨G, UVᵀ⟩_F = Σσᵢ (muon_polar_achieves_nuclear);
• unbeatable — every contraction D has ⟨G,D⟩_F ≤ Σσᵢ (muon_polar_is_max). Compare the SGD/sign rows (steepest_l2_*, steepest_linf_*): same bound+attained shape, one norm up. A Newton–Schulz iteration with convergent coefficients computes UVᵀ in the limit (MuonNewtonSchulz.nsStep_iterate_tendsto_polar); Muon's tuned quintic approximates it to a band (MuonNewtonSchulz.qScalar_iterate_band_half).

The SVD of an invertible matrix, constructed. For invertible G, there are orthogonal U, V (UᵀU = VᵀV = 1) and nonnegative singular values s with G = U (diagonal s) Vᵀ.

The build is spectral, not a black box: A := GᵀG is symmetric positive definite (positive definite ⇐ G invertible), so the spectral theorem gives an orthogonal eigenbasis V and eigenvalues λ with A = V (diagonal λ) Vᵀ, all λᵢ > 0. Set the singular values sᵢ := √λᵢ and U := G V Σ⁻¹ (Σ⁻¹ = diagonal (1/sᵢ), which exists because λᵢ > 0). Then UᵀU = Σ⁻¹ (Vᵀ A V) Σ⁻¹ = Σ⁻¹ (diagonal λ) Σ⁻¹ = 1 and U Σ Vᵀ = G V Vᵀ = G. No matrix square root is needed — only the spectral decomposition, the scalar √, and diagonal inverses. This discharges the SVD hypothesis of muon_polar_achieves_nuclear for full-rank G (the singular case, which needs the orthonormal completion of U, is not proved).

For invertible G, the constructed SVD's polar factor attains the nuclear norm. Combining the constructed SVD (svd_of_isUnit) with the achievability half (muon_polar_achieves_nuclear): for invertible G there exist orthogonal U, V and singular values s ≥ 0 with G = U (diagonal s) Vᵀ and ⟨G, UVᵀ⟩_F = Σ sᵢ. This is achievability only; with muon_polar_is_max it makes UVᵀ the operator-norm steepest-ascent direction.

Powers of a diagonal conjugation become pointwise powers — the spectral-calculus workhorse. For orthonormal W (WᵀW = 1), (W (diagonal d) Wᵀ)^k = W (diagonal dᵏ) Wᵀ: conjugating a diagonal commutes with raising to a power, sending a matrix power to a scalar power of each diagonal entry. This is what lets the Shampoo preconditioners' inverse fourth-roots (GᵀG)^{-1/4}, (GGᵀ)^{-1/4} collapse to diagonal arithmetic in shampoo_eq_muon.

The Shampoo = Muon jewel. Single-step Shampoo preconditions the gradient G by the inverse fourth-roots of its two Gram matrices: G ↦ (GGᵀ)^{-1/4} G (GᵀG)^{-1/4}. This equals Muon's update — the polar factor UVᵀ of G = UΣVᵀ. Two famous optimizers, one geometry.

Given the SVD G = U (diagonal s) Vᵀ (U,V orthonormal, s > 0), the inverse fourth-roots are spectral: with sᵢ^{-1/2} = (√sᵢ)⁻¹, take R := V (diagonal s^{-1/2}) Vᵀ and L := U (diagonal s^{-1/2}) Uᵀ. The three conjuncts are:

• R⁴ · (GᵀG) = 1 — R really is (GᵀG)^{-1/4} (GᵀG = V (diagonal s²) Vᵀ, so R⁴ inverts it);
• L⁴ · (GGᵀ) = 1 — L really is (GGᵀ)^{-1/4};
• L · G · R = U Vᵀ — the jewel. The collapse is the scalar identity s^{-1/2} · s · s^{-1/2} = 1 applied to each singular value (conj_diag_pow turns the matrix fourth-roots into these pointwise powers). Cf. muon_polar_achieves_nuclear (the same UVᵀ, now reached from Shampoo's side instead of the nuclear-norm side).

The Shampoo = Muon jewel, unconditional for any invertible G. Composing the constructed SVD (svd_of_isUnit) with shampoo_eq_muon: for invertible G there are orthonormal U, V and singular values s > 0 (strict, since diagonal s = Uᵀ G V is a unit) with G = U (diagonal s) Vᵀ, such that the two factors R = V (diagonal s^{-1/2}) Vᵀ, L = U (diagonal s^{-1/2}) Uᵀ are the inverse fourth-roots of GᵀG, GGᵀ (R⁴·GᵀG = L⁴·GGᵀ = 1) and Shampoo's preconditioned gradient is Muon's polar factor: L · G · R = U Vᵀ.

Muon's update lands on the orthogonal group. The polar factor UVᵀ is orthogonal — (UVᵀ)ᵀ(UVᵀ) = (UVᵀ)(UVᵀ)ᵀ = 1 — i.e. a point of O(n) (the Stiefel manifold of orthonormal frames). This is the geometric content of "Muon orthogonalizes the gradient": the update is not a vector in flat weight space but a point on the manifold of orthogonal maps. A Newton–Schulz iteration with convergent coefficients reaches it in the limit (MuonNewtonSchulz.nsStep_iterate_tendsto_polar); Muon's tuned quintic approximates it to a band.

Muon's update is the nearest orthogonal matrix to G — the projection of the raw gradient onto O(n) in Frobenius distance: ‖G − UVᵀ‖_F ≤ ‖G − Q‖_F for every orthogonal Q (stated in squared fInner form to avoid √). This is why the polar factor is "the orthogonalized gradient". Expanding ‖G − Q‖_F² = ‖G‖_F² − 2⟨G,Q⟩_F + n (orthogonal Q has ‖Q‖_F² = tr(QᵀQ) = n), minimizing the distance is maximizing ⟨G,Q⟩_F over O(n) ⊆ {contractions} — exactly the von Neumann bound muon_polar_is_max, attained at UVᵀ. The same inequality that makes UVᵀ the steepest direction makes it the nearest orthogonal matrix.