= Solution
Work with canonically normalized real components of the <scalar field>; a complex <representation> can first be written in terms of its real and imaginary components. At a <vacuum expectation value> $\varphi_0$, expansion of the <scalar potential> gives
$$
V(\varphi_0+\eta)=V(\varphi_0)+\frac12\eta^T M^2\eta+\cdots,\qquad M^2_{rs}=\partial_r\partial_s V(\varphi_0).
$$
The linear term vanishes by stationarity. Thus the <scalar mass matrix> is the <Hessian matrix> of the <scalar potential>. Invariance under each <Lie algebra generator> gives the identity $\partial_r V(\varphi)(iT_a\varphi)_r=0$. Differentiating this identity with respect to $\varphi_s$ and evaluating at the stationary point removes the term containing $\partial_rV$, leaving
$$
M^2(iT_a\varphi_0)=0.
$$
The kernel of the map $T\mapsto iT\varphi_0$ is precisely the <Lie algebra> of the <stabilizer subgroup> H. Consequently its restriction to a complementary space of broken <Lie algebra generators> is injective: a linear combination that annihilates $\varphi_0$ would belong both to the stabilizer algebra and its complement, and hence would vanish. There are therefore $\dim G-\dim H$ independent <Goldstone directions in the scalar mass matrix>, tangent to the <vacuum manifold>. \b[There are $\dim G-\dim H$ symmetry-protected massless scalars.]
This <Goldstone theorem> guarantees at least that many zero quadratic masses. The claim that these are all the massless scalars additionally requires nonzero curvature transverse to the vacuum orbit. For example, $V=\kappa(\varphi^T\varphi-v^2)^4$, with $\kappa>0$, has an ordinary spherical <vacuum manifold> but its entire <Hessian matrix> vanishes there. These <accidental massless radial modes beyond Goldstone modes> are not excluded by the hypotheses. For the indicated <orthogonal group> breaking, the protected number is
$$
\boxed{\dim O(n)-\dim O(n-1)=\frac{n(n-1)-(n-1)(n-2)}2=n-1.}
$$
For the <Higgs doublet>, put $r=\sqrt{2\phi^\dagger\phi}=v+H$ and $z=\sqrt2\phi/r$ wherever $r\ne0$. If $z=(z_1,z_2)^T$, then
$$
U=\begin{pmatrix}z_2^*&z_1\\-z_1^*&z_2\end{pmatrix},\qquad U^\dagger U=1,\qquad\det U=1,\qquad U\binom01=z.
$$
This explicitly exhibits the three angular coordinates as an <SU(2)> transformation of a real radial field. In particular, the <polar coordinates for the electroweak Higgs doublet> can locally be chosen in the form
$$
\phi=\exp(-i\boldsymbol n\cdot\boldsymbol\sigma+in_3 I)\frac1{\sqrt2}\binom0{v+H}.
$$
At the vacuum the three variations generated by $-i\sigma_1$, $-i\sigma_2$, and $i(I-\sigma_3)$ acting on $(0,1)^T$ are independent over the reals. They span the three broken directions; the remaining generator $Q=(\sigma_3+I)/2$ annihilates the vacuum. This proves that the exponential supplies local angular coordinates. An inverse local <gauge transformation>, applied also to the <gauge fields>, removes these three fields in <unitary gauge>. They supply the longitudinal polarizations of the massive <W bosons> and <Z boson> through the <Higgs mechanism>. The parameterization is local about a nonzero vacuum, rather than a coordinate system through every possible zero of the <Higgs doublet>.
Use the metric $(+---)$ and define the charged <W bosons> and neutral <gauge fields> by
$$
W_\mu^\pm=\frac{A_\mu^1\mp iA_\mu^2}{\sqrt2},\qquad Z_\mu=c_W A_\mu^3-s_W B_\mu,\qquad A_\mu^\gamma=s_W A_\mu^3+c_W B_\mu,
$$
where $s_W=\sin\theta_W$, $c_W=\cos\theta_W$, and $\tan\theta_W=g'/g$. Direct substitution into the <gauge-covariant kinetic term> gives
$$
D_\mu\phi=\binom{\displaystyle \frac{ig}{2}(v+H)W_\mu^+}{\displaystyle \frac{\partial_\mu H}{\sqrt2}+\frac{i(v+H)}{2\sqrt2}(-gA_\mu^3+g'B_\mu)},
$$
and hence
$$
(D_\mu\phi)^\dagger D^\mu\phi=\frac12(\partial H)^2+\frac{g^2(v+H)^2}{4}W_\mu^-W^{+\mu}+\frac{(v+H)^2}{8}(gA_\mu^3-g'B_\mu)^2.
$$
The <electroweak doublet gauge-boson mass matrix> in the $(A^3,B)$ basis is
$$
M^2_{\rm neutral}=\frac{v^2}{4}\begin{pmatrix}g^2&-gg'\\-gg'&g'^2\end{pmatrix}.
$$
Its normalized zero direction is $(s_W,c_W)$ and its massive direction is $(c_W,-s_W)$. Thus the scalar-vector terms in physical fields are
$$
\mathcal L_{H,\rm vectors}=m_W^2\left(1+\frac H v\right)^2W_\mu^-W^{+\mu}+\frac12m_Z^2\left(1+\frac H v\right)^2Z_\mu Z^\mu,
$$
with the <tree-level electroweak gauge-boson masses>
$$
\boxed{m_W=\frac{gv}{2},\qquad m_Z=\frac{v\sqrt{g^2+g'^2}}2,\qquad m_\gamma=0,\qquad m_W=m_Z\cos\theta_W.}
$$
These are also charge eigenstates. The unbroken <U(1) gauge symmetry> has generator $Q=T_3+Y$; on the <Higgs doublet>, $Y=I/2$. Since $[Q,T_\pm]=\pm T_\pm$, the combinations $W^\pm$ have charges $\pm1$, while Z, the <photon>, and H are neutral. The <photon> disappears from the displayed scalar-vector terms because $gA^3-g'B=\sqrt{g^2+g'^2}Z$. \b[The photon is massless and has no direct tree-level coupling to the remaining radial scalar in this sector.] This does not remove its interactions with charged <W bosons>, or exclude a scalar-to-photon interaction generated by loops.
Finally, retain the printed normalization of the <Higgs field potential>:
$$
V(H)=\frac\lambda2\left(vH+\frac{H^2}{2}\right)^2=\frac{\lambda v^2}{2}H^2+\frac{\lambda v}{2}H^3+\frac\lambda8H^4.
$$
The kinetic term is $\tfrac12(\partial H)^2$, so the <Higgs radial mass for a half-normalized quartic potential> is \b[$\boxed{m_H^2=\lambda v^2}$]. The extra factor of two familiar from a potential with prefactor $\lambda$, rather than $\lambda/2$, does not apply here.
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