We study an oracle subspace-perturbation model for 3D localization. The procedure is given an $s$-dimensional subspace $\widetilde{\mathcal U}$ and a deterministic error bound \[ \varepsilon_{\mathrm{sub}} = \left\|P_{\widetilde{\mathcal U}}-P_{\mathcal U}\right\|_2, \] where $\mathcal{U}$ is an $s$-dimensional subspace of far-field patterns with wavenumber $κ$, and $P_{\mathcal{V}}$ denotes orthogonal projection onto a subspace $\mathcal{V}$. Under explicit separation and conditioning hypotheses for arbitrary point clouds, we prove that the perturbed spherical MUSIC objective \[ \widetilde{q}({\mathbf z}) = 1-\left\|P_{\widetilde{\mathcal{U}}}\varphi_{\mathbf z}\right\|_2^2 \] has a unique strongly convex well in every ball $B_{γ/κ}(x_j)$ and a uniform value gap outside the union of the certified wells. A fixed-step gradient map with $h\asympκ^{-2}$ leaves each well invariant and converges linearly to its unique minimizer. Consequently, thresholding on an $O(κ^{-1})$-mesh, followed by gradient descent and duplicate removal, recovers all relevant minima with localization bound \[ \mathfrak{R}(\varepsilon_{\mathrm{sub}}) \lesssim \frac{\varepsilon_{\mathrm{sub}}}κ. \] The arbitrary-cloud frame analysis yields the sufficient condition \[ κδ_X\gtrsim s^{2/3} \] through an absolute coherence row sum and Gershgorin's theorem. We construct lower-frame counterexamples below the $s^{1/6}$ scale, upper-frame counterexamples below the $s^{1/3}$ scale, and examples showing that $2/3$ is optimal for the absolute-row-sum argument. Finally, for parameter classes containing a uniformly admissible one-point displacement path, we prove \[ \mathfrak{R}(\varepsilon_{\mathrm{sub}}) \sim \frac{\varepsilon_{\mathrm{sub}}}κ. \]

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