We study an oracle subspace-perturbation model in which the 3D localization procedure is given an \(s\)-dimensional subspace \(\widetilde{\mathcal U}\) and a deterministic error bound $\eps_{\rm sub}$ measuring the sine-theta distance between $\widetilde{\mathcal U}$ and an $s$-dimensional subspace $\cU$ of far-field patterns with wavenumber $κ$. Under explicit arbitrary-cloud separation and conditioning hypotheses, we prove that the perturbed spherical MUSIC objective \[ \widetilde q(\bz) = 1-\|P_{\widetilde{\mathcal U}}\varphi_\bz\|_2^2 \] has a unique strongly convex well in each ball \(B_{γ/κ}(x_j)\) and the objective has a uniform value gap outside the union of the certified wells. A fixed-step gradient map with \(h\asympκ^{-2}\) leaves every certified well invariant and converges linearly to its unique minimizer. Consequently, thresholding on an \(O(κ^{-1})\)-mesh, followed by gradient descent from all accepted grid points and duplicate removal, recovers all relevant minima. The arbitrary-cloud frame analysis gives the sufficient condition \[ κδ_X\gtrsim s^{2/3} \] through an absolute coherence row sum and Gershgorin's theorem. We also construct lower-frame counterexamples below the \(s^{1/6}\) scale, upper-frame counterexamples below the \(s^{1/3}\) scale, and examples showing that the exponent \(2/3\) is optimal for the absolute-row-sum argument. The latter is a sharpness result for the proof method and is not a spectral necessity claim. Finally, for parameter classes containing a uniformly admissible one-point displacement path, we prove that the deterministic oracle localization modulus is \[ \mathfrak R(\eps) \asymp \frac{\eps}κ. \]

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