We study far-field discrimination between one and two incoherent point sources in the singular regime of weak and closely spaced emitters. Under ideal alignment, spatial-mode demultiplexing (SPADE) attains the quantum-optimal large-sample Stein exponent, but the finite-photon behavior near the one-source boundary and the effect of realistic imperfections remain less understood. Using singular learning theory, we analyze both the aligned and misaligned problems. In the aligned Gaussian case, we derive the zeta-function poles for direct imaging and SPADE, show that both share the same real log canonical threshold $λ=1/2$ but differ in multiplicity, and obtain the corresponding Bayes free-energy asymptotics. This yields a universal subleading advantage of aligned SPADE in the local prior-weighted regime. In the misaligned setting, we study a physically motivated binary-SPADE reduction that retains the full leading $O(s^2)$ leakage contrast near alignment, with corrections from the detailed higher-mode redistribution entering only at $O(s^4)$. We show that misaligned binary-SPADE and direct imaging acquire nontrivial local power on different intrinsic scales, $s=O(n^{-1/4})$ and $s=O(n^{-1/2})$, respectively. However, finite-$n$ Neyman--Pearson comparisons under common physical conditions reveal that direct imaging is stronger on the plotted grids and that misaligned binary-SPADE exhibits an exact blind separation $s^\ast=2θ$, where its power collapses to $α$. These results identify model singularity as a structural organizing principle for finite-photon quantum discrimination and clarify how ideal aligned SPADE benchmarks can fail to translate into finite-$n$ advantages under misalignment.

翻译：我们研究了弱且紧密间隔的发射器奇异区域内一个与两个非相干点源的远场鉴别问题。在理想对准条件下，空间模式解复用（SPADE）能够实现量子最优的大样本斯坦因指数，但靠近单源边界的有限光子行为以及现实缺陷的影响仍未被充分理解。利用奇异学习理论，我们分析了有对准和无对准两种情况。在对准高斯情况下，我们推导了直接成像和SPADE的zeta函数极点，表明两者具有相同的实对数典范阈值$λ=1/2$，但多重性不同，并得到了相应的贝叶斯自由能渐近式。这揭示了在对准局域先验加权区域中，SPADE具有普适的次主导优势。在无对准情况下，我们研究了一个物理动机驱动的二元SPADE简化模型，该模型保留了对准附近完整的$O(s^2)$主导泄漏对比度，而来自高阶模重新分布的修正仅在$O(s^4)$阶出现。我们表明，无对准二元SPADE和直接成像分别在不同内在尺度上获得非平凡的局域幂次，即$s=O(n^{-1/4})$和$s=O(n^{-1/2})$。然而，在常见物理条件下的有限$n$奈曼-皮尔逊比较显示，在绘制的网格上直接成像更强，且无对准二元SPADE表现出精确的盲分离$s^\ast=2θ$，此时其幂次降至$α$。这些结果识别出模型奇异性作为有限光子量子鉴别的结构组织原则，并阐明理想对准SPADE基准如何可能在无对准条件下无法转化为有限$n$优势。