AI 中文总结
该研究为离壳Kerr几何建立单极小值凸性判据，以模糊暗物质分布为解析基准推导强场径向分布响应，证实强场盒是形式化基准而非自洽旋转标量场解。
AI 中文摘要
我们针对具有正的、非递减质量分布$m(r)$的离壳Kerr族$Δ(r) = r^2 - 2rm(r) + a^2$建立了一个充分的单极小值判据，证明$1 - 2m'(r) - rm''(r) > 0$可保证严格凸性并确定$Δ$的根的数量。利用一个满足该约束的、受模糊暗物质启发的基准模型，我们推导了外视界、极端分支、光子球和阴影泛函在一般形变$m/M_{\text{ADM}} = 1 + \u03b5 h$下的一阶响应。我们证明静态视界和光子响应由分布控制，自旋奇的阴影位移依赖于完备化，标度一致的弱场极限使得局部分布梯度效应可忽略（$\u226a 10^{-20}$），证实了强场盒是一个形式化的径向分布基准，而非自洽的旋转标量场解。
英文摘要
We establish a sufficient one-minimum criterion for the off-shell Kerr family $Δ(r) = r^2 - 2rm(r) + a^2$ with a positive, nondecreasing mass profile $m(r)$, showing that $1 - 2m'(r) - rm''(r) > 0$ ensures strict convexity and determines root counts for $Δ$. Using a fuzzy-dark-matter-inspired benchmark satisfying this bound, we derive first-order responses for the outer horizon, extremal branch, photon sphere, and shadow functional under general deformations $m/M_{\text{ADM}} = 1 + \varepsilon h$. We demonstrate that static horizon and photon responses are profile-controlled, spin-odd shadow displacements are completion-dependent, and scale-consistent weak-field limits render local profile-gradient effects negligible ($\ll 10^{-20}$), confirming the strong-field box as a formal radial-profile benchmark rather than a self-consistent rotating scalar-field solution.
