AI 中文总结
针对(1+1)进化算法，研究期望命中时间对变异率的敏感性，指出形式化陷阱，证明OneMax上的逐状态符号定理，并给出非可合并族的可验证符号证书。
AI 中文摘要
针对采用标准位变异的(1+1)进化算法，我们研究了期望命中时间 $H_p=\mathbb{E}_x T$ 对变异率的敏感性。首先，我们指出一个容易被忽视的形式化陷阱：改进事件在变异掩码上不是单调的，因此Margulis-Russo公式的无符号（总影响）形式不适用；正确的对象是带符号的端点差。其次，我们给出一个精确的三维分离：两个适应度函数共享完整的一步成功率曲线，但它们的期望命中时间是两个不同的精确有理数；因此，一步成功率量不能决定期望命中时间。基于运行时导数 $H'_p=(I-Q_p)^{-1}Q'_p H_p$，我们构造了可计算的二重残差符号证书，证明了OneMax上的逐初始状态符号定理（对于每个非最优初始状态，在 $0<c<1$ 上 $\partial_c H<0$，其中 $p=c/n$；在 $c=1$ 时仅距离为1的状态是平稳的），并将该框架扩展到非可合并的正线性族：一个显式的非可合并性见证，一个覆盖所有状态而无需枚举的块区间二重残差证书，一个在整个区间 $c\in[1/4,1/2]$ 上对所有偶数规模 $n\ge 8$ 的显式族的均匀符号界 $\partial_c\mathbb{E}T\le -9n/16$，以及一个在 $c=1$ 时在63个状态中的57个上满足 $H'_x\le -1/6$ 的异构实例证书。所有有限验证均使用精确有理算术。有界的系统性文献搜索未发现这一确切组合，尽管底层工具已成熟；因此，我们不主张超出所述组合的新颖性。
英文摘要
For the (1+1) evolutionary algorithm with standard bit mutation, we study the sensitivity of the expected hitting time $H_p=\mathbb{E}_x T$ to the mutation rate. We first point out an easily overlooked formalization pitfall: the improvement event is not monotone in the mutation mask, so the unsigned (total-influence) form of the Margulis-Russo formula does not apply; the correct object is the signed endpoint difference. Second, we give an exact three-dimensional separation: two fitness functions share the entire one-step success-rate curve, yet their expected hitting times are two different exact rational numbers; hence one-step success-rate quantities do not determine the expected hitting time. Building on the runtime derivative $H'_p=(I-Q_p)^{-1}Q'_p H_p$, we construct computable double-residual sign certificates, prove a per-initial-state sign theorem on OneMax (for every non-optimal initial state, $\partial_c H<0$ on $0<c<1$, where $p=c/n$; at $c=1$ only the distance-one state is stationary), and extend the framework to non-lumpable positive linear families: an explicit non-lumpability witness, a block-interval double-residual certificate that covers all states without enumerating them, a uniform sign bound $\partial_c\mathbb{E}T\le -9n/16$ over the whole interval $c\in[1/4,1/2]$ for an explicit family at all even scales $n\ge 8$, and a heterogeneous instance certificate $H'_x\le -1/6$ on 57 of 63 states across $c=1$. All finite verifications use exact rational arithmetic. A bounded systematic literature search did not uncover this exact combination, although the underlying tools are well established; we therefore make no novelty claim beyond the stated combination.
