AI 中文总结
该研究分类瞬态冲击模型中非线性项与记忆核的组合顺序，揭示幂律冲击的可操纵性条件，提出核完全正性的等价判据，并量化幂律情形下的相关复杂性。
AI 中文摘要
瞬态冲击模型将非线性项与记忆核组合，组合顺序决定了无价格操纵的判据。我们对两种组合顺序进行分类：若任意瞬时法则 f 在任意非零可积 Volterra 核之前作用于交易速率，且每条有限分段恒定往返的非负成本迫使 f 为仿射函数，对每个非零卷积核则为线性函数。特别地，幂律 sgn(x)|x|^δ（δ>0）与幂律衰减 t^(-γ)（0<γ<1）的组合存在操纵当且仅当δ≠1：平方根冲击在任意衰减指数下均可操纵，Gatheral 的慢速率双块界 δ+γ≥1 所留区域收缩至直线δ=1。早期刚性定理要求核在零点有界；此处的论证是通过两笔基准交易读出的零体积抖动泵，适用于奇异核。若单调读出器在核后作用于冲击状态，则所有输入和所有读出器的安全性等价于核的完全正性，且存在构造性逆；特别地，幂律记忆后的平方根冲击无操纵。一个远程补偿块表明，对尾均匀消失的核，往返安全性与全输入安全性一致；对永久记忆，二者通过显式存储商区分。这些机制对所有双模 Prony 核、一阶非均匀时间记忆及稳定全驱动矩阵记忆进行分类，并量化了幂律情形背后的摩擦、双块相及切换复杂性。校准后的指数对均位于可操纵集：无摩擦时，凹性必须出现在记忆之后而非之前。
英文摘要
Transient impact models compose a nonlinearity with a memory kernel, and the order of composition determines the criterion for absence of price manipulation. We classify both orders. If an arbitrary instantaneous law $f$ acts on the trading rate before any nonzero integrable Volterra kernel, nonnegative cost on every finite piecewise-constant round trip forces $f$ to be affine, and linear for every nonzero convolution kernel. In particular, the power law $\mathrm{sgn}(x)|x|^δ$, $δ>0$, combined with power-law decay $t^{-γ}$, $0<γ<1$, admits manipulation if and only if $δ\ne1$: square-root impact is manipulable at every decay exponent, and the region left open by Gatheral's slow-rate two-block bound $δ+γ\ge1$ collapses to the line $δ=1$. Earlier rigidity theorems require a kernel that is bounded at zero; the argument here is a zero-volume chattering pump read out by two thin baseline trades, and it applies to singular kernels. If instead a monotone readout acts on the impact state after the kernel, safety for all inputs and all readouts is equivalent to complete positivity of the kernel, with a constructive converse; in particular, square-root impact after power-law memory is manipulation-free. A remote compensating block shows that round-trip safety and all-input safety coincide for kernels with uniformly vanishing tails and differ, for permanent memory, by an explicit storage quotient. These mechanisms classify every two-mode Prony kernel, first-order time-inhomogeneous memory, and stable fully actuated matrix memory, and they quantify the friction, the two-block phase, and the switching complexity behind the power-law case. Calibrated exponent pairs all lie in the manipulable set: absent friction, concavity has to enter after the memory, not before it.
