AI 中文总结
针对局域减除产生重整化群单调量的场景差异问题，提出秩匹配方法，通过三个精确测试揭示端点不等式与跑动单调性可分离，明确缺陷-初级维数对b函数单调性的影响。
AI 中文摘要
为何在某些场景下，局域减除会产生重整化群单调量，而在另一些场景中却会失效？我们提出了秩匹配方法。局域抵消项决定了标度导数需要满足多少种方案独立性要求。每一个导数都会增加一个连通插入项，而可用的正性输入是双线性形式或正的二阶变分，因此仅能控制二次数据。一阶是其标度导数仍处于这些输入直接作用范围内的最后一次减除。当Ward恒等式或熵恒等式提供一个带符号的二次形式时，一阶减除可以闭合；更高阶则需要额外的动力学。三个可精确求解的测试展现了两种结果，并表明端点不等式可以成立，而跑动单调性却会失效。有质量标量场在所有奇数维p-球面（p≥3）上产生非单调的滤波自由能。当缺陷-初级维数Δ̂满足1/2≤Δ̂<1时，广义自由表面缺陷b函数是严格单调的；而当0<Δ̂<1/2时，其必然是非单调的。在阈值处，其流动系数在正则谱坐标中是完全单调的，各阶导数符号交替变化。一个局域四维单值性缺陷实现了完整的转变。在大分量熵极限的既定假设下，圆盘熵与球面自由能共享端点和总F损失，但在标度上的分布方式不同。秩匹配将端点排序、跑动单调性以及损失在标度上的分布分离开来。
英文摘要
Why do local subtractions produce renormalization-group monotones in some settings but fail in others? We propose rank matching. Local counterterms fix how many scale derivatives scheme independence requires. Every derivative adds one connected insertion, while the available positivity inputs are bilinear forms or positive second variations and therefore control only quadratic data. Degree one is the last subtraction whose scale derivative stays within their direct reach. First-order subtractions can close when a Ward or entropic identity supplies a signed quadratic form. Higher orders need extra dynamics. Three exactly solvable tests exhibit both outcomes and show that the endpoint inequality can hold while running monotonicity fails. Massive scalars yield nonmonotone filtered free energies on every odd dimensional $p$-sphere with $p\geq3$. A generalized-free surface-defect $b$-function is strictly monotone when the defect-primary dimension $\widehatΔ$ satisfies $1/2\leq\widehatΔ<1$, and necessarily nonmonotone for $0<\widehatΔ<1/2$. At the threshold, its flow coefficient is completely monotone in the canonical spectral coordinate, with derivatives of every order alternating in sign. A local four-dimensional monodromy defect realizes the full transition. Under a stated assumption on the large-component entropy limit, disk entropy and sphere free energy share endpoints and total $F$ loss but distribute it differently over scale. Rank matching separates endpoint ordering, running monotonicity, and the distribution of loss over scale.
