Rank matching in renormalization-group irreversibility: Exact defect and entropic tests

Abstract

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.