Local subtractions and renormalization-group monotonicity: exact defect and sphere flows

We ask when the sphere free energy of a quantum field theory, after subtraction of its local counterterms in a fixed renormalization prescription, decreases monotonically along the renormalization-group (RG) flow. The counterterms fix the required zeros of the subtraction polynomial, but positivity of the spectral data alone does not determine monotonicity. For Gaussian determinants, complete monotonicity of a rescaled integrated heat trace is sufficient. This condition proves monotonicity for Robin boundary flows on every sphere $S^p$, $p\ge 2$. A related identity proves monotonicity for massive Dirac fields on odd spheres $S^d$, $d\ge 3$. Conversely, under the stated endpoint and decay assumptions, an approach to the infrared value from below excludes every endpoint-preserving finite polynomial subtraction. For the double-trace families considered here, driven by ${\mathcal{O}}^2$ on spherical surface defects and, at leading order in large $N$, on $S^3$, a monotone subtraction exists exactly when the dimension of $\mathcal{O}$ is at least $1\mathrm{/}2$ and $(3-\sqrt{37\mathrm{/}13})\mathrm{/}2$, respectively. Below these thresholds no subtraction is monotone, although the spectra remain positive. The same determinants describe double-trace boundary flows of a free massive scalar in AdS${}_3$ and AdS${}_4$, where the classifications carry over. Finally, with contact terms fixed, the leading infrared sphere coefficient of a flow from a three-dimensional conformal field theory to a gapped phase is proportional to a difference of nonnegative stress-tensor spectral moments. In the massive critical $O(N)$ model this difference vanishes at leading order and is negative at order $N^0$.