{
  "abstract": "We ask when local subtractions of the sphere free energy yield a function that decreases monotonically along a renormalization-group flow. The subtractions are polynomials in $D=R d/dR$ that remove the local counterterms and leave fixed-point values unchanged, in a fixed renormalization prescription. We prove that if the subtracted free energy $C$ approaches its infrared value from below, with ordered endpoints and a power-law approach to both fixed points, no such subtraction is monotone, and that further factors with real zeros cannot remove a nonmonotonicity already present. For a three-dimensional flow to a gapped phase, $C-C_{\\rm IR}=4π^2(M_2-3M_0)/R+\\dots$, where $M_0$ and $M_2$ are spectral moments of the flat-space stress-tensor two-point function, so every subtraction fails when $M_2<3M_0$, as for a free conformally coupled massive scalar. In two dimensions the leading term is always negative, so no unitary flow to a gapped phase whose subtracted free energy starts at $c_{\\rm UV}/3$ admits a monotone subtraction; the massive Dirac fermion on $S^2$ is an exact example. In the conformally coupled massive $O(N)$ model, $M_2-3M_0$ is negative at leading order in $N$ for every finite nonnegative quartic coupling. At criticality it vanishes at that order and its first correction is negative. At weak positive coupling it is negative for every curvature coupling and every finite $N$. For the Gross--Neveu flow from an interacting fixed point, it is negative already at leading order. In contrast, we prove monotonicity for Robin boundary flows on every sphere $S^p$ with $p\\ge2$ and for massive Dirac fields on every odd sphere $S^d$ with $d\\ge3$. For two solvable double-trace families we find exact thresholds: a monotone subtraction exists if and only if $\\widehatΔ\\ge1/2$ on a spherical surface defect, and $Δ\\ge(3-\\sqrt{37/13})/2$ on $S^3$ at leading large $N$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02447",
  "author_total": 1,
  "authors": [
    {
      "name": "Francesco Scardino"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/36jdjqh5vbk35.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02447",
  "fulltext_url": null,
  "id": "36jdjqh5vbk35",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02447",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7172413029",
  "pdf_url": "https://arxiv.org/pdf/2608.02447",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/36jdjqh5vbk35.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:11Z",
      "native_id": "2608.02447",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02447"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Local subtractions and renormalization-group monotonicity: exact defect and sphere flows",
  "title_text": "Local subtractions and renormalization-group monotonicity: exact defect and sphere flows",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/36jdjqh5vbk35",
  "venue": "arXiv",
  "versions": []
}