{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"c92b78b802cf93678b67e4083a41b4773443cbbf59e06977c74e482564a0d3cb","created":"2026-10-03T07:18:09Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"014fe09e28c70596c12434044427a515a3593446c5cf604eb96364937a2b52db","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"qft.rg-irreversibility.defect-odd-dim","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Defects such as impurity lines or surfaces inside a critical material carry their own count of degrees of freedom. This count is proven to drop under coarse-graining for line, surface and four-dimensional defects, but not for three-dimensional ones.","posed_since":"2018","precise":"For a conformal defect of dimension $p$ in a unitary d-dimensional CFT with the bulk held fixed, monotonicity under defect RG flows is proven for $p = 1$ (defect entropy; Cuomo, Komargodski, Raviv-Moshe 2021), $p = 2$ (b-theorem; Jensen and O'Bannon 2015, extended to surface defects in 2018) and $p = 4$ (defect a-theorem; Wang 2021). For $p = 3$, prove or refute that the defect contribution to the regularized sphere free energy decreases, as conjectured by Kobayashi, Nishioka, Sato and Watanabe (2018).","problem_ref":null,"references":"","settled_by":"A proof of monotonicity of the defect sphere free energy for all unitary $p = 3$ defect flows, or a unitary counterexample.","status_note":"The naive $b$-theorem was found to fail when the bulk also flows (Shachar, Sinha, Smolkin 2024); the $p = 3$ case with fixed bulk is unproven.","title":"Irreversibility of RG flows on three-dimensional 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