{"id":"2299921e-f042-439e-a751-efb678217d2e","arxiv_id":"2512.07657","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A systematic catalogue of realisable Higgs-family, general-CP, and GOOFy/T-GOOFy symmetries in three-Higgs-doublet models, with new invariant potentials and a corrected U(1)◦V4 group structure.","lead":"This paper classifies the discrete and continuous symmetries that can be imposed on the scalar sector of three-Higgs-doublet models, including new 'GOOFy' transformations that act separately on fields and their conjugates. It provides a reference catalogue of symmetry groups and their allowed scalar potentials for model builders.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness of the 3HDM symmetry catalogue rests on an unproven finite-order scan of U(3) subgroups; a missing subgroup outside Ref. [52] or above order 36 could invalidate the 'most complete' claim.","rationale":"The reader's verdict is already CONDITIONAL, with the weakest assumption identified as the completeness of the scan based on Ref. [52] and saturation. My independent read converges on the same load-bearing concern: the paper's headline 'most complete picture' is not backed by a proof of completeness, only by an empirical saturation argument. The GOOFy restriction is an additional limitation but is explicitly acknowledged as such and does not affect the central HF/GCP tables. Thus no verdict change is needed; the conditional framing already correctly reflects the risk. The concrete test I propose directly targets the completeness gap: an independent exhaustive enumeration of finite subgroups of U(3) up to order 2000, followed by a potential scan for any missing groups, would either confirm saturation or expose a counterexample.","tokens_in":54377,"tokens_out":4811,"duration_ms":48304,"concrete_test":"Cross-check the generator list of Ref. [52] against the complete classification of finite subgroups of SU(3) (Ludl, J. Phys. A 43 (2010) 395204) and the GAP SmallGroup library up to order 2000. For any finite subgroup of U(3) missing from Ref. [52], impose its generators on the most general 3HDM scalar potential and compare the resulting invariant potential with Tables 1–2 (coupling counts and bilinear eigenvalue patterns). If a missing group yields a new potential, the 'most complete' claim is falsified; if none does, the saturation claim is considerably strengthened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that Tables 1–2 provide the most complete overview of realisable HF and GCP symmetries in 3HDMs depends on the brute-force scan over generator families from Ref. [52], which lists finite subgroups of U(3) of order <2000 'with some special exceptions.' The authors explicitly disclaim strict mathematical completeness (Sec. 3) and justify the catalogue by saturation: no new potentials beyond order 36. This is an empirical observation, not a proof. Two concrete gaps follow: (i) Ref. [52] may omit some subgroups below order 2000 due to the 'special exceptions,' and the saturation argument only covers groups actually scanned; (ii) even with a complete list, saturation at order 36 does not logically exclude a larger group (order 37–2000) that produces a new potential, unless the scan is exhaustive in that range. Because the tables are intended as a practical diagnostic for model builders, a single realisable HF or GCP symmetry not listed would break the headline claim. The GOOFy sections have their own stated restriction to Ref. [68] block forms (Sec. 6.2), but the 'most complete' claim is specifically tied to Tables 1–2, so the load-bearing concern is the completeness of the HF/GCP scan.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a systematic, scan-based classification of symmetry groups realisable by the scalar potential of three-Higgs-doublet models. For conventional Higgs-family (HF) and general CP (GCP) transformations, the authors construct invariant potentials by imposing generator families of finite subgroups of U(3) taken from Ref. [52] (order below 2000), and organise the results into Tables 1–3 by numbers of couplings and bilinear eigenvalue patterns. They also analyse GOOFy and T-GOOFy transformations of the block forms (6.11)–(6.12), providing invariant quartic potentials in Sections 6–7 and Appendices D–E, and they correct the earlier U(1)×D4 label by identifying the underlying central-product structure and using cohomology (Appendix B). The authors state explicitly that strict mathematical completeness is not claimed, and that the GOOFy analysis is restricted to the families of Ref. [68].","tokens_in":54601,"tokens_out":10526,"duration_ms":92273,"significance":"The paper has substantial reference value. The explicit scalar potentials and the bilinear eigenvalue-pattern tables give model builders a concrete diagnostic tool, the basis-invariant counting in Appendix A is useful, and the Appendix B analysis of U(1)◦V4 is a genuine improvement over the earlier U(1)×D4 label. The GOOFy sections are, to my knowledge, the first systematic survey of 3HDM scalar potentials invariant under the Ref. [68] block-form transformations. However, the central HF/GCP catalogue is established by an empirical scan rather than by a proof of completeness, and the GOOFy survey is restricted by construction. These caveats should be made explicit in the abstract and conclusions.","major_comments":[{"comment":"The central HF/GCP catalogue is presented as the 'most complete overview to date', but its completeness is not proven. The scan starts from generator families of finite subgroups of U(3) of order below 2000 listed in Ref. [52], a reference that itself contains 'special exceptions', and the saturation criterion is the empirical observation that no new potentials appear beyond order 36 (Sec. 3). This does not logically exclude a realisable group of order 37–2000, nor a missing subgroup below 2000. Because Tables 1–2 are offered as a practical diagnostic, an omitted realisable symmetry would break the headline claim. Please either provide a rigorous completeness argument, or explicitly and consistently downgrade the claims to a scan-based catalogue with completeness left open; the abstract and Sec. 8 currently overstate the status.","section":"Sec. 3; Tables 1–2"},{"comment":"The GOOFy/T-GOOFy analysis is restricted to transformations of the block forms (6.11)–(6.12) from Ref. [68]. The paper admits in Sec. 6.2 that no general mapping from arbitrary h_ij transformations to these forms is provided, and the T-GOOFy survey is limited to quartic terms. The abstract and Sec. 8 describe this as an expansion of the set of symmetries and a 'first systematic study'; this is defensible only within the stated block-form restriction. Please state the restriction prominently in the abstract and treat the GOOFy part as a survey of the Ref. [68] family, not of generalised symmetries beyond that family.","section":"Sec. 6.2; Sections 6–7"},{"comment":"There are internal discrepancies in the summary tables. Table 3 lists S3×Z*2 at N≤10 and Z3⋊Z*2 at N≤15 in the GCP column, but neither appears in the GCP column of Table 1 or in the eigenvalue-pattern list of Table 2. The preamble to Table 1 explains some exclusions (real-parameter restrictions), but the Table 3 entries are not footnoted or cross-referenced. Since these tables are meant to be a practical guide, the authors should either list these groups in Tables 1–2 with the relevant eigenvalue patterns, or remove them from Table 3 with an explanation.","section":"Sec. 5; Tables 1–3"},{"comment":"The physical motivation for GOOFy symmetries rests on all-order RG stability, but the paper states that a full proof is expected in a future publication. The present manuscript therefore classifies potentials invariant under transformations that are not symmetries of the kinetic term, with the dynamical stability claim unproven. Please separate the classification (which is well defined) from the RG-stability claim, and either prove or clearly mark the stability assertions as conjectures based on Ref. [16].","section":"Sec. 6.1"}],"minor_comments":[{"comment":"The claim that V4_G1, V4_G3 and V4_G4 are unitarily inequivalent is asserted without proof. Please give a short argument or cite a computation.","section":"Sec. 7.1, after Eq. (7.11)"},{"comment":"The two generators in Eq. (4.10) use phases θ1, θ2 without stating their ranges or independence. A brief specification would let the reader verify that the generated group is indeed O(2)×U(1).","section":"Sec. 4.1, Eq. (4.10)"},{"comment":"The notation G1, G2, ... is used for both individual generators and generated symmetry groups. Consider using calligraphic labels for groups, or add a sentence disambiguating the notation.","section":"Sec. 7.1, Eqs. (7.3)–(7.13)"},{"comment":"In several rows the 'GHF' sub-column lists the identity matrix, which is confusing. Leaving those cells empty or marking them as 'trivial' would improve readability.","section":"Sec. 6.3, Table 4"},{"comment":"The scan-based 'discarding cases that reproduce previously obtained potentials' step is not described in enough detail to be independently reproduced. Publishing the GAP scripts or a pseudo-code description of the potential-comparison algorithm would strengthen confidence in the catalogue.","section":"Sec. 2 and Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and makes a useful contribution, but the advertised completeness of the HF/GCP tables is stronger than the evidence in the manuscript. The GOOFy sections are interesting but should be framed as restricted to the Ref. [68] block forms. I would be comfortable with acceptance after the completeness claims are made precise and the cross-table inconsistencies are fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you work on multi-Higgs-doublet models. The paper is a systematic, explicit catalogue of realisable symmetries in 3HDMs, and two things stand out. First, it corrects the previously used U(1)×D4 label to a central product U(1)◦V4, with a cohomological argument in Appendix B that looks right. Second, the GOOFy and T-GOOFy sections are genuinely new: the 3HDM GOOFy potential families and the T-GOOFy classification are not in the cited literature. The authors also say clearly which parts consolidate earlier work, which earns credit.\n\nThe tables are the main practical payoff. Tables 1–3 list the symmetry groups, coupling counts, and bilinear eigenvalue patterns; they also give parameter-count upper bounds with the centraliser logic in Appendix A. For model builders these are useful diagnostics. The paper ships explicit generators for the new classes, and the eigenvalue patterns are computed, not imposed, so there is no circularity in the classification itself.\n\nThe soft spots are exactly where the authors are candid. Completeness of the conventional HF/GCP catalogue rests on a brute-force scan over the finite subgroups of U(3) of order below 2000 from Ref. [52], and that list has 'some special exceptions.' The authors state flatly that strict mathematical completeness cannot be claimed; the saturation at order 36 is an empirical observation, not a proof. A subgroup outside the scanned set would dent the 'most complete' wording. I would not call this fatal — the catalogue is still the best available and the claim is explicitly empirical — but a referee should make sure the abstract and summary do not imply a proof that was not supplied. The GOOFy part is also restricted, by the authors' own admission, to quartic terms and to the block-form transformations of Ref. [68]; a general map for arbitrary hij transformations is not provided. And the all-order RG-stability argument that motivates GOOFy physics is deferred to a future paper, so the physical relevance is conditional.\n\nWho should read it: anyone building 3HDM models who needs to identify or impose a symmetry, and anyone working on generalised CP or GOOFy applications. It deserves a serious referee; I would send it to review. My own verdict would be accept after a moderate revision that sharpens the proven-versus-scanned distinction and notes the exact scope of the GOOFy classification in the abstract.","headline":"Careful, honest 3HDM symmetry catalogue that fixes the U(1)×D4 label and adds new GOOFy/T-GOOFy sections; referee it, but push back on the scan-based 'most complete' phrasing.","tokens_in":55207,"tokens_out":2791,"would_cite":true,"duration_ms":28196,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A systematic catalogue of realisable symmetries in three-Higgs-doublet models, including newly studied GOOFy transformations, is presented as a practical reference for model builders.","keywords":["3HDM","Higgs family symmetries","general CP transformations","GOOFy symmetries","T-GOOFy transformations","scalar potential classification","bilinear formalism","discrete symmetries"],"falsifier":"Finding a 3HDM scalar potential whose symmetry group is not listed in Tables 1–2 and whose bilinear eigenvalue pattern is not among those tabulated would falsify the completeness claim. Concretely, a discrete subgroup of U(3) with order above 36 that yields a new quartic potential, or a GOOFy transformation not expressible in the (U(1), U(2)) block form producing a distinct quartic potential, would invalidate the classification.","tokens_in":54208,"feed_emoji":"🌌","tokens_out":1226,"duration_ms":13360,"temperature":0.7,"pith_summary":"This paper aims to provide the most complete classification to date of realisable symmetries in three-Higgs-doublet models (3HDMs). It revisits conventional Higgs-family and general-CP transformations, identifies limitations in previous approaches, and compiles tables of realisable groups with their coupling counts and bilinear eigenvalue patterns. The authors then extend the analysis to GOOFy transformations, which act differently on Higgs doublets and their conjugates and can flip kinetic-term signs, providing the first systematic study of GOOFy- and T-GOOFy-invariant scalar potentials (restricted to quartic terms). If correct, model builders gain a practical diagnostic reference for identifying the symmetry of any 3HDM scalar potential, including previously overlooked structures such as U(1)∘V4.","feed_headline":"Catalogue of realisable 3HDM symmetries, now with GOOFy","feed_subtitle":"Tables of coupling counts and eigenvalue patterns give model builders a practical way to identify any scalar-sector symmetry.","key_machinery":"The central machinery is the extended 2N-dimensional Higgs space H = (h_i, h_i*) and the SU(2) bilinear singlets h_ij = h_i† h_j, together with the brute-force procedure of imposing symmetry generators iteratively on the most general 3HDM potential. Bilinear eigenvalue patterns (e.g., 8^1, 2^2 4^1) serve as basis-invariant diagnostics for identifying symmetry classes. For GOOFy/T-GOOFy transformations, the key structural device is the pair of independent unitary blocks U^(1), U^(2) acting on doublets and conjugates, with the requirement that the product of a transformation and its Hermitian conjugate be diagonal with ±1 eigenvalues.","core_discovery":"The central claim is that, by systematically imposing generator families of finite subgroups of U(3) (and continuous limits) on the most general 3HDM scalar potential, one obtains a complete practical catalogue of realisable symmetry groups, summarised in Tables 1–3. The paper also establishes that GOOFy HF-like transformations admit only two unique generator types, Z2 and Z4, and that GCP-like GOOFy transformations likewise reduce to Z2 and Z4, while T-GOOFy transformations (with independent unitary rotations of doublets and conjugates) yield additional distinct quartic potentials that can be organised by bilinear eigenvalue patterns.","pith_inferences":["If the catalogue is taken as complete, it suggests that the space of realisable 3HDM symmetries is closed under the operations considered — a useful constraint for future searches for exotic symmetries.","The GOOFy analysis hints that the distinction between 'realisable symmetry' and 'accidental parameter relation' may be more fluid than previously assumed, since GOOFy transformations enforce tree-level relations that are not symmetries of the full Lagrangian in the usual sense.","The restriction to quartic terms in the GOOFy study leaves open the question of whether bilinear-sector constraints could be classified systematically for arbitrary U^(1), U^(2) pairs; a general mapping from hij transformations to the block form is not provided.","The observation that U(1)∘V4 admits multiple finite lifts selected by the Yukawa sector suggests a general principle: scalar-sector symmetries may define equivalence classes of group extensions, with fermion embeddings choosing a representative."],"forward_implications":["Tables 1–3 give model builders a direct way to identify the symmetry of a 3HDM scalar potential by comparing coupling counts and bilinear eigenvalue patterns.","The paper consolidates previously scattered results and corrects the earlier U(1)×D4 identification to the central-product structure U(1)∘V4, clarifying when different finite lifts (D4, Q8, P1) give the same potential.","GOOFy symmetries can eliminate bilinear terms while leaving quartic couplings intact, yielding Gildener–Weinberg-like scale-invariant scalar sectors and potentials that decouple a Higgs doublet at tree level.","The classification reveals that eigenvalue patterns alone do not uniquely identify a symmetry, motivating additional basis-invariant checks.","The GOOFy framework points toward new model-building directions, including the possibility of auxiliary scalar fields and RG-stable parameter relations without a conventional symmetry."],"fun_headline_variants":["3HDM symmetries: GOOFy transformations added","Systematic 3HDM symmetry catalogue with GOOFy","Re-examining 3HDM symmetries via GOOFy","3HDM symmetries expanded by GOOFy analysis","New clarity on 3HDM symmetry groups"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The catalogue's completeness rests on the assumption that the generator families of finite subgroups of U(3) of order below 2000, together with their continuous limits, exhaust all physically realisable 3HDM symmetries — an assumption the authors explicitly state cannot be claimed in a strict mathematical sense.","fun_headline_variants_meta":{"raw":{"variants":["3HDM symmetries: GOOFy transformations added","Systematic 3HDM symmetry catalogue with GOOFy","Re-examining 3HDM symmetries via GOOFy","3HDM symmetries expanded by GOOFy analysis","New clarity on 3HDM symmetry groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1173,"prompt_tokens":683,"completion_tokens":490,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":427,"completion_tokens_details":{"reasoning_tokens":417}},"tokens_in":427,"tokens_out":490,"duration_ms":4172,"temperature":1.0,"reasoning_tokens":417,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T17:51:11.921850+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Finding a 3HDM scalar potential whose symmetry group is not listed in Tables 1–2 and whose bilinear eigenvalue pattern is not among those tabulated would falsify the completeness claim. Concretely, a discrete subgroup of U(3) with order above 36 that yields a new quartic potential, or a GOOFy transformation not expressible in the (U(1), U(2)) block form producing a distinct quartic potential, would invalidate the classification.","supporting_citations":[],"review_version":1}