{"id":"2a89f006-29de-45f0-8bd3-2db85e3489fd","arxiv_id":"2607.09131","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Discrete Z_N-protected majoron dark matter excludes Z_5, leaves Z_7/Z_11/Z_13 viable, and predicts a 1–10 MeV Z_7 majoron testable by COSI through 511 keV and γγ lines.","lead":"A minimal majoron dark-matter model protected by a discrete Z_N gauge symmetry predicts viable MeV-scale majorons for Z_7 (and heavier for larger N), while Z_5 is already excluded by neutrino-decay bounds. Upcoming COSI MeV gamma-ray data can test the Z_7 case via the 511 keV line and a monochromatic photon line.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged O(1) Planck-coefficient and pure-misalignment assumptions.","rationale":"The paper is a standard, carefully executed hep-ph phenomenology study. Its strongest claim is a viability map under explicit, stated assumptions (discrete Z_N selection rules, leading Planck operator, pure pre-inflationary misalignment). Those assumptions are the natural soft spot, and the reader already identified them. Extending the κ range or adding a competing UV operator is the cleanest single check that would quantify how much the MeV window depends on the O(1)-coefficient premise. Because that premise is already flagged and the rest of the argument holds, no verdict adjustment is warranted.","tokens_in":32195,"tokens_out":516,"duration_ms":6100,"concrete_test":"Recompute the Z_7 mass band of Table 1 / Fig. 2 by extending the κ scan to 10^{-4}–10^2 (or by allowing one additional higher-dimensional operator of comparable size). If the resulting m_J interval moves outside the 1–10 MeV window for more than half the scanned points while still satisfying the abundance and isocurvature conditions, the specific COSI-target claim weakens; otherwise the claim is robust under the paper's own UV assumptions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Z_5 excluded by neutrino-decay bounds; Z_7/Z_11/Z_13 viable, with Z_7 in the 1–10 MeV window testable by COSI) is internally consistent once the paper's stated premises are granted. The mass formula (Eq. 7), abundance expressions (Eqs. 14–15), and decay widths (Eqs. 9–12) follow from the discrete-gauge selection rules and the pre-inflationary misalignment setup. The reader's weakest_assumption correctly isolates the softest point: the assumption that the leading operator is precisely the lowest Z_N-allowed Planck-suppressed term with κ scanned only over 0.01–1 and that production is pure misalignment with O(1) θ_0. No stronger internal inconsistency, missing cancellation, or calculational error that would overturn the viability map or the Z_5 exclusion appears under scrutiny. The COSI projections are already labeled indicative (backgrounds, positron propagation, D-factor), so they do not constitute an independent load-bearing flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs a minimal majoron dark-matter model by extending the SM with three right-handed neutrinos and a complex scalar Φ, with the theory defined by an exact discrete gauge symmetry Z_N ⊂ U(1)_{B-L}. The global U(1)_{B-L} then emerges only accidentally at low energies; for N = 5, 7, 11, 13 the leading Planck-suppressed operator Φ^n generates a controlled majoron mass (Eq. 7, Table 1). Production is studied via pre-inflationary misalignment in both radiation-dominated and early-matter-dominated cosmologies (Eqs. 14–15). The resulting parameter space is confronted with isocurvature, CMB/BBN/Ly-α, and indirect-search bounds. The central claim is that Z_5 is excluded by the dominant J \to \nu\nu decay, while Z_7, Z_11 and Z_13 remain viable; Z_7 predicts m_J ∼ 1–10 MeV and is testable by COSI through the 511 keV line (J \to e⁻e⁺, with Sommerfeld enhancement) and the monochromatic line at E_γ = m_J/2 (J \to γγ).","tokens_in":32554,"tokens_out":1219,"duration_ms":12626,"significance":"If the stated premises hold, the work supplies a predictive, discrete-gauge realization of majoron dark matter that cleanly separates viable from excluded Z_N choices and isolates a concrete MeV-scale target for COSI. Strengths include the controlled mass formula from Z_N selection rules, standard and carefully derived decay widths (tree-level \nu\nu, one-loop e⁻e⁺, two-loop γγ), an explicit Sommerfeld treatment in Appendix A, and a parameter scan of the K_ij coefficients (Appendix B) that yields falsifiable ranges for the MeV signals. The simultaneous 511 keV + monochromatic-line prediction for Z_7 is a distinctive, near-term observational handle.","major_comments":[{"comment":"Sec. 2, Eq. (7) and Table 1: the mass predictions (and therefore the Z_5 exclusion and the Z_7 1–10 MeV window) rest on the assumption that the leading explicit breaking is precisely the lowest Z_N-allowed Planck-suppressed operator with cutoff m_pl and coefficient κ scanned only over 0.01–1. The manuscript should state more explicitly that additional UV operators of comparable size, or a lower cutoff, would shift the mass bands and could reopen or close the windows; a short sensitivity discussion would make the claim robust rather than premise-dependent.","section":null},{"comment":"Sec. 3.2.4 and right panel of Fig. 4: the J \to γγ width is evaluated in an electron-loop-only approximation, with hadronic contributions argued to be suppressed by m_J^{2}/m_had^{2}. For the upper end of the Z_7 window (m_J ∼ O(10) MeV) this is only marginally safe; a quantitative estimate or a clear statement that the exclusion of m_J ≳ O(10) MeV is indicative would strengthen the figure and the associated COSI projection.","section":null}],"minor_comments":[{"comment":"Sec. 3.1 and Figs. 2–3: the magenta hatched abundance band uses 0.5 ≤ θ_0 ≤ 5; a brief remark on how the band changes for θ_0 \to O(0.1) or near the anharmonic regime would help readers assess robustness.","section":null},{"comment":"Sec. 4.1: the COSI 511 keV sensitivity is labeled indicative (RoI definition, positron escape, neglected backgrounds). Making this caveat more prominent in the figure caption of Fig. 6 (left) would avoid over-reading the orange band.","section":null},{"comment":"Appendix B: the scan sets Im z_i = 0. A one-sentence note that moderate imaginary parts mainly affect the tails (as already suggested) would clarify that the quoted 99.8 % interval is a baseline, not a full prior.","section":null},{"comment":"Notation: f ≡ v_Φ/N is introduced after Eq. (14); stating it once in Sec. 2 when the nonlinear representation is defined would reduce later ambiguity for n \neq N cases.","section":null},{"comment":"Table 1 caption: the phrase “equivalently m_J / [κ^{1/2}(v_Φ/10^{10} GeV)^{n/2−1}]” is slightly awkward; rephrasing as the numerical prefactor for that normalization would improve readability.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid, self-contained hep-ph phenomenology paper. The two major points are premise-clarification rather than calculational errors; either can be handled with short additions. Fit for a standard particle-physics journal is good; no novelty or citation concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a solid, usable hep-ph paper. The real addition is not “majoron DM exists,” but a minimal discrete-gauge construction that makes the mass prediction sharp enough to exclude Z_5 by the neutrino channel and leave Z_7 sitting in the 1–10 MeV window with two correlated COSI lines.\n\nWhat they do well: the selection rules (Table 1 + Eq. 7) are transparent; the RD and early-MD abundance maps (Eqs. 14–15) are standard and correctly applied; the tree/one-loop/two-loop widths (Eqs. 9–12) and the Sommerfeld treatment in Appendix A are careful; the parameter scan for |K11−K22−K33| is honest about priors. The Z_5 exclusion is robust once you accept the setup. The Z_7 MeV target is the part people will actually use.\n\nSoft spots, in proportion: the mass and abundance still rest on O(1) Planck coefficients (κ scanned 0.01–1) and pure pre-inflationary misalignment with O(1) θ0. That is the usual quality-control assumption for discrete-gauge majorons, not a hidden fit, but it is load-bearing. The COSI “covers the green band” statements are already labeled indicative—backgrounds, positron propagation, D-factor—so they do not overclaim. Electron-loop-only γγ is fine for the MeV window they care about; they flag the hadronic caveat. No internal contradiction or missing cancellation that overturns the viability map.\n\nWho it is for: people working on seesaw/majoron DM, MeV gamma-ray forecasts, or discrete-gauge quality control of global symmetries. Citation pattern is normal for the subfield. Math and logic check out. I would send it to referees; it is ready for that level of scrutiny. Worth engaging if you care about MeV targets or discrete B−L remnants.","headline":"Clean minimal Z_N majoron setup that cleanly kills Z_5 and hands COSI a concrete MeV target; soft only where the authors already say it is soft.","tokens_in":33237,"tokens_out":510,"would_cite":true,"duration_ms":6461,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["95.35.+d","14.60.St","12.60.-i","98.80.Cq"],"model":"grok-4.5","headline":"A discrete Z_N gauge symmetry turns a light majoron into predictive dark matter: Z_5 is excluded, while Z_7 lands in the 1–10 MeV window that COSI can test with two line signals.","keywords":["majoron dark matter","discrete gauge symmetry","Z_N","misalignment mechanism","seesaw","MeV gamma rays","B-L symmetry","COSI"],"falsifier":"A null COSI search that fails to detect both an off-bulge 511 keV excess and a monochromatic line at E_γ = m_J/2 across 1–10 MeV, at fluxes corresponding to the paper’s central 99.8 % scan of |K_11−K_22−K_33|, would exclude the bulk of the viable Z_7 parameter space under the stated production assumptions.","tokens_in":33052,"feed_emoji":"🔭","tokens_out":1295,"duration_ms":27534,"temperature":0.7,"pith_summary":"This paper constructs a minimal majoron dark-matter model by adding three right-handed neutrinos and one complex scalar to the Standard Model, but defining the theory with an exact discrete gauge symmetry Z_N rather than a fundamental global B−L symmetry. Global B−L then emerges only accidentally at low energies; for nontrivial N (5, 7, 11, 13) the first allowed Planck-suppressed operators generate a controlled majoron mass, so the pseudo-Nambu–Goldstone boson can be dark matter. Abundance is computed from post-inflationary misalignment in both radiation-dominated and early-matter-dominated histories, then confronted with isocurvature, cosmological, and indirect-search bounds. Z_5 is already ruled out by limits on its dominant decay into neutrinos, while Z_7, Z_11 and Z_13 remain viable. The Z_7 case is the most immediate target: it predicts a 1–10 MeV mass whose decays produce a 511 keV line and a monochromatic gamma-ray line that future MeV telescopes, especially COSI, can probe.","feed_headline":"Z_7 majoron dark matter lands at 1–10 MeV","feed_subtitle":"Discrete gauge control plus misalignment leave a COSI-testable window of 511 keV and monochromatic gamma lines.","key_machinery":"The discrete gauge symmetry Z_N that forbids all Φ^n operators with n ≤ 4 and allows a leading Planck-suppressed term Φ^N/m_pl^{N−4}. That operator alone fixes the majoron mass scaling m_J ∝ v_Φ^{N/2−1}/m_pl^{N/2−2} and thereby predicts the mass window for each N.","core_discovery":"In the minimal majoron model with three right-handed neutrinos and a complex scalar, an exact discrete gauge symmetry Z_N ⊂ U(1)_{B−L} makes global B−L accidental. For Z_5, Z_7, Z_11 and Z_13 the leading Planck-suppressed operators set a light majoron mass. Misalignment production after inflation leaves Z_5 excluded by neutrino-decay limits while Z_7, Z_11 and Z_13 remain open; Z_7 specifically predicts a 1–10 MeV majoron that yields both a 511 keV line from e⁻e⁺ and a monochromatic line at E_γ = m_J/2, both accessible to COSI.","pith_inferences":["Simultaneous detection of an off-bulge 511 keV excess and a monochromatic line at half the same mass would favor an electron-loop–dominated majoron over many other MeV dark-matter candidates whose rates are not correlated that way.","The same discrete-gauge control of Planck-suppressed operators can be applied to other accidental global symmetries (for example axion quality problems) to generate predictive light pseudo-Goldstone masses.","A COSI null result would mainly exclude the bulk of the scanned Z_7 coupling window; smaller K_ij cancellations or early-matter-domination corners could still survive.","Requiring pre-inflationary breaking to avoid Z_N defects can tighten the allowed reheating window more than the paper’s conservative thermal-restoration bounds once concrete inflation models are specified."],"forward_implications":["Z_5 majoron dark matter is excluded by existing cosmological and neutrino-line limits on J→νν.","Z_7 majoron dark matter is predicted to lie in the 1–10 MeV range for both radiation-dominated and early-matter-dominated misalignment.","COSI can cover most of the representative Z_7 coupling range via the 511 keV line (including near the e⁻e⁺ threshold thanks to the Sommerfeld effect) and the monochromatic line at half the majoron mass.","Conventional high-scale thermal leptogenesis remains compatible with radiation-dominated misalignment for the viable Z_N models; early-matter domination requires lower-scale or flavored leptogenesis.","Z_11 and Z_13 push the majoron to lighter masses and weaker couplings that stay cosmologically allowed but are harder to test soon."],"fun_headline_variants":["Z_7 majoron DM at 1–10 MeV yields COSI 511 keV and gamma lines","Discrete Z_N majorons: Z_5 out, Z_7–13 open with MeV mass window","Misalignment majoron from Z_7 survives limits at 1–10 MeV","Planck-suppressed Z_N sets light majoron DM testable by COSI","Z_7 majoron predicts e+e- 511 keV line and monochromatic gamma"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The mass and abundance rest on the claim that the majoron mass comes only from the lowest Z_N-allowed Planck-suppressed operator with a coefficient of order one, and that dark matter is produced solely by pre-inflationary misalignment with an order-one initial angle.","fun_headline_variants_meta":{"raw":{"variants":["Z_7 majoron DM at 1–10 MeV yields COSI 511 keV and gamma lines","Discrete Z_N majorons: Z_5 out, Z_7–13 open with MeV mass window","Misalignment majoron from Z_7 survives limits at 1–10 MeV","Planck-suppressed Z_N sets light majoron DM testable by COSI","Z_7 majoron predicts e+e- 511 keV line and monochromatic gamma"]},"model":"grok-4.5","effort":"low","cost_usd":0.004434,"raw_usage":{"total_tokens":1417,"prompt_tokens":921,"num_sources_used":0,"completion_tokens":112,"cost_in_usd_ticks":44340000,"prompt_tokens_details":{"text_tokens":921,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":384,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":921,"tokens_out":112,"duration_ms":4139,"temperature":1.0,"reasoning_tokens":384,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T05:10:52.606328+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"A null COSI search that fails to detect both an off-bulge 511 keV excess and a monochromatic line at E_γ = m_J/2 across 1–10 MeV, at fluxes corresponding to the paper’s central 99.8 % scan of |K_11−K_22−K_33|, would exclude the bulk of the viable Z_7 parameter space under the stated production assumptions.","supporting_citations":[],"review_version":1}