{"id":"59a7c2b1-f14c-41d6-bd29-1e984ffda4e2","arxiv_id":"2610.36060","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A hybrid type-III, inverse, and linear seesaw for neutrinos in an SU(2)_1 x SU(2)_2 x U(1)_Y gauge model with a majoron.","lead":"This paper constructs a neutrino mass model in an extended gauge symmetry where the Standard Model electroweak group emerges as the diagonal subgroup of two SU(2) factors. The model yields a massless majoron that enables observable lepton flavor violating decays, particularly muon to electron plus majoron.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Seesaw expansion condition (mixing matrices B_{L,R} << 1) is not explicitly verified for scanned points that predict observable mu -> e J, risking the validity of the derived neutrino mass and majoron coupling.","rationale":"The reader flagged the seesaw expansion condition as the weakest assumption, but concluded it was likely satisfied. My stress test identifies that the lack of explicit numerical verification is a concrete gap: a small number of otherwise viable parameter points could in principle violate the condition, especially near r~1 where the matrix W can have entries that are not small relative to the hierarchy (though the mixing P remains sqrt(m_nu/mu)-suppressed as long as the seesaw formula holds). The test is straightforward and would increase confidence. However, the paper's hierarchy (v_eta ≪ v_H ≪ v_Phi, mu ~ TeV) ensures the condition generically holds; I do not expect it to fail for any reasonable choice. Thus the verdict remains unchanged, but the test is a valuable check the reader could request.","tokens_in":33015,"tokens_out":4135,"duration_ms":35931,"concrete_test":"For each point in Fig. 4a that yields BR(mu→e J) > 10^{-12}, evaluate the mixing matrix P = M_D^* (M^{-1})^dagger (Eq. 63). If max_{i,j} |P_{ij}| > 0.1 for any point, the seesaw expansion is unreliable for that point and the corresponding BR prediction should be discarded. Replot Fig. 4 after removing such points.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that mu->e J can be observable relies on the seesaw expansion (Appendix B, Eqs. 115–116) to derive both the light neutrino mass matrix M_nu = -W W^T (Eq. 64) and the one-loop majoron coupling (Eq. 82). The expansion assumes mixing matrices B_{L,R} ≪ 1. While the paper argues that the mixing is suppressed by sqrt(m_nu/mu) (Eq. 69), this estimate itself follows from the expansion and does not constitute an independent check. The numerical scan (Fig. 4) explores regions where cancellations (e.g., r ≃ 1) alter the hierarchy; the paper does not report whether the condition ||P|| ≪ 1 holds for the points that yield observable branching ratios. If any such point violates the condition, the factorized form and the majoron coupling derived from it are invalid, undermining the central phenomenological conclusion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper studies neutrino mass generation in an SU(3)_c × SU(2)_1 × SU(2)_2 × U(1)_Y model in which the SM SU(2)_L is the diagonal subgroup of the two SU(2) factors and a global U(1)_{B-L} is spontaneously broken by a scalar bidoublet VEV. The authors derive the tree-level spectrum, including a massless majoron, and show that the light neutrino mass matrix takes the factorized hybrid form M_ν = −W W^T, combining type-III, inverse, and linear seesaw contributions in a ratio controlled by one parameter r. They then study electroweak precision constraints, the nearly degenerate W′/Z′ bosons, and lepton-flavor-violating decays ℓ_α → ℓ_β J. The central phenomenological claim is that μ → e J can be within reach of COMET and Mu3e, and that an observation would imply new charged fermions close to the TeV scale currently probed by colliders. The technical derivation is organized around small-parameter expansions in ε = v_H/v_Φ and δ, with block-diagonalization and scalar mass matrices relegated to appendices.","tokens_in":33436,"tokens_out":9097,"duration_ms":101899,"significance":"If correct, the paper is a valuable contribution: it connects a deconstructed SU(2)_1 × SU(2)_2 electroweak structure with a spontaneously broken global U(1)_{B-L} and shows that the resulting majoron can provide the most distinctive observable signal of the model. The analytic work is careful: expansions are stated, the seesaw block-diagonalization is spelled out in Appendix B, and the scalar and gauge sectors are treated consistently. The paper also avoids circular reasoning: parameter space is scanned subject to neutrino oscillation data and external constraints, not fitted to claim a phenomenological success. The degeneracy of W′ and Z′ and the correlation between μ → e J observability and the mass scale of the new charged leptons are sharp, falsifiable predictions. The main caveats are that the seesaw consistency conditions are imposed but not reported numerically for the benchmark points, and that the lower bound on μ is imported from canonical type-III searches despite the model-specific structure of the charged fermions.","major_comments":[{"comment":"The central predictions rely on the seesaw expansion: Eq. (64) for M_ν and Eq. (82) for the one-loop majoron coupling are both derived under the assumption that the light-heavy mixing matrices satisfy (B_{L,R})_{ij} ≪ 1, as stated in Eqs. (115)–(116). The scan leading to Fig. 4 is said to ‘impose the seesaw conditions’, but no quantitative information about the maximal entries of B_L, B_R, or P is reported for the points that predict observable μ → e J. The estimate P ∼ sqrt(m_ν/μ) in Eq. (69) is itself obtained within the expansion and is not an independent check. I therefore ask the authors to report, for the scanned points, the maximal norm or largest entries of the mixing matrices, or to plot where the points predicting BR(μ → e J) near the COMET sensitivity sit relative to the condition P ≪ 1. This is load-bearing because both the factorized neutrino mass formula and the majoron coupling would be invalid if any of those points violated the expansion.","section":"§4.3, Fig. 4, and Appendix B","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The quantitative conclusion that an observable μ → e J signal requires the new charged leptons to lie near the currently explored collider range uses the bound μ ≳ 0.1–1 TeV, taken from conventional type-III seesaw searches. The authors acknowledge in footnote 2 and in the final outlook that this bound is approximate because their charged fermion sector has a more complex structure. Since the same bound enters Eq. (71), the estimate v_Φ ≳ (y_N/Y) 10^{5–6} TeV, the upper bound in Eq. (94), and the exclusion estimates in Eqs. (96)–(98), the main falsifiable claim would be more solid if the paper either provided a model-specific recast of the ATLAS type-III search or presented the conclusions as conditional on the approximate limit. At minimum, a short sensitivity study showing how the central relation shifts when μ is varied within a plausible range would make the robustness of the claim transparent.","section":"§3.3.2 and §4.3, Eqs. (71), (94)–(98)","authors_acknowledge":"A dedicated analysis of the collider limits on the type-III charged fermions, taking into account the more involved structure of this model, would also be required to turn our approximate bound onµinto a precise one.","authors_acknowledge_p":0.89},{"comment":"The one-loop majoron coupling is imported from model C2b of Ref. [41] with the replacements y_χ → y_χ/√2 and v_{φ_s} → √2 v_Φ. This is acceptable because the reference is published and the mapping is stated. However, the phenomenological claim in Fig. 4 depends sensitively on the matrix Θ in Eq. (83) and on the sign/relative size of the y_N and y_χ terms. The text should state explicitly what values of Θ are used in the scan, since the plotted points are generated with M_η = 10^{−1}/√2 v_Φ to make Θ ≈ 1. Without this information the reader cannot reproduce the figures or check whether the cancellation regions in the 2y_N² − y_χ² Θ term are probed.","section":"§4.3, Eq. (82)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The majoron profile in the main text has third component δ/2, while the more detailed expression in Eq. (146) gives v_η²/(2 v_Φ²) (1 − M_η²/(λ′_Φ v_Φ²)). The two expressions should be reconciled or the approximation level at which the simplified profile is valid should be stated.","section":"§3.2.1, Eq. (43), and Appendix C.2, Eq. (146)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The equalities M²_{P1} = λ′_Φ v_Φ⁴/v_η² δ = M_η² and the analogous relation for S_4 are compact but potentially confusing; a short sentence explaining why δ = M_η² v_η²/(λ′_Φ v_Φ⁴) converts the first equality into the second would help the reader.","section":"§3.2.1, Eqs. (42) and (47)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The scan generating Fig. 4 is described only qualitatively. Please state the number of scanned points, the ranges of y_N, y_χ, Y, v_Φ, and v_η, and how the best-fit neutrino oscillation parameters are imposed, so that the coverage of the plane shown in Fig. 4 can be assessed.","section":"§4.3, Fig. 4","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The inequality in Eq. (95) is only meaningful when the combination 2y_N² − y_χ² Θ is positive. The text should state that the scan enforces this condition, or discuss the regions where the effective coefficient changes sign.","section":"§4.3, Eq. (95)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The reader's circularity concern does not land: the model is not fitted to a claimed success, and the scan imposes external constraints. The stress-test concern is partially mitigated by the statement that the scan 'imposes the seesaw conditions', but the absence of numerical reporting for the mixing matrices leaves a genuine verification gap in the central prediction. The approximate μ bound is acknowledged by the authors but is nevertheless load-bearing for the main phenomenological conclusion. I see no reason to reject, and the requested checks are within the scope of a revision. The paper is likely to be a useful contribution once the seesaw consistency and the μ-bound sensitivity are reported explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Two things to know. The construction is genuine: the 3221 gauge structure with a bidoublet VEV breaking both the extra SU(2) and a global U(1)_{B-L}, and producing a majoron that is mostly SM-singlet, is new in this combination. The neutrino mass analysis has a clean core. The factorization M_nu = -W W^T, with the three contributions in the ratio 1 : 2r : r^2, is good algebra that actually works: it shows why the linear term never dominates, and why the low-scale promise of the inverse seesaw does not transfer here, since mu sets both the inverse scale and the charged triplet masses. That last point is genuinely useful.\n\nThe most distinctive claim is the correlation: an observable mu -> e J forces v_Phi around 10^4-10^6 TeV with mu near a TeV, so the new charged leptons must sit within collider reach. I think that argument survives reading, and it is also where the soft spots cluster.\n\nStrengths: the expansions in delta and epsilon are stated and used consistently; the block-diagonalization in Appendix B is standard and applied correctly; Eq. (82) is imported from the authors' own published JHEP work with the convention mapping stated, which is legitimate. The paper is unusually honest, flagging the approximate collider bounds, the uncomputed N_eff effect, and the r = -1 cancellation that would improve the model but is not symmetry-enforced.\n\nOn the stress-test concern about unverified seesaw conditions in the scan: it does not land as a load-bearing flaw. The paper states that the scan imposes the seesaw conditions, and the algebraic relation between mixing and light mass puts ||P|| ~ sqrt(m_nu/mu) ~ 10^-7 for the observable points, forcing the conditions to hold unless one engineers the r = -1 pathology the authors explicitly exclude. The residue is a reporting gap: they should print the norms of P and B_L for their benchmark points. Minor.\n\nThe real softness is different, and the authors know it. The one visible signature, mu -> e J, is a one-loop majoron effect largely inherited from the singlet model of Ref. [41]; the gauge structure is pushed to 10^4-10^6 TeV by the neutrino mass constraint, so W' and Z' are out of reach and the approximate type-III bound on mu carries more weight than a factor-of-few estimate should. The model's appeal is therefore as a well-posed existence proof with a sharp falsifiability relation, not as a directly testable framework. Minor: one supporting argument, the domain-wall point, cites a preprint rather than published work.\n\nWho it is for: model builders working on seesaw mechanisms and majoron phenomenology. It deserves a serious referee; the requested revision should document the seesaw-mixing check, deliver the deferred N_eff estimate, and ideally treat the charged-lepton collider bounds properly. I would read it carefully and push on those three.","headline":"Genuine 3221 construction with a clean hybrid-seesaw factorization; the mu->e J correlation argument holds up, but the model's own neutrino constraints push its gauge structure out of experimental reach.","tokens_in":34003,"tokens_out":8289,"would_cite":true,"duration_ms":92646,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["12.60.Cn","14.60.Pq","13.35.Bv"],"model":"deepseek/deepseek-v4-flash","headline":"Light neutrino masses come from a hybrid of type-III, inverse, and linear seesaw mechanisms, and the massless majoron left behind makes the decay of a muon to an electron plus a majoron the model's most testable signature.","keywords":["hybrid seesaw","majoron","neutrino mass generation","lepton flavor violation","mu to e J decay","type-III seesaw","inverse seesaw","linear seesaw"],"falsifier":"Evaluate Eqs. (115)–(116) at the Fig. 4 scan points that reproduce the neutrino oscillation data and give $\\mathrm{BR}(\\mu\\to e J)$ within COMET reach: any entry of $B_L$ or $B_R$ of order one there would invalidate the seesaw expansion on which $M_\\nu = -W W^T$ and the derived bounds rest. Experimentally, a null search for $\\mu\\to e J$ at COMET's projected sensitivity $\\mathrm{BR} < 4.6\\times10^{-9}$ would close the order-one-Yukawa window $v_\\Phi \\lesssim 10^{5\\text{--}6}$ TeV that the paper identifies as the observable region.","tokens_in":34546,"feed_emoji":"⚛️","tokens_out":20744,"duration_ms":153576,"temperature":0.7,"pith_summary":"This paper proposes a specific extension of the Standard Model gauge group — SU(3)_c × SU(2)_1 × SU(2)_2 × U(1)_Y, in which the observed weak force $SU(2)_L$ emerges as the diagonal subgroup of the two $SU(2)$ factors — and shows that light neutrino masses are then produced at tree level by three classic seesaw mechanisms at once: type-III (heavy fermion triplet), inverse (small lepton-number-violating mass term), and linear (mixing with a heavy singlet). The three contributions are not independent: expanding the factorized mass matrix $M_\\nu = -W W^T$, they appear in the fixed ratio $1:2r:r^2$ controlled by a single parameter, so one combination of couplings and scales decides which mechanism dominates. Spontaneous breaking of the global $U(1)_{B-L}$ symmetry leaves a massless Nambu-Goldstone boson, the majoron, which is mostly a Standard Model singlet but couples to charged leptons through a dimension-five operator. The paper's central phenomenological claim is that the decay $\\mu \\to e J$ this induces is within reach of the COMET and Mu3e experiments, and that observing it would place the model's new charged leptons near the TeV scale, making the whole neutrino-mass machinery testable through one rare decay.","feed_headline":"One rare muon decay could lay bare the neutrino mass mechanism","feed_subtitle":"If seen, mu to e J would put the new heavy charged leptons at the TeV scale, within collider reach.","key_machinery":"The argument runs on three pieces. First, the fermion bidoublet $\\chi$, which splits under the low-energy $SU(2)_L$ into a singlet $\\chi_s$ and a triplet $\\chi_t$: the singlet carries the inverse and linear seesaw terms (via $y_\\chi L \\tilde\\eta \\chi_s$ and $Y N^c \\phi_s^\\dagger \\chi_s$), the triplet carries the type-III term, and the same matrix $\\mu$ controls both sectors, which is why the mechanisms cannot be tuned independently. Second, the factorization $M_\\nu = -W W^T$ itself, which forces the $1:2r:r^2$ hierarchy among the three contributions and makes the number of massive neutrinos equal to the number of heavy families. Third, the majoron $J$, the exactly massless Nambu-Goldstone boson of the spontaneously broken $U(1)_{B-L}$; its derivative couplings generate the dimension-five operator $(c_{ij}/v_\\Phi)\\,\\partial_\\mu J\\, \\bar L_i \\gamma^\\mu L_j$, which on-shell becomes $J\\, \\bar\\ell_i (c_{ij} M_\\ell/v_\\Phi) \\ell_j$ and is the source of the $\\mu \\to e J$ rate. A supporting element is the tiny bidoublet VEV splitting $\\delta \\simeq M_\\eta^2 v_\\eta^2/(\\lambda'_\\Phi v_\\Phi^4)$, fixed by the vacuum to about $10^{-9}$ at reference parameters, which preserves an approximate custodial $SU(2)$ so that the $\\rho$-parameter correction $\\rho-1 \\simeq 2\\delta^2/\\epsilon^2$ is automatically tiny and the heavy $W'$ and $Z'$ bosons stay nearly degenerate.","core_discovery":"On the paper's own terms: in the 3221 model with $SU(2)_L$ as the diagonal subgroup of $SU(2)_1 \\times SU(2)_2$ and the global $U(1)_{B-L}$ broken by the bidoublet VEV, the light neutrino mass matrix takes the factorized form $M_\\nu = -W W^T$ with $W = \\frac{v_\\eta}{2} y_\\chi \\mu^{-1/2} + \\frac{2 v_H}{v_z+v_w} y_N (Y^T)^{-1} \\mu^{1/2}$, and expanding $W W^T$ produces the type-III, linear, and inverse contributions in the ratio $1:2r:r^2$, with the linear term never dominant on its own. Since $\\mathrm{rank}(M_\\nu) = n_F$, the minimal choice of two heavy lepton families gives two massive neutrinos and one massless, matching the two measured mass splittings. The same mass parameter $\\mu$ that sets the inverse-seesaw scale also fixes the type-III triplet mass, and the paper shows that the light–heavy mixing is then suppressed by the light neutrino mass itself, removing the large-mixing feature normally associated with low-scale seesaws; reproducing neutrino data requires either very small Yukawa couplings or a high breaking scale $v_\\Phi \\gtrsim (y_N/Y)\\,10^{5\\text{--}6}$ TeV for $\\mu \\gtrsim 0.1\\text{--}1$ TeV. The majoron mediates flavor-violating decays $\\ell_\\alpha \\to \\ell_\\beta J$ with couplings of order $M_\\ell/v_\\Phi$, and the paper finds that the scale window in which $\\mu\\to e J$ is observable with order-one couplings ($v_\\Phi \\lesssim 10^{5\\text{--}6}$ TeV) overlaps the window required by neutrino masses, so a signal would pin the new charged leptons near $\\mu\\sim 0.1\\text{--}1$ TeV, within reach of upcoming searches.","pith_inferences":["The coincidence that the neutrino-mass lower bound on $v_\\Phi$ roughly matches the upper bound for an observable $\\mu\\to e J$ rate is the paper's key testable link; a natural hardening step the paper does not report would be to tabulate the actual entries of $B_L$ and $B_R$ (Appendix B, Eqs. 115–116) at every surviving scan point rather than only imposing the seesaw condition.","A null result at COMET would not falsify the model — the pure inverse-seesaw limit $y_\\chi\\to 0$ and the $r\\gg 1$ regime would remain viable — but it would close what the paper calls its most promising signature and push the BSM structure beyond $10^{5\\text{--}6}$ TeV, out of reach of any near-term probe.","The structural feature that one mass parameter $\\mu$ sets both the type-III triplet mass and the inverse-seesaw breaking strength looks like a general model-building lesson: in any hybrid seesaw where a single operator carries both mechanisms, the low-scale enhancement of light–heavy mixing is lost, and flavor-violating decays rather than collider fermion production become the discovery channel.","The paper identifies multilepton-plus-missing-energy final states from $W'/Z'$ decays into $\\chi$ pairs but does not quantify their search reach; a dedicated analysis could turn the nearly degenerate $W'/Z'$ pair into a sharper test of the breaking pattern than Drell-Yan searches."],"forward_implications":["The three seesaw contributions stand in the ratio $1:2r:r^2$, so the model interpolates between a type-III-dominated regime ($r\\ll 1$) and an inverse-dominated one ($r\\gg 1$) with a single knob, and the linear term never dominates alone.","With the minimal choice $n_F = 2$ of heavy families the lightest neutrino is massless, and the scan points reproduce the best-fit normal-ordering parameters while respecting $\\sum m_\\nu < 0.16$ eV.","Electroweak precision is safe almost automatically: the tree-level $\\rho$ correction is of order $10^{-16}$ at reference values, and the only severe constraint is $Z$–$Z'$ mixing, which bounds $v_\\Phi \\gtrsim 3$ TeV for $g_1 = g_2$.","For order-one Yukawas, an observed $\\mu \\to e J$ at COMET sensitivity ($\\mathrm{BR} \\lesssim 4.6\\times10^{-9}$) would require the new charged leptons to lie at $\\mu \\lesssim 1$ TeV, accessible to near-future colliders; only a narrow region with very small $y_N$ and $v_\\eta$ escapes this conclusion.","The invisible Higgs decay $h\\to JJ$ is predicted to be unobservably small (branching ratio at most about $10^{-3}$, typically $10^{-6}$–$10^{-8}$), because the majoron scale is tied to gauge-symmetry breaking rather than being a free lepton-number-breaking scale."],"supporting_citations":[{"why":"Supplies the type-III seesaw mechanism, realized here through the triplet component of the fermion bidoublet.","marker":"[23]"},{"why":"Supplies the inverse seesaw mechanism, whose lepton-number-violating scale enters the second term of the factorized matrix $W$.","marker":"[24]"},{"why":"Establishes the linear seesaw mechanism, the third ingredient of the $1:2r:r^2$ hierarchy.","marker":"[25, 26]"},{"why":"Provides the LHC lower bound on type-III charged-lepton masses that anchors the estimate of the observable window.","marker":"[35]"},{"why":"Supplies the one-loop majoron–charged-lepton coupling formula from which the paper computes $\\mathrm{BR}(\\mu\\to e J)$.","marker":"[40]"},{"why":"Defines the simplified model whose singlet-sector majoron coupling the paper adapts to the present construction with the stated projection factors.","marker":"[41]"},{"why":"Gives the current experimental limit $\\mathrm{BR}(\\mu\\to e J) \\lesssim 10^{-5}$ used to normalize the LFV rates.","marker":"[49]"},{"why":"Provides the Mu3e projected sensitivities that define the near-future reach for $\\mu\\to e J$.","marker":"[50, 51]"},{"why":"Provides the COMET projected sensitivity ($\\mathrm{BR}\\lesssim 4.6\\times10^{-9}$) that sets the upper bound on $v_\\Phi$ for observable LFV.","marker":"[52, 53]"}],"fun_headline_variants":["Muon decay to majoron would pin down neutrino masses","Majoron in muon decay ties neutrino masses to TeV-scale leptons","One rare muon decay could expose the hybrid seesaw","Mu-to-e-majoron signal would place new leptons within reach","Hybrid seesaw leaves its mark in muon decays to majorons"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The whole derivation is a seesaw expansion: it assumes, as the paper states in Appendix B (Eqs. 115–116), that the light–heavy mixing matrices $B_L$ and $B_R$ are much smaller than one — a hierarchy among the Yukawa couplings, the VEVs, and the scales $\\mu$ and $v_\\Phi$ that the paper says it imposes in its numerical scan but whose actual values at the surviving points are not reported, and on which the factorized formula $M_\\nu = -W W^T$ and every derived bound rest.","fun_headline_variants_meta":{"raw":{"variants":["Muon decay to majoron would pin down neutrino masses","Majoron in muon decay ties neutrino masses to TeV-scale leptons","One rare muon decay could expose the hybrid seesaw","Mu-to-e-majoron signal would place new leptons within reach","Hybrid seesaw leaves its mark in muon decays to majorons"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"cost":0.0001761032,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001761032,"upstream_inference_prompt_cost":3.37792e-05,"upstream_inference_completions_cost":0.000142324},"total_tokens":2070,"prompt_tokens":1288,"completion_tokens":782,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1146,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":704}},"tokens_in":1288,"tokens_out":782,"cost_source":"openrouter_usage","duration_ms":19299,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":704,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-28T16:48:06.557934+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Evaluate Eqs. (115)–(116) at the Fig. 4 scan points that reproduce the neutrino oscillation data and give $\\mathrm{BR}(\\mu\\to e J)$ within COMET reach: any entry of $B_L$ or $B_R$ of order one there would invalidate the seesaw expansion on which $M_\\nu = -W W^T$ and the derived bounds rest. Experimentally, a null search for $\\mu\\to e J$ at COMET's projected sensitivity $\\mathrm{BR} < 4.6\\times10^{-9}$ would close the order-one-Yukawa window $v_\\Phi \\lesssim 10^{5\\text{--}6}$ TeV that the paper identifies as the observable region.","supporting_citations":[{"cited_title":"Seesaw Neutrino Masses Induced by a Triplet of Leptons,","cited_arxiv_id":null,"evidence_quote":"Supplies the type-III seesaw mechanism, realized here through the triplet component of the fermion bidoublet."},{"cited_title":"Neutrino Mass and Baryon Number Nonconservation in Superstring Models,","cited_arxiv_id":null,"evidence_quote":"Supplies the inverse seesaw mechanism, whose lepton-number-violating scale enters the second term of the factorized matrix $W$."},{"cited_title":"The majoron coupling to charged leptons","cited_arxiv_id":"2311.10145","evidence_quote":"Supplies the one-loop majoron–charged-lepton coupling formula from which the paper computes $\\mathrm{BR}(\\mu\\to e J)$."},{"cited_title":"The Type-I Seesaw family,","cited_arxiv_id":null,"evidence_quote":"Defines the simplified model whose singlet-sector majoron coupling the paper adapts to the present construction with the stated projection factors."},{"cited_title":"Majoron emission in muon and tau decays revisited","cited_arxiv_id":"0902.0525","evidence_quote":"Gives the current experimental limit $\\mathrm{BR}(\\mu\\to e J) \\lesssim 10^{-5}$ used to normalize the LFV rates."}],"review_version":1}