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Modular Δ(96) Littlest Seesaw yields 35 viable lepton-mixing patterns with new fixed PMNS columns beyond TM₁, all sharply testable by JUNO, DUNE and T2HK.

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2026-07-10 16:25 UTC pith:6AWEUXXV

load-bearing objection Solid first catalogue of modular Δ(96) Littlest Seesaw patterns with real new fixed PMNS columns; the multi-modulus premise is the main caveat, not a hidden flaw. the 2 major comments →

arxiv 2607.07865 v1 pith:6AWEUXXV submitted 2026-07-08 hep-ph

Lepton mixing from the Delta(96) Modular Littlest Seesaw

Hai-Zhi Hao , Li-Na Yan , Xiang-Gan Liu , Cai-Chang Li This is my paper
classification hep-ph PACS 14.60.Pq11.30.Hv12.60.Jv
keywords modular symmetryΔ(96)Littlest Seesawvector-valued modular formsPMNS matrixneutrino mixingfixed pointsCSD(n)
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether the finite modular group Δ(96) can generate realistic neutrino masses and lepton mixing when the three sectors of the Minimal Seesaw are controlled by modular forms evaluated at fixed points. The authors construct all relevant vector-valued modular forms, list the inequivalent residual-symmetry fixed points, and scan every assignment of residual symmetries in the charged-lepton, atmospheric and solar sectors. They find 35 inequivalent breaking patterns that survive current oscillation data (21 normal-ordering, 14 inverted-ordering). The resulting Dirac mass matrices are more general than the usual constrained sequential-dominance textures, produce previously unseen fixed columns of the PMNS matrix, and fix the three light-neutrino masses, the three mixing angles and both CP phases to narrow numerical windows controlled by only three real parameters. Those windows lie within reach of the next round of precision oscillation experiments, so the catalogue can be decisively confirmed or excluded in the coming decade.

Core claim

An exhaustive, model-independent scan of residual symmetries and low-weight triplet vector-valued modular forms of modular Δ(96) produces exactly 35 phenomenologically viable Modular Littlest Seesaw patterns (21 normal ordering, 14 inverted ordering). Their Dirac neutrino mass matrices go beyond conventional CSD(n) alignments, generate new fixed PMNS columns, and therefore impose novel sum rules among the mixing angles and the Dirac phase that are narrower than those of the classic TM₁ paradigm.

What carries the argument

Vector-valued modular forms (VVMFs) of modular Δ(96) evaluated at the 92 inequivalent fixed points: once the three moduli are fixed at residual-symmetry points, the VVMFs supply algebraic vacuum alignments for the two columns of the Dirac matrix, fixing one column of the PMNS matrix and leaving only three free real parameters.

Load-bearing premise

The whole construction assumes that three independent moduli can be stabilised at three different modular fixed points without large cross-sector couplings that would spoil the residual symmetries.

What would settle it

A high-precision measurement of sin² heta₁₂ by JUNO that falls outside the narrow intervals predicted by every remaining viable pattern (especially the eight TM₁ patterns), or a combined DUNE/T2HK determination of sin² heta₂₃ and δ_CP that lies outside all 35 predicted bands.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper presents the first comprehensive, model-independent study of Modular Littlest Seesaw models based on the finite modular group Δ(96). It constructs the vector-valued modular forms (VVMFs) for all irreducible representations of modular Δ(96), classifies the 92 inequivalent symmetry-preserving fixed points via double cosets, and derives the alignments of the lowest- and next-to-lowest-weight triplet VVMFs at those points. An exhaustive scan over residual symmetries in the charged-lepton, atmospheric, and solar sectors (with three independent moduli τℓ, τatm, τsol stabilized at fixed points) identifies 35 phenomenologically viable inequivalent breaking patterns (21 NO, 14 IO). The resulting Dirac neutrino mass matrices go beyond conventional CSD(n), producing new fixed PMNS columns and sum rules beyond TM1. With only three continuous parameters (ma, r, η), the models yield narrow, experimentally testable ranges for neutrino masses, mixing angles, and CP phases.

Significance. If the multi-modulus fixed-point assumption can be realized in a UV completion, this work substantially enlarges the Modular Littlest Seesaw landscape beyond S4/A5 constructions. The algebraic rigidity of fixed-point VVMF alignments (Eqs. 3.11–3.14, 4.7–4.16) produces a finite set of exact directions and new fixed PMNS columns (Table 6, Appendix C) that are independent of the fitted parameters. The exhaustive scan over 89662 patterns, the NuFIT-based χ² analysis, the explicit sum rules (Appendix D), and the confrontation with JUNO/DUNE/T2HK sensitivities constitute a concrete, falsifiable catalogue. The group-theory and VVMF appendices (A–B) are a reusable resource for future modular Δ(96) model building.

major comments (2)
  1. The entire catalogue of 35 patterns rests on the multi-modulus premise that three independent moduli can be stabilized at distinct fixed points preserving different residual symmetries (§4, Eq. (4.2); Table 2). A single common modulus would force the residual symmetries to be conjugate under the same modular transformation, collapsing most of the new fixed columns in Table 6 and Appendix C. The paper itself flags this as an open UV question (Conclusion; refs. [42,43,45,54]). The claim that the 35 patterns are “phenomenologically viable Modular Littlest Seesaw models” should be qualified more carefully as a classification under the multi-modulus assumption, with a clearer statement of the conditions under which the residual symmetries survive.
  2. The restriction to lowest- and next-to-lowest-weight triplet VVMFs is presented as a minimality choice (§3, Table 1). While reasonable for predictivity, the paper does not quantify how many additional viable patterns would appear if higher-weight triplets (or the sextet) were admitted. A short discussion of the completeness of the weight cut, or an argument that higher weights generically reintroduce continuous freedom or lose residual-symmetry protection, would strengthen the claim that the 35-pattern list is exhaustive within the modular framework.
minor comments (5)
  1. Table 6 and Appendix C: the analytical fixed-column expressions are dense; a short numerical cross-check (e.g., that the listed vectors are unit-normalized and orthogonal to the two Yukawa directions) would help the reader verify the algebra.
  2. Figures 2–3: the symbol coding for the 21 NO / 14 IO patterns is hard to parse without a legend that maps each marker to the case labels N1–N21 / I1–I14 of Table 6.
  3. Eq. (4.16) and Appendix D: the sum-rule formulae for the non-TM1 patterns are lengthy; stating the numerical values of the fixed-column components used in each derivation would improve reproducibility.
  4. A few minor typos appear (e.g., “dirreducible” for “d-irreducible” in §2; occasional missing spaces around “τ”). A light copy-edit pass would clean these.
  5. The relation of the Δ(96) modular construction to the earlier tri-direct CP Δ(96) analysis of Ref. [27] is noted in §5.1; a short table comparing the vacuum alignments that are shared versus those that are modular-specific would clarify the complementarity claimed in the text.

Circularity Check

1 steps flagged

No significant circularity: fixed PMNS columns and sum rules are algebraic consequences of modular covariance at fixed points; continuous parameters are fitted in the standard phenomenological way.

specific steps
  1. fitted input called prediction [Sec. 5.1, Tables 7–9; abstract claim of ‘narrow ranges’]
    "The viable models are highly predictive, giving narrow ranges for neutrino masses, mixing parameters and CP phases… For any given set of input parameters, the model predicts the neutrino masses, mixing parameters, and the corresponding χ² value… the three leptonic mixing angles, the two CP-violating phases, the neutrino masses, and the effective Majorana mass… are all confined to narrow regions."

    ma, r, η are fitted by minimizing χ² against the same six oscillation observables whose best-fit values and 3σ ranges are then reported as model ‘predictions.’ The narrowness of those ranges is partly forced by the requirement that all fitted observables already lie inside the experimental 3σ intervals. This is only a mild, standard phenomenological presentation, not a definitional loop: the fixed PMNS columns and sum rules (Eq. 4.16, App. C–D) remain independent algebraic content that genuinely constrain the fit.

full rationale

The derivation chain is self-contained and non-circular under the stated criteria. VVMFs for all irreps of modular Δ(96) are constructed from MLDEs and tensor products (Sec. 3, App. B); inequivalent fixed points and residual stabilizers are classified by double cosets (Eqs. 3.11–3.15, Table 2); vacuum alignments of lowest- and next-to-lowest-weight triplets are eigenvectors of residual generators (Table 3). The null vector ˆv_fix = vatm × vsol / |…| (Eq. 4.7) and the fixed PMNS column Pℓ U†ℓ ˆv_fix (Eqs. 4.13–4.16) are therefore fixed algebraically by modular covariance once the residual symmetries and VVMF weights are chosen—independent of the continuous parameters ma, r, η. Those three parameters are then scanned against NuFIT 6.1 via a global χ² (Sec. 5.1); the resulting best-fit values and 3σ ranges for mixing angles, phases, masses and mee are the model’s allowed region under that fit, which is ordinary model-building practice rather than a definitional reduction. Self-citations to the multi-modulus Modular Littlest Seesaw setup ([42,43] and related works) supply the framework assumption, not a uniqueness theorem that forbids alternatives or forces the 35-pattern catalogue. The multi-modulus premise is a genuine UV assumption (as the paper itself notes in the Conclusion), but that is a correctness/realizability concern, not circularity. No step reduces a claimed first-principles result to its own input by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central catalogue rests on standard modular-form mathematics, the type-I seesaw with two RHNs, the multi-modulus fixed-point ansatz, and three continuous parameters fitted per pattern. No new particles beyond the standard modular LS field content are introduced. The load-bearing modeling choice is independent stabilization of three moduli at fixed points; the free parameters are the usual seesaw scale, ratio, and relative phase.

free parameters (3)
  • ma (overall light-neutrino mass scale) = O(few–12) meV depending on pattern (Table 7)
    Positive real parameter in the rank-2 light-neutrino mass matrix (Eq. 4.5–4.6); fitted per pattern to Δm² and mixing data via χ² minimization.
  • r (solar/atmospheric Yukawa–mass ratio) = O(0.15–4.5) depending on pattern (Table 7)
    Positive real ratio |y_sol² Matm / y_atm² Msol| (Eq. 4.6); fitted per pattern.
  • η (relative CP phase) = O(0.1–1.9)π depending on pattern (Table 7)
    Physical phase Arg(y_sol² Matm / y_atm² Msol) ∈ [0, 2π) (Eq. 4.6); fitted per pattern.
axioms (6)
  • domain assumption Yukawa couplings are vector-valued modular forms transforming under a finite modular group Gf ≅ Γ/ker(ρ) with the automorphy factor (cτ+d)^k.
    Standard modular flavor symmetry framework (Feruglio 2019; Liu–Ding 2022); invoked throughout §2–3.
  • domain assumption Light neutrino masses arise from the type-I seesaw with exactly two right-handed neutrinos (rank-2 Mν, one massless neutrino).
    Minimal/Littlest Seesaw setup (§4, Eq. 4.5); excludes three-RHN or other mass mechanisms.
  • ad hoc to paper Three independent moduli τℓ, τatm, τsol can be stabilized at distinct modular fixed points preserving different residual symmetries.
    Multi-modulus Modular Littlest Seesaw ansatz (§4; motivated by multiple modular symmetries or factorizable tori). Not derived from a UV completion in this work.
  • ad hoc to paper Only lowest- and next-to-lowest-weight triplet VVMFs are used for Yatm and Ysol.
    Minimality cut stated in §3; higher-weight forms are excluded without dynamical justification.
  • domain assumption Kähler potential is the minimal modular-invariant form, yielding canonical kinetic terms after τ acquires a VEV.
    Standard assumption (Eq. 2.16); non-minimal Kähler terms could alter mass matrices.
  • domain assumption NuFIT 6.1 3σ intervals define phenomenological viability.
    External experimental input (Table 5); viability is data-dependent.

pith-pipeline@v1.1.0-grok45 · 63153 in / 3603 out tokens · 40546 ms · 2026-07-10T16:25:42.592597+00:00 · methodology

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read the original abstract

We perform the first comprehensive and model independent study of Modular Littlest Seesaw models based on the finite modular group $\Delta(96)$. We construct the vector-valued modular forms (VVMFs) for all irreducible representations of modular $\Delta(96)$, classify the inequivalent symmetry-preserving fixed points, and derive the corresponding alignments of the low-weight and next-to-lowest-weight triplet VVMFs. These results allow an exhaustive scan over the residual symmetries in the charged lepton, atmospheric neutrino, and solar neutrino sectors. We identify 35 phenomenologically viable and inequivalent breaking patterns, including 21 with normal ordering and 14 with inverted ordering. The resulting Dirac neutrino mass matrices go beyond the conventional CSD$(n)$ structure, yielding new fixed PMNS columns and novel correlations among the lepton mixing parameters beyond the TM$_1$ paradigm. The viable models are highly predictive, giving narrow ranges for neutrino masses, mixing parameters and CP phases, and can be stringently tested by upcoming experiments such as JUNO, DUNE and T2HK.

Figures

Figures reproduced from arXiv: 2607.07865 by Cai-Chang Li, Hai-Zhi Hao, Li-Na Yan, Xiang-Gan Liu.

Figure 1
Figure 1. Figure 1: A compactified fundamental domain DK of K = ker(∆(96)) and the images of the 92 inequivalent fixed points. The tiles are generated from the fundamental domain D of Γ by the determinant￾one matrix representatives in Eq. (A.4). Pink squares denote the images of τST = ω, blue triangles denote the images of τS = i, and red circles denote the cusp images of τT = i∞. In practice, it is therefore sufficient to de… view at source ↗
Figure 2
Figure 2. Figure 2: The best fit results for 21 viable breaking patterns with NO neutrino mass spectrum, showing χ 2 minima, lepton mixing angles, and CP violation phases. Red dashed lines indicate best fit values, light blue bands show the 1σ and 3σ ranges from NuFIT 6.1 with Super-Kamiokande atmospheric data [78]. The pale green band shows the predicted 3σ range for sin2 θ12 after 6 years of JUNO data [63, 64]. The faint gr… view at source ↗
Figure 3
Figure 3. Figure 3: The best fit χ 2 for all 14 viable breaking patterns with IO neutrino mass spectrum, covering the three lepton mixing angles and CP violation phases. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Contours of δCP /π in the sin2 θ13–sin2 θ23 plane for the representative TM1 mixing pattern and five non-TM1 viable patterns with normal neutrino mass ordering and χ 2 min < 5. The red regions show the model predictions obtained from a scan over the input parameter space, subject to the requirement that all lepton mixing parameters and neutrino mass squared differences remain within their experimentally al… view at source ↗
Figure 5
Figure 5. Figure 5: Profile likelihoods for the six oscillation observables in the 15 NO lepton mixing patterns with χ 2 min ≤ 6. The tan dotted curves denote the likelihoods obtained from NuFIT [78]. In the sin2 θ12 panel, the green solid line shows the expected likelihood after six years of JUNO operation [63, 64]. The green solid line in the sin2 θ23 and δCP panels correspond to the anticipated sensitivities of DUNE [65] a… view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗

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