Pith. sign in

REVIEW 2 major objections 5 minor 43 references

The neutrino cuboid's exact alignment predicts the mass ordering, the atmospheric octant, and all three neutrino masses from two measured inputs.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 03:35 UTC pith:NPJCHQ2R

load-bearing objection Exact, honest phenomenology of a prior ansatz; the algebra is clean and the paper is admirably transparent, but the alignment at its core remains an unjustified assumption. the 2 major comments →

arxiv 2607.23100 v1 pith:NPJCHQ2R submitted 2026-07-25 hep-ph

Exact Phenomenology of the Neutrino Cuboid: Normal Mass Ordering, the Tribimaximal Limit, and Cosmological Constraints

Jianlong Lu This is my paper
classification hep-ph PACS 14.60.Pq
keywords neutrino cuboidmass–mixing alignmenttribimaximal mixingnormal mass orderingatmospheric octantabsolute neutrino massneutrino mass sum rulesreactor oscillation data
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.

The paper develops the exact consequences of the neutrino cuboid, a geometric way of writing the three neutrino masses as m1 = m0 sinξ, m2 = m0 cosξ sinζ, m3 = m0 cosξ cosζ. Its load-bearing move is identifying the two geometric angles with the measured solar and atmospheric mixing angles (ξ = θ12, ζ = θ23), an alignment assumption taken from an earlier proposal. Under that identification, two closed-form identities remove the usual freedom in the lightest neutrino mass: the absolute scale is fixed by m0² = (Δm21² + Δm31²)/(1 − 3 sin²θ12), and the atmospheric angle is predicted by sin²θ23 = [sin²θ12 + η(1 − 2 sin²θ12)]/[(1 − sin²θ12)(1 + η)]. Because the measured solar angle lies below 1/3, the first identity forces normal mass ordering and excludes inverted ordering, while the second forces θ23 below 45° (the lower atmospheric octant); the individual masses, their sum, and the effective masses for beta decay and neutrinoless double-beta decay then become predictions instead of fitted parameters — about 0.094, 0.094, and 0.106 eV, with a sum near 0.29–0.33 eV. The author shows that expansion around the tribimaximal limit is numerically unreliable for the solar splitting ratio, and that current oscillation data are only mildly uncomfortable with the predicted atmospheric angle, whereas standard (ΛCDM) cosmology strongly disfavors the predicted mass sum.

Core claim

Under the exact alignment ξ = θ12, ζ = θ23, the cuboid ceases to be a pure reparametrization and predicts the whole neutrino mass spectrum. Two identities carry the argument: m0² = (Δm21² + Δm31²)/(1 − 3 sin²θ12) fixes the absolute scale, and sin²θ23 = [sin²θ12 + η(1 − 2 sin²θ12)]/[(1 − sin²θ12)(1 + η)] predicts the atmospheric angle. Because the measured sin²θ12 is below 1/3, positivity of m0² excludes inverted ordering, and a derived identity forces θ23 into the lower octant. Representative current inputs give sin²θ23 ≈ 0.440, (m1, m2, m3) ≈ (0.094, 0.094, 0.106) eV, and Σν ≈ 0.294 eV. The first-order expansion about the tribimaximal point mispredicts η by nearly a factor of two, so exact

What carries the argument

The neutrino cuboid parametrizes the masses as m1 = m0 sinξ, m2 = m0 cosξ sinζ, m3 = m0 cosξ cosζ — a pure change of variables whose cubic point reproduces the tribimaximal angles (sin²θ12 = 1/3, sin²θ23 = 1/2). The mechanism that produces physics is the alignment hypothesis ξ = θ12, ζ = θ23. Inserting it yields two exact identities: m0² = (Δm21² + Δm31²)/(1 − 3 sin²θ12), fixing the absolute scale, and sin²θ23 = [sin²θ12 + η(1 − 2 sin²θ12)]/[(1 − sin²θ12)(1 + η)], predicting the atmospheric angle. A third identity, sin²θ23 − 1/2 = (3 sin²θ12 − 1)(1 − η)/[2(1 − sin²θ12)(1 + η)], turns the measured solar angle into the ordering and octant theorems. The residuals Rξ = ξg − θ12 and Rζ = ζg − θ23

Load-bearing premise

The load-bearing premise is that the cuboid's two geometric angles are exactly equal to the measured solar and atmospheric mixing angles (ξ = θ12, ζ = θ23) — an identification assumed outright rather than derived, and one the cuboid parametrization itself does not require. If the geometric angles are not literally the mixing angles, whether because of charged-lepton corrections, θ13 effects, or a different underlying flavor-symmetry relation, every ordering, octant, and absol

What would settle it

A definitive experimental demonstration of inverted neutrino mass ordering would falsify the ansatz outright, because the measured sin²θ12 < 1/3 in the paper's Eq. (12) requires Δm21² + Δm31² > 0. Short of that, an oscillation measurement establishing sin²θ23 > 1/2 at high significance contradicts the octant identity, and a cosmology-independent laboratory bound on Σν or mβ below the predicted values (roughly 0.3 eV and 0.1 eV) would exclude the spectrum. The cleanest single check: measure sin²θ23 precisely and compare it with the value computed from sin²θ12 and the mass-splitting ratio alone.

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

If this is right

  • Inverted mass ordering is excluded outright: with the measured sin²θ12 < 1/3, positivity of m0² demands Δm21² + Δm31² > 0, a condition only the normal ordering satisfies.
  • θ23 must lie in the lower octant (sin²θ23 < 1/2), so a definitive upper-octant measurement would rule the ansatz out.
  • The absolute mass scale stops being free: the model outputs concrete numbers — masses near 0.09–0.12 eV, a mass sum near 0.29–0.33 eV, mβ near 0.09–0.11 eV, and a nonzero neutrinoless-double-beta-decay envelope — that upcoming laboratory and cosmological measurements can directly confront.
  • First-order expansions around the tribimaximal point mispredict the solar-to-atmospheric splitting ratio by nearly a factor of two (0.056 versus 0.029), so only the exact relations are reliable for precision tests.
  • Standard (ΛCDM) cosmology, with mass-sum limits below roughly 0.12 eV in recent analyses, strongly disfavors the predicted spectrum, while weaker limits in extended cosmological models leave it viable — cosmology, not oscillation data, currently applies the decisive pressure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exact alignment is a kinematic hypothesis with no stated origin: if a deeper flavor symmetry is ever built to produce it, that symmetry must also explain the reactor angle θ13, which the cuboid leaves entirely free — so the natural next step the author leaves implicit is a dynamical mechanism for both the alignment and θ13.
  • The residual-plane formulation suggests a presentation strategy for future global fits: plot the (Rξ, Rζ) trajectory swept out by the lightest mass in each ordering, and treat 'exact alignment' as a measurable distance from the origin rather than a binary model choice.
  • The predicted near-degenerate spectrum — two states separated by less than a milli-eV near 0.094 eV — sits exactly where next-generation laboratory mass searches are heading; a laboratory bound below roughly 0.1 eV would test the ansatz without any cosmological modelling.
  • Because the predicted mass sum diverges as sin²θ12 approaches 1/3 from below, modest future improvements in the solar-angle measurement carry disproportionately large leverage: a σx ≈ 0.001 input would already pin the atmospheric prediction to about 0.1°, making the correlation a near-term target for reactor and accelerator data.

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. This paper derives the exact phenomenological consequences of the 'neutrino cuboid' parametrization m1=m0 sin(ξ), m2=m0 cos(ξ) sin(ζ), m3=m0 cos(ξ) cos(ζ), combined with the mass–mixing alignment ξ=θ12, ζ=θ23 proposed in Ref. [23]. The central results are the closed-form relations m0^2=(Δm21^2+Δm31^2)/(1−3 sin^2 θ12) (Eq. 12) and sin^2 θ23=[x+η(1−2x)]/[(1−x)(1+η)] with η=Δm21^2/Δm31^2 (Eq. 19), together with the corollary sin^2 θ23 − 1/2 = (3x−1)(1−η)/[2(1−x)(1+η)] (Eq. 22). From the measured x<1/3 and 0<η<1, the authors conclude that, within the ansatz, inverted ordering is excluded and θ23 lies in the lower octant, and that the absolute spectrum is fixed: representative inputs give sin^2 θ23≃0.440, (m1,m2,m3)≃(0.0938,0.0942,0.1063) eV, and Σν≃0.294 eV. The paper further shows that the first-order expansion about the tribimaximal point is numerically unstable for η because of a leading-order cancellation, compares the exact prediction with oscillation data (approximate pulls 0.9–1.9σ), confronts Σν with cosmological limits (strong but model-dependent tension), and introduces geometric alignment residuals (Rξ, Rζ) as null-test observables.

Significance. If the alignment ansatz is taken at face value, the paper delivers an unusually sharp set of predictions: two mixing angles and two mass-squared differences determine the full mass spectrum, the atmospheric octant, and the mass sum, all in closed form and without expansion. The algebra was independently checked and is internally consistent; the numerical results reproduce from the stated inputs, and Appendix A specifies the Monte Carlo recipe (generator, seed, sampling order, rejection rules, quantile definition) essentially to the level of exact reproducibility. The falsifiable structure — inverted ordering, an upper atmospheric octant, or Σν below roughly 0.25 eV would each exclude the ansatz — is a genuine strength. The paper is also honest about its limitations: the benchmark inputs are explicitly hybrid and illustrative, the likelihoods are not fully profiled, the cosmological limits are treated as sum-only benchmarks, and the predictive content is entirely conditional on Eq. (5). The central weakness is the absence of any dynamical or symmetry motivation for that alignment; this is an assumption burden rather than an internal error, and the residual framework is a reasonable

major comments (2)
  1. [Sec. II-B, Eq. (5)] The entire predictive program is conditional on the exact identification of the geometric angles with the measured PMNS angles, an assumption imported from Ref. [23] and acknowledged by the authors as carrying all predictive content (Secs. I and II-A). The paper does not quantify how the derived results degrade under approximate alignment. Since the ordering theorem (Eqs. 20–21), the octant theorem (Eq. 22), and the absolute-mass predictions (Eqs. 12–15, 86) all inherit this premise, I recommend adding a short misalignment-sensitivity analysis: for small delta_xi = xi − theta_12 and delta_zeta = zeta − theta_23, compute the induced shifts in m0^2, sin^2 theta_23, and Sigma_nu, and locate the point on the (R_xi, R_zeta) residual trajectories (Eqs. B6–B7) closest to the current global fit. This would convert the ansatz into a quantitatively testable deformation rather than a binary assumpt
  2. [Sec. IV-C, Eq. (77)] The approximate Gaussian pulls combine the upper-side uncertainty of the predicted sin^2 theta_23 with the lower-side uncertainty of the global result because y_pred < y_glob. The quoted global determinations are asymmetric in the opposite direction (Eq. 76: NuFIT +0.017/−0.013; Capozzi +0.023/−0.013), so this pairing is orientation-dependent and tends to minimize the apparent tension. The paper correctly calls the pulls non-formal, but I recommend also giving the opposite pairing, or a symmetric chi^2 profile, so that the headline 'mild tension' conclusion can be assessed independent of the sign convention.
minor comments (5)
  1. [Appendix A.4] The residuals r0, rx, r21, r31, ry are defined and asserted to be at floating-point roundoff level, but their achieved numerical magnitudes are never reported. Reporting the maximum residual values across the retained samples would complete the reproducibility claim.
  2. [Sec. IV-A and Table II] Both benchmarks use the same preliminary atmospheric-scale input (Eq. 67); a footnote to Table II should restate the hybrid nature of the benchmarks so that the intervals in Eqs. (74)–(75) are not misread as consistent global-fit contours.
  3. [Sec. III-C, Eq. (51)] The caveat that eta[2] is the ratio of consistently truncated second-order splittings rather than a genuine Taylor expansion of eta should be repeated immediately after Eq. (51) or in the Table I caption, since Table I quotes eta[2].
  4. [Sec. V-B, Eq. (96)] The sum-only, central-splittings caveat is stated for the 0.0642 eV row only; a sentence before Eq. (96) noting that all rows inherit these assumptions would prevent over-interpretation of the translated sin^2 theta_12 bounds.
  5. [Appendix A.2 heading] Typo: 'T ransformation to predicted observables' contains a stray space between 'T' and 'ransformation'.

Circularity Check

0 steps flagged

No significant circularity: the central ansatz is an openly stated alignment hypothesis, and all derived mass-mixing relations follow by exact algebra and are tested against independent data.

full rationale

The paper's derivation chain is conditional but not circular. It explicitly states that the cuboid parametrization Eq. (1) is 'an exact reparametrization of three positive masses' whose 'predictive content arises only after a relation between the geometric angles ξ and ζ and independently measurable flavor-mixing parameters is imposed' (Sec. II.A). The only load-bearing step is the alignment Eq. (5), ξ=θ12 and ζ=θ23, which is imported from Ref. [23] (Z.-z. Xing, not an author of the present paper) and repeatedly labelled an ansatz or hypothesis. From that assumed identification, the paper derives the exact relations m0^2=(Delta m21^2+Delta m31^2)/(1-3 sin^2 theta12) and y=[x+eta(1-2x)]/[(1-x)(1+eta)] by purely algebraic manipulation of Eq. (7). The target observables sin^2 theta23, Sigma_nu, m_beta, and m_beta_beta are not used to set any of the input parameters; the inputs are measured values of theta12, Delta m21^2, and Delta m31^2, and the predictions are then compared with independent global fits, KATRIN, KamLAND-Zen, and cosmological bounds. The alignment residuals R_xi and R_zeta are defined precisely as null-test variables, with the exact ansatz at (0,0); this is a falsifiable hypothesis and not an output smuggled into the inputs. The first- and second-order expansion analysis in Sec. III is a convergence diagnostic, not a load-bearing identification, and the paper explicitly uses the exact formulas for its phenomenological conclusions. The self-citations [16,17] appear only in the introductory remark that tribimaximal mixing can arise from discrete symmetries; they do not carry the mass-mixing derivation. No fitted parameter is renamed as a prediction, no uniqueness theorem is invoked, and no result is equivalent to its input by construction. The main vulnerability is the plausibility of the alignment assumption itself, but that is an acknowledged assumption burden, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No numerical parameter is fitted to the target observables: sin²θ23 and the mass spectrum are predicted from external θ12 and Δm² inputs. The entire predictive content rests on the ad hoc alignment Eq. (5), which is the key axiom. Cosmological and 0νββ comparisons introduce standard assumptions that the paper explicitly flags.

axioms (5)
  • domain assumption The three light neutrinos are the only mass states and the standard three-flavor PMNS parametrization applies.
    The paper uses m1, m2, m3 and standard oscillation formulas throughout; no sterile neutrinos or non-standard decay channels are considered.
  • ad hoc to paper The geometric angles ξ and ζ are exactly equal to the PMNS angles θ12 and θ23 (Eq. (5)).
    This is the physical alignment hypothesis imported from Ref. [23]; it is not derived and carries all the predictive content of the paper.
  • standard math The cuboid parametrization Eq. (1) is an exact reparametrization of three positive masses.
    Positivity and unique inversion are stated in Sec. II.A; boundary cases with vanishing mass are treated only as limits.
  • domain assumption For neutrinoless double-beta decay, standard light-neutrino exchange with two arbitrary Majorana phases is assumed.
    Used in Eq. (103) and the phase-envelope derivation; the paper explicitly notes nuclear-matrix-element and mechanism dependence.
  • domain assumption Cosmological mass limits are interpreted as sum-only benchmarks at central mass-splitting values.
    Sec. V.B states the published limits are compared algebraically rather than through a joint cosmological likelihood; the paper acknowledges this approximation.

pith-pipeline@v1.3.0-alltime-deepseek · 21566 in / 14096 out tokens · 140697 ms · 2026-08-01T03:35:27.219767+00:00 · methodology

0 comments
read the original abstract

We investigate the exact phenomenology of a geometric neutrino cuboid defined by $m_1=m_0\sin\xi$, $m_2=m_0\cos\xi\sin\zeta$, and $m_3=m_0\cos\xi\cos\zeta$, whose cubic point corresponds to mass degeneracy and the tribimaximal values of the solar and atmospheric mixing angles. Imposing the mass-mixing alignment $\xi=\theta_{12}$ and $\zeta=\theta_{23}$, we derive closed-form consistency relations without relying on an expansion about the cubic limit. These relations show that the observed condition $\sin^2\theta_{12}<1/3$ selects normal mass ordering and, in the resulting normal-ordering regime, the physical condition $0<\Delta m_{21}^2/\Delta m_{31}^2<1$ requires $\theta_{23}$ to lie in the lower octant. Representative JUNO-based inputs give $\sin^2\theta_{23}\simeq0.440$, $(m_1,m_2,m_3)\simeq(0.0938,0.0942,0.1063)\mathrm{eV}$, and $\sum_i m_i\simeq0.294\mathrm{eV}$. We demonstrate that the first-order expansion around the tribimaximal point is numerically unstable for the solar-to-atmospheric mass-splitting ratio because of a leading-order cancellation. Present atmospheric-angle data yield only mild tension with exact alignment, whereas stringent neutrino-mass bounds in baseline $\Lambda$CDM cosmology strongly disfavor it; its viability under extended cosmological models remains model dependent. We finally formulate geometric alignment residuals as general null-test observables, providing a systematic framework for testing the neutrino cuboid with future oscillation and absolute-mass measurements.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

43 extracted references · 20 linked inside Pith

  1. [1]

    Input sampling and physical domain We collect the three inputs that determine the exact-alignment predictions into p≡ x,∆m 2 21,∆m 2 31 .(A1) For a general covariance matrixC, the Monte Carlo samples are drawn according to p(k) ∼ N(p,C), k= 1, . . . , NMC.(A2) 29 In the illustrative analysis presented in the main text, we useN MC = 106 and set C= diag σ2 ...

  2. [2]

    and ζ∗ = π/4, one obtains m1 = m2 = m3 = m0/ √

  3. [3]

    This correspondence concerns the two large mixing angles; the reactor angle θ13 is not fixed by the cuboid construction itself

    The same two geometric angles coincide with the solar and atmospheric values of the tribimaximal pattern, namelyθ12 = arctan(1/ √ 2) and θ23 = π/4. This correspondence concerns the two large mixing angles; the reactor angle θ13 is not fixed by the cuboid construction itself. Moreover, the geometric parametrization alone is a change of variables and carrie...

  4. [4]

    The exact cuboid ansatz corresponds to the point ( Rξ, Rζ) = (0, 0), while oscillation and absolute-mass measurements constrain the surrounding residual plane. This formulation separates the kinematic geometry from the dynamical hypothesis of exact alignment and permits controlled tests of approximate alignment, experimental uncertainties, and possible sy...

  5. [5]

    The arrays for x, ∆m2 21, and ∆ m2 31 are generated in that order

    We use numpy.random.Generator with the PCG64 generator and the fixed seed 20260724. The arrays for x, ∆m2 21, and ∆ m2 31 are generated in that order. Samples failing the physical-domain conditions specified in Appendix A are discarded without replacement, and empirical quantiles are evaluated using linear interpolation. A complete covariance matrix cover...

  6. [6]

    Evaluation of Eq

    and profiling χ2 align = min p χ2 input(p) + ∆χ2 23 F x, ∆m2 21 ∆m2 31 .(79) The goodness of fit of the alignment hypothesis should then be assessed relative to the standard fit in which sin2 θ23 is allowed to vary independently. Evaluation of Eq. (79) requires the experimental or global-fit likelihood profiles and their relevant correlations, rather than...

  7. [7]

    26 Two qualitative consequences follow immediately

    These equations are exact and do not rely on an expansion about the cubic point. 26 Two qualitative consequences follow immediately. Since the observed solar angle satisfies x <1/3, positivity ofm 2 0 requires ∆m2 21 + ∆m2 31 >0.(113) In inverted ordering, ∆ m2 31 < 0 and |∆m2 31|> ∆m2 21, so this condition cannot be satisfied. Inverted ordering is theref...

  8. [8]

    Y.-F. Li, J. Cao, Y. Wang, and L. Zhan, Phys. Rev. D88, 013008 (2013), arXiv:1303.6733 [hep-ex]

    Pith/arXiv arXiv 2013

  9. [9]

    (A9) is obtained in radians and is converted to degrees only after the nonlinear transformation has been performed

    T ransformation to predicted observables For every accepted sample, the atmospheric prediction is evaluated from the exact map y(k) pred =F x(k), η(k) = x(k) +η (k) 1−2x (k) (1−x (k)) (1 +η (k)) ,(A8) followed by θpred 23 (k) = arcsin q y(k) pred.(A9) The angle in Eq. (A9) is obtained in radians and is converted to degrees only after the nonlinear transfo...

  10. [10]

    The quantiles are evaluated using numpy.quantile withmethod=’linear’

    Quantile-based uncertainty intervals For an observable O, let qp(O) denote the empirical quantile such that a fraction p of the accepted samples lies at or below it. The quantiles are evaluated using numpy.quantile withmethod=’linear’. The central 68% interval is reported as O=q 0.50(O) +[q0.84(O)−q0.50(O)] −[q0.50(O)−q0.16(O)],(A14) and the central 95% i...

  11. [11]

    In the following expressions, the Monte Carlo sample label (k) is suppressed for readability

    Exact numerical consistency checks For each accepted sample, the numerical implementation can be checked using residuals of identities that must hold analytically. In the following expressions, the Monte Carlo sample label (k) is suppressed for readability. We define the dimensionless normalization residual r0 ≡ m2 1 +m 2 2 +m 2 3 m2 0 −1 ,(A21) the solar...

  12. [12]

    Majorana-phase envelope For completeness, the exact phase envelope ofm ββ can be derived geometrically. With z≡A+Be iα21 +Ce iα31, m ββ =|z|,(A27) define L≡A+B+C, L max ≡max{A, B, C}.(A28) The triangle inequality gives mββ ≤L,(A29) with equality when all three contributions have the same phase. The generalized reverse triangle inequality gives mββ ≥max{2L...

  13. [13]

    Normal-ordering trajectory For normal ordering, ∆m 2 31 = ∆ andµ=m 1. The three masses are m1 =µ, m 2 = p µ2 +δ, m 3 = p µ2 + ∆.(B2) The squared norm of the mass vector is m2 0 = 3µ2 +δ+ ∆.(B3) The corresponding geometric angles satisfy sin2 ξNO g (µ) = µ2 3µ2 +δ+ ∆ ,(B4) and tan2 ζ NO g (µ) = µ2 +δ µ2 + ∆,sin 2 ζ NO g (µ) = µ2 +δ 2µ2 +δ+ ∆ .(B5) Thus, th...

  14. [14]

    Inverted-ordering obstruction For inverted ordering, ∆m2 31 =−∆ andµ=m 3. The spectrum is m1 = p µ2 + ∆, m 2 = p µ2 + ∆ +δ, m 3 =µ,(B17) and the squared norm of the mass vector is m2 0 = 3µ2 + 2∆ +δ.(B18) The geometric angles obey sin2 ξIO g (µ) = µ2 + ∆ 3µ2 + 2∆ +δ ,(B19) and tan2 ζ IO g (µ) = µ2 + ∆ +δ µ2 ,sin 2 ζ IO g (µ) = µ2 + ∆ +δ 2µ2 + ∆ +δ .(B20) ...

  15. [15]

    If instead Σ max ν ≤ ΣNO ν,min, the complete normal-ordering trajectory is excluded under the same sum-only interpretation

    Restrictions from absolute-mass information For normal ordering, the mass sum along the residual trajectory is ΣNO ν (µ) =µ+ p µ2 +δ+ p µ2 + ∆.(B26) It is strictly increasing because dΣNO ν dµ = 1 + µp µ2 +δ + µp µ2 + ∆ >0.(B27) The minimum mass sum along the trajectory occurs atµ= 0 and is ΣNO ν,min = √ δ+ √ ∆.(B28) If a cosmological constraint is repres...

  16. [16]

    Fukudaet al.(Super-Kamiokande Collaboration), Phys

    Y. Fukudaet al.(Super-Kamiokande Collaboration), Phys. Rev. Lett.81, 1562 (1998), arXiv:hep-ex/9807003 [hep-ex]

    Pith/arXiv arXiv 1998

  17. [17]

    Q. R. Ahmadet al.(SNO Collaboration), Phys. Rev. Lett.89, 011301 (2002), arXiv:nucl- ex/0204008 [nucl-ex]

    arXiv 2002

  18. [18]

    F. P. Anet al.(Daya Bay Collaboration), Phys. Rev. Lett.108, 171803 (2012), arXiv:1203.1669 [hep-ex]

    Pith/arXiv arXiv 2012

  19. [19]

    Esteban, M

    I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. P. Pinheiro, and T. Schwetz, JHEP12(2024), 216, arXiv:2410.05380 [hep-ph]

    Pith/arXiv arXiv 2024

  20. [20]

    Capozzi, W

    F. Capozzi, W. Giar` e, E. Lisi, A. Marrone, A. Melchiorri, and A. Palazzo, Phys. Rev. D111, 093006 (2025), arXiv:2503.07752 [hep-ph]

    Pith/arXiv arXiv 2025

  21. [21]

    L. Zhan, Y. Wang, J. Cao, and L. Wen, Phys. Rev. D78, 111103 (2008), arXiv:0807.3203 [hep-ex]

    Pith/arXiv arXiv 2008

  22. [22]

    L. Zhan, Y. Wang, J. Cao, and L. Wen, Phys. Rev. D79, 073007 (2009), arXiv:0901.2976 [hep-ex]

    Pith/arXiv arXiv 2009

  23. [23]

    Xing, Neutrino cuboid for normal mass ordering and tribimaximal flavor mixing (2026), arXiv:2607.07311 [hep-ph]

    Z.-z. Xing, Neutrino cuboid for normal mass ordering and tribimaximal flavor mixing (2026), arXiv:2607.07311 [hep-ph]

    Pith/arXiv arXiv 2026

  24. [24]

    Anet al.(JUNO Collaboration), J

    F. Anet al.(JUNO Collaboration), J. Phys. G43, 030401 (2016), arXiv:1507.05613 [physics.ins- det]

    Pith/arXiv arXiv 2016

  25. [25]

    Abuslemeet al.(JUNO Collaboration), Nature654, 343 (2026), arXiv:2511.14593 [hep-ex]

    A. Abuslemeet al.(JUNO Collaboration), Nature654, 343 (2026), arXiv:2511.14593 [hep-ex]

    arXiv 2026

  26. [26]

    Y. Wang, JUNO, Plenary talk at the XXXII International Conference on Neutrino Physics and Astrophysics (Neutrino 2026), University of California, Irvine (2026), presented on 22 June 2026

    2026

  27. [27]

    P. F. Harrison, D. H. Perkins, and W. G. Scott, Phys. Lett. B530, 167 (2002), arXiv:hep- ph/0202074 [hep-ph]

    arXiv 2002

  28. [28]

    He and A

    X.-G. He and A. Zee, Phys. Lett. B560, 87 (2003), arXiv:hep-ph/0301092 [hep-ph]

    Pith/arXiv arXiv 2003

  29. [29]

    K. S. Babu, E. Ma, and J. W. F. Valle, Phys. Lett. B552, 207 (2003), arXiv:hep-ph/0206292 40 [hep-ph]

    Pith/arXiv arXiv 2003

  30. [30]

    Xing and Z.-h

    Z.-z. Xing and Z.-h. Zhao, Rept. Prog. Phys.79, 076201 (2016), arXiv:1512.04207 [hep-ph]

    Pith/arXiv arXiv 2016

  31. [31]

    (23) follows immediately from this map whenx <1/3 and 0< η <1

    The exact lower-octant result in Eq. (23) follows immediately from this map whenx <1/3 and 0< η <1. It is useful to compare two solar-input benchmarks while using the same atmospheric-scale input. The first is based on the published analysis of the first 59 .1 days of JUNO data, which gives [10] Dpub :x= 0.3092±0.0087, ∆m2 21 = (7.50±0.12)×10 −5 eV2. (65)...

  32. [32]

    J. Lu, A. H. Chan, and C. H. Oh, PoSICHEP2024, 189 (2025)

    2025

  33. [33]

    J. Lu, A. H. Chan, and C. H. Oh, Universe10, 50 (2024)

    2024

  34. [34]

    Fritzsch and Z.-z

    H. Fritzsch and Z.-z. Xing, Phys. Lett. B372, 265 (1996), arXiv:hep-ph/9509389 [hep-ph]

    Pith/arXiv arXiv 1996

  35. [35]

    R. Jora, S. Nasri, and J. Schechter, Int. J. Mod. Phys. A21, 5875 (2006), arXiv:hep-ph/0605069 [hep-ph]

    Pith/arXiv arXiv 2006

  36. [36]

    Spinrath, J

    M. Spinrath, J. Phys. Conf. Ser.888, 012176 (2017), proceedings of Neutrino 2016, arXiv:1609.07708 [hep-ph]

    Pith/arXiv arXiv 2017

  37. [37]

    Antusch, P

    S. Antusch, P. Huber, S. F. King, and T. Schwetz, JHEP04(2007), 060, arXiv:hep-ph/0702286 [hep-ph]

    Pith/arXiv arXiv 2007

  38. [38]

    Buccella, M

    F. Buccella, M. Chianese, G. Mangano, G. Miele, S. Morisi, and P. Santorelli, JHEP04(2017), 004, arXiv:1701.00491 [hep-ph]

    Pith/arXiv arXiv 2017

  39. [39]

    Akeret al.(KATRIN Collaboration), Science388, 180 (2025), arXiv:2406.13516 [nucl-ex]

    M. Akeret al.(KATRIN Collaboration), Science388, 180 (2025), arXiv:2406.13516 [nucl-ex]

    arXiv 2025

  40. [40]

    Abeet al.(KamLAND-Zen Collaboration), Phys

    S. Abeet al.(KamLAND-Zen Collaboration), Phys. Rev. Lett.135, 262501 (2025), arXiv:2406.11438 [hep-ex]

    arXiv 2025

  41. [41]

    Aghanimet al.(Planck Collaboration), Astron

    N. Aghanimet al.(Planck Collaboration), Astron. Astrophys.641, A6 (2020), erratum: Astron. Astrophys. 652, C4 (2021), arXiv:1807.06209 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  42. [42]

    Elberset al.(DESI Collaboration), Phys

    W. Elberset al.(DESI Collaboration), Phys. Rev. D112, 083513 (2025), arXiv:2503.14744 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  43. [43]

    Jimenez, C

    R. Jimenez, C. Pe˜ na Garay, F. Simpson, and L. Verde, From evidence to evident: Decisive cosmological evidence for the normal neutrino mass hierarchy (2026), arXiv:2606.18987 [astro- ph.CO]. 41

    Pith/arXiv arXiv 2026