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This paper proposes that the neutrino mass-squared differences obey the exact ratio (√Δm²31 + √Δm²21)/(√Δm²31 − √Δm²21) = √2, which would fix the absolute neutrino masses if the lightest neutrino is massless.

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2026-08-03 07:52 UTC pith:YDT7UJMO

load-bearing objection A genuinely new numerical coincidence in neutrino mass-squared differences, honestly presented, but the paper promotes an assumption to a relation without quantifying how unlikely the coincidence is. the 3 major comments →

arxiv 2601.18781 v2 pith:YDT7UJMO submitted 2026-01-26 hep-ph

Another relation among the neutrino mass-squared differences?

I. Alikhanov This is my paper
classification hep-ph PACS 14.60.Pq
keywords neutrino massesmass-squared differencesnormal mass orderingabsolute neutrino mass scalesum of neutrino masseseffective Majorana massalgebraic mass relationvanishing lightest neutrino mass
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 argues that the ratio of the sum and difference of the square roots of the two neutrino mass-squared differences is exactly √2. The observed oscillation data already sit within about 0.3% of that value, so the author promotes the coincidence to an exact physical equality. If the ansatz is right, only one mass-squared difference is independent, and the entire spectrum is fixed. Under the additional interpretation that the lightest neutrino mass m1 is zero, the absolute masses become m2 ≈ 0.00866 eV, m3 ≈ 0.05029 eV, and the sum of masses ≈ 0.05895 eV — a scale consistent with cosmological and laboratory bounds and low enough that next-generation direct searches could probe it. The relation also reduces to tan(π/8) = √(m2/m3) when m1 = 0 and mirrors the structural form of the known charged-lepton mass relation.

Core claim

The author's central claim is that the ratio λ = (√Δm²31 + √Δm²21)/(√Δm²31 − √Δm²21) is exactly √2, not merely close to it. Assuming λ = √2, the paper derives that one mass-squared difference fixes the other two, that (Δm²21 + Δm²31)/(√Δm²21 + √Δm²31)² = 3/4, and that (Δm²32)²/(Δm²21 Δm²31) = 32. The author reads the last identity as evidence that ν1 is massless, which gives the mass ratio √(m2/m3) = tan(π/8) and the absolute predictions m2 ≈ 0.00866 eV, m3 ≈ 0.05029 eV, mνe ≈ 0.00892 eV, Σmi ≈ 0.05895 eV, and an effective Majorana mass between 0.00141 and 0.00383 eV depending on the unknown phases.

What carries the argument

The load-bearing object is the dimensionless ratio λ(Δm²21, Δm²31) defined in Eq. (1), built from the square roots of the two measured mass-squared differences. The entire paper is the algebra that follows from setting this ratio exactly to √2: the relation is rearranged into a sum-to-square ratio of 3/4, a product identity (Δm²32)² = 32 Δm²21 Δm²31, and, once m1 = 0 is assumed, into m3/m2 = (√2+1)/(√2−1), equivalently √(m2/m3) = tan(π/8). Because √2 is a fixed irrational number, the relation constrains the two independent splittings to a one-parameter family and ties absolute masses to a single input.

Load-bearing premise

The results stand or fall on the premise that the observed ratio's proximity to √2 — about 0.3% at best-fit values — is an exact physical equality rather than a numerical coincidence, together with the assumed normal mass ordering.

What would settle it

Measure Δm²21 and Δm²31 with sub-0.1% precision and form λ = (√Δm²31 + √Δm²21)/(√Δm²31 − √Δm²21); if the central value differs from √2 by more than the combined uncertainty — for instance a measured λ of 1.419 or 1.409 with errors near 10⁻⁴ — the exact relation is ruled out. Such a measurement is within the stated reach of upcoming reactor-neutrino experiments.

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

If this is right

  • A single measured mass-squared difference determines the other two, so future precision on either Δm²21 or Δm²31 directly predicts the third.
  • If m1 = 0, the absolute neutrino masses are predicted: m2 ≈ 0.00866 eV, m3 ≈ 0.05029 eV, and Σmi ≈ 0.05895 eV, a value near the lower limit allowed by cosmology.
  • The predicted effective electron-neutrino mass mνe ≈ 0.00892 eV is within range of proposed quantum-technology beta-decay experiments aiming at ~0.01 eV sensitivity.
  • The predicted effective Majorana mass lies between about 0.0014 and 0.0038 eV, a target for next-generation neutrinoless double-beta decay searches once their sensitivity crosses this band.
  • The same algebraic structure that yields λ = √2 also produces a charged-lepton-mass-like relation with value 3/4, offering a possible link between neutrino and charged-lepton mass generation.

Where Pith is reading between the lines

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

  • The 0.27% agreement across fits is not a calibrated test: a scan over many simple algebraic functions of the two splittings would show how many combinations land within 0.3% of a simple constant, which would calibrate how surprising this particular coincidence is.
  • The demonstrated 0.03σ shift of Δm²21 that reproduces √2 to seven digits is a reminder that the relation is an ansatz rather than a fit; future tests should compare the predicted ratio against the full correlated uncertainties of both splittings, not against each best-fit point separately.
  • If m1 is not exactly zero but small, the relation still holds while the absolute masses scale differently; a cosmological measurement of Σmi near 0.059 eV would support m1 = 0, while a value noticeably above that would favor a lighter-but-nonzero m1 within the relation.
  • A derivation of the values 3/4 and √2 from a single mass-generation mechanism would unify this relation with the charged-lepton mass formula; searching for such a mechanism is a natural theoretical next step.

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

3 major / 5 minor

Summary. The manuscript proposes an exact relation, Eq. (2), between the solar and atmospheric neutrino mass-squared differences: (sqrt(Δm²31)+sqrt(Δm²21))/(sqrt(Δm²31)-sqrt(Δm²21)) = sqrt(2). It reports that this ratio evaluated at best-fit points from several global fits lies within 0.27% of sqrt(2), and that a 0.03σ shift of Δm²21 produces a seven-decimal match. The paper derives algebraic consequences, including a Koide-like relation, Eq. (4), and interprets the relation as pointing toward m1 = 0. For m1 = 0 it obtains m2 ≈ 0.00866 eV, m3 ≈ 0.05029 eV, Σmi ≈ 0.05895 eV, mνe ≈ 0.00892 eV, and a range for mββ. The normal mass ordering is assumed throughout.

Significance. If Eq. (2) were exact and m1 = 0, the neutrino mass spectrum would be fixed by a single mass-squared difference, and the predicted absolute mass scale would sit near the lower edge of cosmological bounds. The algebra from Eq. (2) to Eq. (8) is simple and correct, and the paper is transparent in stating that Eq. (2) is an assumption rather than a derived law. The relation is falsifiable with future JUNO and other oscillation data. However, the quantitative evidence presented is only a numerical coincidence: no statistical test is performed, no null hypothesis is defined, and no account is taken of the post hoc choice of the ratio and the target value sqrt(2). Consequently, the paper's central claim is not yet quantitatively supported, and the 'predictions' in Section IV mostly restate the input data plus the ansatz rather than providing independent checks.

major comments (3)
  1. [Section II, Eq. (2), Table I] The only evidence for Eq. (2) is that λ evaluated at several best-fit points lies in [1.4140, 1.4181], a 0.27% spread. This is not a statistical test. The paper does not compute the probability that two measured quantities would yield a ratio this close to sqrt(2) under a plausible null hypothesis, nor does it account for the look-elsewhere effect of selecting this particular functional form and target constant after inspecting the data. I request a quantitative evaluation: a chi-square or posterior probability for λ = sqrt(2) using the full (correlated) covariance matrix of Δm²21 and Δm²31, and an estimate of the trials factor involved in choosing this ratio. This is load-bearing, since Eq. (2) is the basis for all subsequent results.
  2. [Section III, Eqs. (6)-(7)] The inference that Eq. (7) implies m1 = 0 is not justified. Equations (6) and (7) are algebraic rewritings of Eq. (2); the left-hand side Δm²21 Δm²31 depends on m1 in general, and expressing it as (1/32)(m3² - m2²)² follows from the assumed ratio, not from vanishing m1. The statement 'we attribute this observation to the masslessness of ν1' is an interpretation, not a derivation. To support m1 ≈ 0, the author should solve Eq. (2) for m1 using the measured values of Δm²21 and Δm²31 (including uncertainties) or otherwise demonstrate that the data favor m1 = 0. As written, the 'solutions' in Section IV impose m1 = 0 by hand.
  3. [Section II, seven-decimal agreement] The demonstration that shifting Δm²21 to 7.5065×10⁻⁵ eV², within 0.03σ of the reference best fit, yields λ = 1.41421359 does not add evidential weight: for any continuous observable and any target value, one can normally find a point inside a confidence interval that satisfies the target exactly. The sentence 'the accuracy can be made arbitrarily high' is therefore misleading. In addition, Table I reports λ without uncertainties, and the various global fits are not independent; their scatter cannot be interpreted as a distribution of λ. I recommend propagating the errors of a single reference fit (including correlations) and de-emphasizing the tuned seven-decimal match.
minor comments (5)
  1. [Eq. (3)] Please state how the uncertainty 0.0045 is obtained and whether the uncertainties in Δm²21 from JUNO and Δm²31 from NuFIT are treated as uncorrelated. Also clarify that these two inputs come from different datasets.
  2. [Section II, near Eq. (2)] The phrase 'the relatively small error' should refer to a specific quantity, ideally the propagated uncertainty in λ or the deviation of the ratio from sqrt(2).
  3. [Section IV] Typographical: 'Zpole' should be 'Z pole'.
  4. [Table I] The two rows for NuFIT-6.0 [49] and the two rows for NuFIT-6.1 [17] should be labeled to indicate which dataset or ordering they correspond to, since they currently appear as duplicate entries.
  5. [General] The paper uses 'best fit' inconsistently: sometimes it means global-fit central values, sometimes a value shifted within 1σ. A uniform terminology with explicit error bars would improve readability.

Circularity Check

3 steps flagged

The central 'predictions' for neutrino masses are the fitted mass-squared differences renamed under the assumption m1 = 0; the ansatz (2) is adopted because the same data already satisfy it, making its 'consistency' checks algebraic tautologies.

specific steps
  1. fitted input called prediction [Section IV, after Eq. (9); numerical solutions]
    "Our interpretation implies the solutions m1 ≃0, m2 = sqrt(Δm2_21), m3 = sqrt(Δm2_31), Σ_i m_i = sqrt(Δm2_21)+sqrt(Δm2_31) ... By adopting the first JUNO results [20] for Δm2_21 ... and taking Δm2_31 ... from the NuFIT-6.1 global analysis [17], we numerically find m1 ≃0, m2 = 0.00866±0.00007 eV, m3 = 0.05029±0.00021 eV ... Σ_i m_i = 0.05895±0.00022 eV"

    With m1 = 0, the identities Δm2_21 = m2^2 and Δm2_31 = m3^2 make m2, m3, and Σm_i literally the square roots and sum of the very same fitted inputs used to motivate and calibrate relation (2). The numbers quoted as 'predictions' are therefore not new results but a relabeling of the input global-fit values; any agreement with external constraints is inherited automatically from the choice of inputs.

  2. self definitional [Section II, Eq. (2) and following 'consistency' checks; Section III, Eqs. (4), (6), (8)]
    "Inspired by these observations, we assume that (sqrt(Δm2_31)+sqrt(Δm2_21))/(sqrt(Δm2_31)-sqrt(Δm2_21)) = sqrt(2) ... From (2), we also find [Eq. (4)] ... which is consistent with the global fits as well. For example, with the data from [50], the left-hand side is 0.7501, in excellent agreement with the predicted value of 3/4."

    Equation (2) is adopted because the global-fit best-fit points already make the left-hand side close to √2; then Eq. (4) is algebraically equivalent to Eq. (2), and Eq. (8) is a rearranged form of the same relation. Checking these equations against the same fitted values is therefore a check of the ansatz against the data that generated the ansatz, not an independent test. The 'excellent agreement' is built into the choice of (2).

  3. fitted input called prediction [Section II, seven-figure agreement; footnote 2]
    "For example, a minor shift in just Δm2_21 to 7.5065×10^-5 eV^2 – a value within 0.03σ of the best fit [50] – leads to the result λ(7.5065×10^-5 eV^2, 2.55×10^-3 eV^2) = 1.41421359, which approximates √2 to seven (!) decimal places."

    The 'seven-decimal' match is manufactured by selecting a point inside the 1σ interval of the fitted Δm2_21 specifically to make λ equal √2. Since any continuous function can be matched to any target value by moving within a confidence interval, this is not evidence for (2); it is an explicit tuning of the input to the claimed output.

full rationale

The paper is honest that Eq. (2) is an ansatz ('we assume'), not a derived law, so the choice of the relation itself is not presented as a prediction. However, the subsequent 'physical predictions' — m2, m3, Σm_i — are obtained by setting m1 = 0 and then taking square roots of the exact same Δm2_21 and Δm2_31 inputs that were used to select Eq. (2). This is a textbook case of fitted input renamed as prediction: the numerical outputs are identities once m1 = 0. The consistency checks in Eqs. (4), (6), and (8) are algebraic rearrangements of Eq. (2), and Eq. (2) was itself chosen because the best-fit values already satisfy it to ~0.3%; the seven-figure agreement is produced by a deliberate 0.03σ shift of Δm2_21. There is no self-citation load-bearing chain and no uniqueness theorem imported from the authors; the circularity is in the reduction of the quantitative predictions to the fitted inputs. Score 7 reflects that the core number-output of the paper reduces by construction, while the paper's one genuinely new content — the specific algebraic form with √2 as a phenomenological hypothesis — remains an unproven ansatz rather than a tested prediction. The external comparisons (KATRIN, DESI, Koide extensions) are only consistency comments and do not rescue the predictions from being restatements of the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The paper introduces no new particles or forces. Its free parameters are the hand-tuned Δm²21 value used to showcase the √2 match and the assumed m1 = 0. The central relation (2) itself is an ad hoc ansatz, and the normal-ordering and standard-framework assumptions are inherited from current neutrino physics.

free parameters (2)
  • Δm²21 = 7.5065×10⁻⁵ eV² (shifted best-fit) = 7.5065e-5 eV²
    Chosen within 0.03σ of the de Salas best fit to make λ equal √2 to seven decimal places; a hand-picked value used to demonstrate 'surprising' precision (Sec. II).
  • m1 (lightest neutrino mass) = 0 eV (assumed)
    Set to zero to convert relation (2) into absolute mass predictions; the paper states this is an attribution, not a consequence of (2).
axioms (5)
  • domain assumption Standard three-flavor neutrino oscillation framework with unitary PMNS mixing.
    Invoked throughout as background (Sec. I); all definitions of Δm² and mixing angles assume this framework.
  • domain assumption Normal mass ordering (m1 < m2 < m3) with Δm²31 > 0.
    Explicitly assumed in Sec. I ('From now on, we will assume the normal mass ordering'); the relation (2) is not defined for inverted ordering.
  • domain assumption The measured best-fit values for Δm²21 and Δm²31 from global fits are correct and can be used as exact inputs.
    All numerical evaluations use cited best-fit values; errors are only partially propagated (e.g., Eq. (3)) and often ignored when fixing the ratio.
  • ad hoc to paper The ratio λ is exactly √2 (Eq. 2).
    This is the central ansatz; it is not derived and is only consistent with data at approximately the 1σ level.
  • ad hoc to paper m1 = 0 (lightest neutrino mass vanishes).
    Assumed to derive m2 = √Δm²21, m3 = √Δm²31 and the sum of masses; the paper says 'we attribute this observation to the masslessness of the ν1 state'.

pith-pipeline@v1.3.0-alltime-deepseek · 8488 in / 15428 out tokens · 161065 ms · 2026-08-03T07:52:59.364589+00:00 · methodology

0 comments
read the original abstract

Determining the absolute neutrino mass scale remains a compelling challenge in particle physics. Establishing precise mathematical connections among the neutrino masses could significantly facilitate this task and shed additional light on the underlying mass-generation mechanism. We highlight the possible existence of a simple algebraic relation between the mass-squared differences, $(\sqrt{\Delta m^2_{31}}+\sqrt{\Delta m^2_{21}})/(\sqrt{\Delta m^2_{31}}-\sqrt{\Delta m^2_{21}})=\sqrt{2}$, which exhibits excellent agreement with the central values of recent global fits of neutrino oscillation data. We investigate the implications of adopting this ansatz and how it reduces the number of independent parameters in the neutrino sector. Notably, for a vanishing $\nu_1$ mass, this expression directly simplifies to the elegant Fritzsch relation: $\sqrt{m_2/m_3}=\tan{(\pi/8)}$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Integral representation of the neutrino mass-squared differences

    hep-ph 2026-07 reject novelty 2.0

    The claimed m1 < 0.0023 eV bound is an artifact of assuming the trapezoid error is small enough to force the result; the integral representation gives no independent constraint.

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