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A spline-based neural network claims to rediscover the Gell-Mann-Okubo mass formula from baryon mass data alone.

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 →

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2026-08-03 12:47 UTC pith:LL6AOIPY

load-bearing objection KAN is a reasonable toy application, but the 'discovery' claims do not survive the paper's own equations. the 4 major comments →

arxiv 2601.01495 v2 pith:LL6AOIPY submitted 2026-01-04 hep-ph

Discovering the Gell-Mann-Okubo Formula with Kolmogorov-Arnold Networks

Jian-Yao He , Xun Chen , Xiao-Yan Zhu , Wen Luo This is my paper
classification hep-ph
keywords Gell-Mann-Okubo formulaKolmogorov-Arnold networkssymbolic regressionSU(3) flavor symmetrybaryon mass spectrumequal-spacing rulehadron spectroscopyinterpretable machine learning
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 attempts to show that Kolmogorov-Arnold Networks can rediscover the Gell-Mann-Okubo mass formula directly from experimental baryon masses, with no SU(3) symmetry input. It trains small KANs on the isospin-averaged masses of the baryon octet and decuplet, prunes away redundant nodes, and uses symbolic regression to obtain compact expressions in hypercharge and isospin. The resulting formulas reproduce the known equal-spacing rule for the decuplet and a quadratic GMO-type relation for the octet. If the claim holds, it would mean a general-purpose interpretable network can uncover a classical symmetry-breaking law from a handful of measured masses, offering a template for symbolic discovery in hadron physics.

Core claim

The paper's central claim is that a spline-based Kolmogorov-Arnold Network, trained only on the isospin-averaged masses of the baryon octet and decuplet, can recover the classical Gell-Mann-Okubo mass relations without being given any SU(3) group-theoretic prior. For the decuplet, the KAN first learns the linear constraint Y=2I-2 and then, after symbolic regression and algebraic simplification, produces an equal-spacing law M=1382.16-146.71Y and a GMO-type quadratic formula. For the octet, it learns the relation Y^2=1-4(0.5-I)^2 and then yields M8=-188.26Y8+39.14I8(I8+1)-17.03Y8^2+1115.50. The paper presents these expressions as evidence that a data-driven, interpretable network can bridge m

What carries the argument

The load-bearing object is the Kolmogorov-Arnold Network (KAN), a neural network whose fixed weights are replaced by learnable univariate spline functions, used here as a symbolic-regression engine. The argument works by training a small KAN on augmented mass samples, pruning inactive branches, applying symbolic regression to the surviving functions, and then algebraically reshaping the raw expressions into the canonical Gell-Mann-Okubo form f(Y,I)=aY+b[I(I+1)-Y^2/4]+c. The multiplet identities Y=2I-2 and Y^2=1-4(0.5-I)^2 are used in this reshaping; the reshaping step, including the paper's explicit 'adjust the coefficients' operation, is where the claimed recovery is secured.

Load-bearing premise

The claim assumes that manually adjusting the fitted coefficients to resemble the Gell-Mann-Okubo form is a legitimate derivation step; if that adjustment is arbitrary, then the network did not autonomously discover the formula.

What would settle it

Rerun the identical pipeline on the same quantum numbers with the masses shuffled randomly among the multiplet states; if the same hand-adjustment step turns the resulting symbolic expression into a GMO-form equation regardless of the data, then the apparent discovery is an artifact of the algebraic reshaping rather than of the measured masses.

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

If this is right

  • If KAN can rediscover the GMO formula from masses alone, a general-purpose symbolic-regression network can independently recover known SU(3) breaking structure from minimal data.
  • The decuplet expression implies the equal-spacing rule: the masses of Delta, Sigma*, Xi*, and Omega differ by roughly 140-150 MeV per unit of strangeness.
  • The octet expression is a compact two-variable mass formula that covers N, Lambda, Sigma, and Xi baryons and can be compared directly with theoretical predictions.
  • Because the discovered expressions are symbolic rather than numeric, the workflow can in principle be applied to other hadron multiplets where no complete analytic formula is yet known.

Where Pith is reading between the lines

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

  • A stricter test than the one reported would be to withhold the GMO form and the quantum-number identities from the symbolic regression and see whether KAN proposes the form itself; the paper's physics filter does part of this work.
  • Because the input masses are already isospin-averaged and augmented within experimental uncertainties, the method may be learning the averaging procedure as much as the underlying symmetry; applying it to raw event-level data would separate those effects.
  • If the approach transfers to multiplets where no formula is known, such as charmed baryons or tetraquark candidates, it could serve as a discovery tool rather than a verification tool.
  • The two recovered quantum-number relations are group-theoretically fixed for these representations, so their rediscovery is a consistency check rather than evidence of new physics; the paper's value rests on the mass formulas themselves.

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

4 major / 5 minor

Summary. The manuscript applies a Kolmogorov-Arnold network with symbolic regression to the four baryon octet masses and four decuplet masses taken from PDG. It claims that, without imposing theoretical priors, the KAN autonomously recovers the Gell-Mann-Okubo (GMO) mass formula for both multiplets, extracts the associated SU(3) symmetry-breaking parameters, and reproduces the decuplet equal-spacing rule. The pipeline consists of data augmentation by resampling masses within quoted uncertainties while fixing I and Y, a 'Physics Filter' that restricts symbolic forms, and a series of algebraic rearrangements that convert raw KAN outputs into the final formulas Eqs. (9), (11), (12), and (17).

Significance. If the claims were substantiated, this would be a useful, interpretable-machine-learning demonstration in hadron spectroscopy. The linear equal-spacing fit in Eq. (11) is good (R^2 ≈ 0.9995), and the KAN framework is a plausible tool for fitting small data sets. However, the central 'autonomous discovery' claim is not supported by the manuscript. The octet result Eq. (17) is a generic four-parameter quadratic that does not have GMO structure, the decuplet result is obtained by manually altering coefficients after symbolic regression, and the analysis imposes an explicit target form via the Physics Filter. No code, data, or machine-checked derivations are provided. The paper therefore demonstrates polynomial interpolation followed by hand rearrangement, not the discovery of a physical law.

major comments (4)
  1. [§5 (Eqs. (6)–(9))] The decuplet derivation is not algebraically valid. Eq. (6) contains the I-dependent term -14.01I; rewriting the I^2 terms as I(I+1) gives 20.66 I(I+1) - 14.01I, but Eq. (7) drops the -14.01I term and shifts the constant by 14.01, which is only correct if I=1. Then Eq. (8) says 'adjust the coefficients' and replaces the fitted coefficients by entirely different numbers; Eq. (9) changes them again. These later coefficients are not derived from the KAN output or from data, but are chosen so that the final quadratic passes through the four decuplet masses. Any quadratic can do that. The explicit admission that coefficients are adjusted by hand directly contradicts the abstract's claim of autonomous recovery.
  2. [§6 (Eq. (17))] The octet result is a generic quadratic with four independent coefficients, not the GMO form. Eq. (17) reads M8 = -188.26Y8 + 39.14 I8(I8+1) - 17.03Y8^2 + 1115.50. The target form Eq. (1) requires the coefficient of I(I+1) to be -4 times the coefficient of Y^2. Here 39.14/(-17.03) ≈ -2.30, not -4. Thus the KAN output does not satisfy the GMO coefficient relation; it has been relabeled as GMO without imposing the required algebraic constraint. The preceding relation Eq. (12), Y8^2 = 1 - 4(0.5 - I8)^2, is an exact identity for the four octet points (Y^2 = 4I - 4I^2), so it is not an empirical discovery.
  3. [§2 (Model Selection / Physics Filter)] The claim of recovering GMO 'without imposing theoretical priors' is contradicted by the manuscript's own methodology. In the Model Selection subsection, symbolic regression is 'Constrained by the Physics Filter' and yields Eq. (1), which is exactly the GMO functional form with free parameters a, b, c. Similarly, the decuplet analysis inserts the relation Y10 = 2I10 - 2 from Eq. (3) as a constraint before deriving Eq. (9). The final formulas are therefore consequences of imposed priors and manually adjusted coefficients, not evidence that KAN autonomously discovered SU(3) symmetry breaking.
  4. [§2 (Data augmentation)] The data-augmentation procedure resamples the four masses in each multiplet within their experimental uncertainties while holding Y and I fixed. Because the uncertainties are small and the underlying function is smooth, these 1000 samples add no independent physical information; they are effectively noisy copies of the same four points. As a result, the four-parameter quadratics in Eqs. (9) and (17) are not tested against independent data, and the high R^2 values reflect interpolation rather than discovery. This weakens the statistical claims and the generalization statements in §3.
minor comments (5)
  1. [General] There are numerous typographical errors, e.g., 'efficiency', 'difficult', 'reaffirms', and 'the the relationship'. The English should be carefully edited.
  2. [Figures 2–5] The symbolic expressions printed in the figure boxes are inconsistent with the equations in the text (e.g., Fig. 2 shows F(Y,I) = -871.53(...)^2 + ..., while Eq. (4) gives a different expression). This makes it difficult to verify the training and symbolic-regression pipeline.
  3. [Eq. (15)] The text says 'temporarily substitute Y8^2 for 0.25Y8^2'; the intended identity is 0.25Y8^2 = I8 - I8^2 (since Y8^2 = 4I8 - 4I8^2). This should be stated correctly and the substitution should be made on I8^2, not on Y8^2.
  4. [Table I / References] The PDG reference [50] is from 2010; current 2024/2025 PDG values would be more appropriate for a modern analysis. The mass uncertainties for Σ are quoted as ±4 MeV, which is much larger than the current PDG value; this should be checked.
  5. [Reproducibility] The paper does not provide the trained KAN architectures, hyperparameters, data-augmentation realizations, or code. For a symbolic-regression claim, the exact fitting pipeline and random seeds should be documented.

Circularity Check

3 steps flagged

The claimed 'autonomous recovery' of the GMO formulas is imposed by the target form and hand-adjusted coefficients, so the output reduces to the input by construction.

specific steps
  1. fitted input called prediction [Section 2 (Model Selection), Eq. (1)]
    "Constrained by the Physics Filter, the regression avoided overly complex functional forms, yielding a physically interpretable expression: f (Y, I ) = a Y + b [ I(I + 1) − 1 4 Y 2 ] + c. (1)"

    The target expression is exactly the Gell-Mann–Okubo form. The regression is constrained to this family before seeing the data, so 'recovering' Eq. (1) is just fitting the parameters a, b, c. The abstract's claim of 'without imposing theoretical priors' is contradicted by this explicit Physics Filter; the discovery is built into the chosen functional form.

  2. self definitional [Section 5, Eqs. (6)–(9)]
    "To achieve a more compact expression, adjust the coefficients of terms involving Y10 and I10(I10 + 1): F (Y10, I10) = − 90.5Y 2 10 − 672.27Y10 + 689.73 + 348.03I10(I10 + 1). = − 253.5Y 2 10 − 1650.27Y10 − 614.27 + 1000.03I10(I10 + 1). To obtain a form similar to Eq. ( 1), and taking into account the constraint Y10 = 2 I10 − 2, we can simplify the above expression to the following form: M10 = − 5550.27Y10 − 903.5Y 2 10 + 3600.03I10(I10 + 1) − 5814.27."

    The transition from Eq. (6) to Eq. (7) drops the −14.01I term and changes constants; the text then explicitly says 'adjust the coefficients', replacing every fitted coefficient with arbitrary numbers. The final Eq. (9) is not algebraically derived from the KAN expression; it is hand-chosen to resemble Eq. (1). Any quadratic through four points can be reparameterized this way, so the reported 'recovered' decuplet GMO formula is a manual fit, not an autonomous KAN discovery.

  3. renaming known result [Section 6, Eq. (17)]
    "Finally, replace F (Y8, I8) with M8 to obtain the final form: M8 = − 188.26Y8 + 39.14I8(I8 + 1) − 17.03Y 2 8 + 1115.50. (17)"

    If this were the GMO form of Eq. (1), the coefficient of I(I+1) would have to be −4 times the coefficient of Y^2. Here 39.14 / (−17.03) ≈ −2.30, not −4. The expression is therefore a generic quadratic, not the Gell-Mann–Okubo formula. The paper labels it as the octet GMO formula solely because the known octet identity 0.25Y^2 = I − I^2 was used during manipulation; that identity is a definitional relation among quantum numbers, not a mass formula derived from data. The final formula is a renamed fit, so the claimed recovery is by construction.

full rationale

The paper's central claim is that KAN autonomously rediscovers the Gell-Mann–Okubo mass formulas from baryon masses 'without imposing theoretical priors'. The paper's own equations show the opposite. First, the regression is explicitly constrained by a 'Physics Filter' to the GMO functional form of Eq. (1), making the target the input. Second, for the decuplet, the raw KAN expression is carried through an algebraically invalid reduction that drops a term, and then the text openly states 'adjust the coefficients', replacing the data-derived numbers with arbitrary ones chosen to resemble Eq. (1); the final M10 is therefore a hand-imposed fit, not a KAN output. Third, for the octet, the final expression Eq. (17) does not satisfy the coefficient relation required by the GMO form (the I(I+1) and Y^2 coefficients should be in the ratio −4, but are ≈ −2.30), so the formula is a generic quadratic relabeled as GMO. In both multiplets, the 'recovered' formulas are either the pre-selected target form or a manually reparameterized fit of the same four training masses. This is not a case of self-citation circularity; it is a direct fitted-input-called-prediction structure: the output is equivalent, by construction, to the target form and hand-adjusted parameters. Consequently, the autonomous-discovery claim is unsupported by the derivation chain.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The central claim rests on four fitted coefficients per multiplet and on hand-imposed algebraic simplifications; the only data are eight PDG masses, so the symbolic results are essentially interpolations. The 'Physics Filter' and target GMO form are priors despite the abstract's assertion.

free parameters (5)
  • a, b, c in target GMO form Eq. (1) = not stated (fitted)
    The symbolic regression target includes three free parameters fitted to baryon masses; used as the assumed form.
  • Octet GMO coefficients (Eq. 17) = -188.26, 39.14, -17.03, 1115.50
    Four coefficients for four octet masses; exact interpolation rather than a predictive law.
  • Decuplet GMO coefficients (Eq. 9) = -5550.27, -903.5, 3600.03, -5814.27
    Coefficients produced by hand 'adjust[ing]' the earlier fit; not derived from data.
  • Equal-spacing fit M0, a (Eq. 11) = 1382.16, -146.71
    Linear fit to four decuplet masses; reproduces the known equal-spacing rule.
  • Data augmentation perturbation scale = experimental uncertainties from PDG
    Resampling 1000 samples within 1-sigma intervals; no new information, but treated as additional training data.
axioms (6)
  • standard math Kolmogorov-Arnold representation theorem supports KAN symbolic regression.
    Assumed valid; not proved in this paper.
  • domain assumption PDG isospin-averaged masses and quantum numbers (I, Y) for the octet and decuplet are correct.
    Used as the only input data; if assignments are wrong, the results fail.
  • ad hoc to paper A Physics Filter and target form Eq. (1) are consistent with 'no theoretical priors'.
    The paper claims no priors, yet imposes a physics filter and uses Eq. (1)'s structure as the regression target.
  • domain assumption Mass perturbations within experimental uncertainties preserve SU(3) symmetry structure.
    Data augmentation assumes this, but it is not demonstrated.
  • ad hoc to paper Algebraic coefficient adjustment in Eqs. (8)-(9) is a valid simplification.
    No rule governs the coefficient changes; they are chosen to make the expression look like GMO.
  • domain assumption The decuplet relation Y = 2I - 2 can be imposed as an input constraint.
    This relation is known for the decuplet but is inserted into the simplification, reducing the autonomy of the discovery.

pith-pipeline@v1.3.0-alltime-deepseek · 12807 in / 15705 out tokens · 145907 ms · 2026-08-03T12:47:02.251744+00:00 · methodology

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read the original abstract

Uncovering physical laws from experimental data is a fundamental goal of theoretical physics. In this work, we apply the spline-based, interpretable Kolmogorov-Arnold Network (KAN) to explore the algebraic structure underlying the baryon octet and decuplet mass spectra. Within a symbolic regression framework and without imposing theoretical priors, KAN autonomously recovers the classical Gell-Mann-Okubo mass relations and accurately extracts the associated SU(3) symmetry-breaking parameters. Compared to conventional fitting approaches, this method achieves comparable predictive accuracy while offering substantially improved interpretability and analytic transparency. Our results demonstrate the potential of KAN as a powerful tool for symbolic discovery in hadron physics and for bridging data-driven modeling with fundamental physical laws.

Figures

Figures reproduced from arXiv: 2601.01495 by Jian-Yao He, Wen Luo, Xiao-Yan Zhu, Xun Chen.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

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