Pith. sign in

REVIEW 4 major objections 5 minor 11 references

Evidence of Relationships Among Fundamental Constants of the Standard Model

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The paper claims that the six quark masses plus the W and Z masses can be expressed through a short chain of formulas starting from the electron mass and the CKM phase δ.

desk verdict An honest, transparent search for simple mass formulas with a genuinely new dimensional-analysis twist, but the statistical argument for 'not random' collapses under its own self-anchored null model. read the letter →

arxiv 2509.07713 v1 pith:J2R22A6E submitted 2025-09-09 hep-ph

S. V. Chekanov , H. Kjellerstrand This is my paper · ORCID
classification hep-ph PACS 11.15.-q12.15.-y12.38.-t14.65.-q14.70.-e06.20.Jr
keywords StandardModelfundamentalconstantsgeneticprogrammingsymbolicregressionquarkmassesbosondimensionalanalysismasshierarchy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the masses of the six quarks and the W and Z bosons are not isolated numbers: a genetic-programming search finds simple analytic formulas connecting them, and the simplest such system is a chain in which each mass is built from the previous one, starting from the electron mass and the CP-violating phase δ. The authors argue that the analytic complexity of these relations is far lower than what random smearing around the measured constants typically produces, with estimated probabilities around half a percent. If correct, the finding would mean the Standard Model's mass sector is not random but has a hidden hierarchical mathematical structure that a future theory might reproduce. The paper presents two full systems: one relating all eight masses directly to the Higgs mass, and a preferred chain that reduces the needed inputs to just m_e and δ.

What carries the argument

The load-bearing mechanism is genetic programming used as symbolic regression, upgraded with automatic dimensional analysis to keep every candidate expression in units of MeV. The search is steered by an analytic-rank measure: expressions with small rank (few operations and constants) and reuse of previously encountered constants are preferred. The central object is the hierarchical chain of equations, Eqs. (9)–(16), where each mass is a function of a lighter mass, and the quantitative evidence is the total complexity C = (total analytic rank) × (number of free parameters), whose low value for the SM is compared against random pseudo-experiments.

What would settle it

One concrete test would be to hold out the Z mass from the fit, re-derive the chain from the other seven masses, and check whether the predicted m_Z lands within the experimental 2 MeV uncertainty; the paper's own closure result suggests it would not.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the simplest symbolic expressions connecting SM masses form a hierarchical chain, Eqs. (9)–(16), in which m_u = m_e(π+1), and each subsequent mass is obtained from the preceding one, ending with m_Z = m_W δ cos(δ−1). This system has total analytic rank 118, lower than any system that ties each mass directly to the Higgs mass, and it expresses eight masses through only two input parameters. A second GP run, Eqs. (17)–(24), restores agreement with all measured masses after adjusting δ within its uncertainty, at total rank 126. Random pseudo-experiments produce complexity scores C whose distribution has mean 890 and width 125, while the SM's value is 565, giving a one-sided probability around 0.46%; for the chain alone, C_b = 118 gives 0.72%. The paper concludes that the GP relationships are unlikely to be numerical artifacts and carry a signature of an underlying unifying theory.

Load-bearing premise

The conclusion stands on the premise that the low complexity score is an unbiased measure of structure, even though that score is exactly what the search is designed to minimize, and the preferred chain failed to close (giving m_Z = 87827 MeV) until δ was re-adjusted in a second fit.

Editorial extensions

If this is right

  • If the central claim holds, the eight quark and boson masses would cease to be independent free parameters; the mass sector could be described by two inputs (m_e and δ), with m_H appearing in the alternative family of relations.
  • The specific functional forms — repeated appearance of π, (δ±1), and factors like (π+1/7) — would become targets that any future mass-generating theory would need to reproduce.
  • Higher-precision measurements of m_t, m_W, m_Z, or δ would provide direct tests: if the chain relations are genuine, updated values should keep the same low-rank structure; if not, the complexity score will jump.
  • The paper's finding that hierarchical chains are simpler than direct Higgs-proportional formulas suggests a structural principle: masses may be generated sequentially from lighter ones, which is a different organizing idea from typical GUT mass relations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The authors stop at saying the relations are unlikely artifacts; one step further, the chain predicts that improved measurements of δ or m_e will shift the whole mass spectrum along a one-parameter family, a pattern the paper does not quantify.
  • A direct extension would be a hold-out test: fit the chain to seven masses, predict the eighth, and repeat for each mass; the paper's own closure failure suggests at least one prediction will miss, which would reframe the claim as 'approximate structure' rather than exact relations.
  • Because θ13 dominates the Higgs-proportional solution and δ dominates the chain solution, the paper's results, if true, would give flavor model builders specific new empirical targets, but the authors do not develop that connection.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript applies genetic programming, augmented with dimensional analysis, to PDG values of SM masses and CKM parameters and searches for low-complexity analytic relations. It presents two systems: Eqs. (1)–(8), in which each quark and boson mass is proportional to the Higgs mass, and an alternative chain Eqs. (9)–(16), in which each mass is expressed through the preceding mass using only m_e and δ, followed by a revised chain Eqs. (17)–(24) obtained after a second GP run. The authors argue that the chain structure is preferred by the analytic rank, and they use a random-sampling test in Sec. 6 to claim that the observed complexity is unlikely to arise by chance. The conclusion is that the found relationships are unlikely to be numerical artifacts and may carry the signature of an underlying unified theory.

Significance. If established, the central claim would be striking: six quark masses and two boson masses reduced to two input parameters, with evidence for hidden hierarchical structure in the SM. The paper has genuine strengths: the search strategy is described in detail, dimensional consistency is enforced, the benchmark closure test in Sec. 5 is a reasonable sanity check, and the generated GP snippets are made available in a public repository. However, the statistical evidence is built on a null distribution that is conditioned on the SM values themselves and on a complexity measure that is exactly the objective optimized by the search. The preferred chain also fails a closure test, and the adjusted chain is a post-hoc refit. Accordingly, the paper does not support its headline conclusion, although it does not disprove the possibility that such relations exist.

major comments (4)
  1. [Section 4, Eqs. (9)–(16)] The preferred chain solution fails as a model of the Standard Model: the text explicitly states that using m_e = 0.510998 MeV and δ = 1.147 in Eqs. (9)–(16) gives m_Z = 87827 MeV, which is about 3.4 GeV below the PDG value m_Z = 91188.0 ± 2.0 MeV. The chain therefore does not reproduce the measured masses when evaluated consistently. Equations (17)–(24) were produced by a second GP run after this failure, with δ=1.1471 chosen to bring the values into agreement; their low total rank of 126 is thus a post-hoc selection and cannot serve as independent confirmation of the hierarchical structure. Since the paper's central conclusion relies on the improbability of obtaining Eqs. (9)–(16) by chance, the acknowledged closure failure removes the central candidate from the statistical argument.
  2. [Section 6, random sampling test and Fig. 1] The null model is not a null model over arbitrary mass spectra: each pseudo-experiment draws values uniformly from narrow bands [d_i - X_i ε_i, d_i + X_i ε_i] around the PDG constants, with X_i deliberately chosen to preserve the mass hierarchy and with uncertainties rescaled to the SM relative errors. Consequently the test asks whether the SM point has low complexity relative to its own perturbed neighborhood, not whether low-complexity relations arise by chance among unconstrained spectra. Moreover, the statistic C is the product of the analytic rank and the number of free parameters, which is precisely the objective the GP search is designed to minimize. The reported p-values of 0.46% and 0.72% are therefore not evidence that the discovered relations are unlikely to be numerical artifacts.
  3. [Section 6, incomplete systems and small sample] When a relation is missing, the authors compute C from an incomplete system by incrementing the number of free parameters by one and treating the missing relation as a hidden variable; this is an ad hoc adjustment rather than a pre-specified test statistic. The one-sided p-values are also obtained by fitting a normal distribution to only 15 pseudo-experiments, which is a fragile small-sample assumption. The text further states that random data were not used to reconstruct the connected system Eqs. (17)–(24), even though those equations are the ones shown as the 'adjusted' standard model point in Fig. 1b. Together, these issues mean the random-sampling test cannot bear the weight of the paper's conclusion.
  4. [Section 7, Eqs. (26)–(27)] The lepton sector is not brought within the claimed reduction: the electron mass requires the arbitrary 'refinement by precision reduction' of Sec. 2, and Eqs. (26)–(27) give m_μ = 105.659 MeV and m_τ = 1776.14 MeV, which the paper acknowledges are slightly outside the modern experimental uncertainties. Since the title and conclusion concern the fundamental constants of the Standard Model as a whole, this limitation should be stated in the abstract and conclusion, and the claim should be explicitly restricted to quark and boson masses unless the lepton relations are brought within uncertainties.
minor comments (5)
  1. [Table 1] The table contains presentation errors: the fine-structure constant is labeled 'PI' instead of π with a numerical value 3.14159, the α_s entry appears as '0 αS', and the PACS code 06.20.Jr is listed twice.
  2. [Section 2, item 4] The 'refinement by precision reduction' rule is described only qualitatively; the manuscript should state how much the precision is reduced and why that specific reduction is justified, since this rule directly affects the lepton-sector results.
  3. [Section 5] The benchmark test of Eq. (25) generates data from the same hierarchical structure that the GP is then asked to find, so its recovery of Eq. (25) is an internal consistency check; it does not quantify the false-positive rate for the Standard Model expressions.
  4. [Section 6, Fig. 1] The filled symbols in Fig. 1 include the 'adjusted' solution Eqs. (17)–(24), but no random test was run for that adjusted system; the figure and the surrounding text should make clear that this point has no corresponding null distribution.
  5. [Abstract and Section 8] The abstract states that the solution depends on only two input parameters, but the lepton masses in Section 7 require additional inputs and do not satisfy the experimental uncertainties; the scope of the claim should be stated consistently throughout.

Circularity Check

4 steps flagged · score 7.0 of 10

The Sec. 6 random-sampling null is built from the SM constants themselves and uses the GP objective as the test statistic, so the reported low p-values do not provide independent evidence against chance.

  1. fitted input called prediction [Section 6, 'Random sampling test' (null construction)]
    "We created 15 pseudo-experiments. Each experiment had 19 parameters with the same names as in Table 1. The values of the CKM constants and the masses of quarks and bosons were generated using a uniform random distribution in the range [d_i−X_i ε_i, d_i +X_i ε_i], where d_i are the nominal SM constants, ε_i are the corresponding experimental uncertainties, and X_i are scaling factors chosen to ensure sufficient random smearing."

    The null distribution is built by perturbing the exact PDG values d_i that were used as GP inputs for Eqs. (1)–(24). Comparing the SM point with its own narrow smeared neighborhood tests local optimality, not whether low-complexity relations arise by chance across the space of independent mass spectra. The paper further states that the smearing 'preserves the hierarchical structure of masses,' so the chain structure offered as evidence is baked into the null. The resulting p-values (0.46% and 0.72%) therefore do not support the conclusion that the relations are unlikely to be numerical artifacts.

  2. self definitional [Section 6, Fig. 1, and Section 2, search strategy criterion 3]
    "The Y-axis shows the total complexity of the system of equations, C, defined as the product of the total analytic rank and the number of free parameters. ... Simplicity criterion: Among all possible solutions, we choose the ones that are the simplest, i.e. with the smallest rank defined in [5]."

    C is the same objective the GP search is designed to minimize: the search selects the smallest analytic rank and smallest number of free parameters. A low C for the SM values is therefore partly guaranteed by the fitting procedure, and using C to compare the SM point with random perturbations compares a fitted minimum against non-fitted neighbors. The test statistic and the search objective are identical by construction, so the comparison does not constitute independent statistical evidence.

2 more flagged steps
  1. fitted input called prediction [Section 4, 'Alternative solution', adjusted Eqs. (17)–(24)]
    "The solution for the above problem can be the following. Since we already know how the structure of the relations should look, one can re-run the GP method a second time, when using the masses as input for each relation in the chain Eqs. (9)-(16), and vary the input parameter δ within its allowed experimental range."

    The preferred chain Eqs. (9)–(16) fails its own closure test: iterating the chain gives m_Z = 87827 MeV, far from the PDG value. The adjusted set Eqs. (17)–(24) was produced by a second GP run after this failure, with δ tuned to 1.1471 within its uncertainty. The low complexity of the adjusted set is therefore a post-hoc fit, not a prediction, and any statistical comparison that treats these adjusted equations as the SM point inherits this selection effect.

  2. fitted input called prediction [Section 2, search strategy step 4, 'Refinement by precision reduction']
    "Refinement by precision reduction: If no acceptable solution is found for a particular mass, we reduce the precision required for that mass and repeat the GP search."

    This search rule explicitly permits loosening the target tolerance until a simple expression exists. The paper also notes that the electron mass precision 'should be reduced' because the original precision produced no acceptable GP solutions. The reported relations are therefore guaranteed to agree with the data only to the loosened precision by construction, which weakens the claim that their closeness to the PDG values is an independent discovery.

full rationale

The central statistical claim, that the GP relations are 'unlikely to be numerical artifacts,' rests on the Section 6 random-sampling test. That test is circular in two linked ways. First, the null is generated by smearing the nominal SM constants within narrow bands (with scaling factors X_i from 20 to 180) that explicitly preserve the mass hierarchy, so the chain structure under test is built into the null. Second, the test statistic C is defined as the product of analytic rank and number of free parameters, which is exactly the objective the GP search minimizes; comparing the SM point with its own perturbed neighborhood therefore tests local optimality of a fitted point rather than chance occurrence among independent constants. This is compounded by post-hoc selection: the preferred chain Eqs. (9)–(16) fails closure, giving m_Z = 87827 MeV, and the adjusted Eqs. (17)–(24) are a second fit made after that failure, with δ tuned to 1.1471. The precision-reduction rule in Section 2 similarly allows the search to continue until a fit is found, so the final expressions are fitted inputs renamed as relationships. The toy-model benchmark in Section 5 is a genuinely external check that the GP method can recover a known relation, and the paper contains no single equation that is simply a restatement of its input by symbolic identity. Nevertheless, the paper's main evidential pillar, the p-value-based 'unlikely by chance' conclusion, is not independent of the fitted values and the search objective. The result is therefore substantially circular, though not equivalent to the input by definition.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The central claim rests on GP-generated fits, a hand-designed ranking function, a last-resort precision reduction, and a null model built from the very values being tested. No new physical entity is introduced, but the 'underlying theory' invoked in the conclusion is an invented explanatory construct without independent evidence.

free parameters (3)
  • Exponents and coefficients in Eqs. (1)-(8) = 1/3, 3/2, 1/5, 1/6, 9 theta_13(1-theta_13), (delta+delta/5), arctan((10-delta)^2), delta cos(delta-1)
    These are GP-discovered forms that reproduce each PDG mass within uncertainty; the specific numeric coefficients and exponents are outputs of the fit.
  • Exponents and coefficients in Eqs. (17)-(24) = fourth root of delta, (delta+1), 9 tan(delta), (4 pi+1), (pi+1/7), pi(pi+10), 1/(delta+1), delta cos(delta-1)
    This chain is a second fit made after the first chain failed closure; the new functional forms are introduced specifically to force agreement with measured masses.
  • Lepton relation denominators in Eqs. (26)-(27) = alpha^{-1} sqrt(5)+alpha^{-1} and 1/alpha^{-1}+1/(alpha^{-1}+8)
    GP output approximating m_mu and m_tau; not derived from any principle and the resulting masses are slightly outside experimental uncertainties.
assumptions (4)
  • ad hoc to paper The analytic rank r defined in [5] is a valid measure of equation simplicity and theoretical relevance.
    All selection criteria (Step 3 and Section 5) rest on this ranking function; no independent justification is given beyond the authors' prior paper.
  • domain assumption All acceptable solutions must vanish as m_H -> 0.
    Derived from SM Yukawa coupling arguments, but used as a hard filter to exclude whole classes of solutions, limiting the search space.
  • ad hoc to paper Reducing the precision requirement for a mass when no solution is found is acceptable.
    Step 4 of the search strategy permits the algorithm to fit to looser tolerances, which weakens the claim that the relations match the measured values.
  • ad hoc to paper Pseudo-experiments generated by smearing SM values with hand-scaled uncertainties form a valid null distribution.
    The null model is centered on the SM values and uses scaling factors Xi chosen without a formal justification, so the resulting p-values are not a calibrated hypothesis test.
invented entities (1)
  • An unspecified underlying dynamical mechanism or symmetry pattern
    purpose: Explains why the found numerical relations exist
    The conclusion invokes a hidden mechanism that is not defined, not independently measured, and has no falsifiable handle outside this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Evidence of Relationships Among Fundamental Constants of the Standard Model." pith.science (2026). https://pith.science/paper/J2R22A6E

@misc{pith2026250907713,
  author       = {Pith},
  title        = {Pith review of: Evidence of Relationships Among Fundamental Constants of the Standard Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J2R22A6E}},
  note         = {Machine review of arXiv:2509.07713}
}
read the original abstract

This paper presents an approach to reducing the number of fundamental parameters in the Standard Model (SM) using genetic programming, a machine learning technique based on evolutionary algorithms. We outline the core principles of our method and identify the simplest analytic relationships among SM parameters. Our results suggest that the SM parameters associated with quark and boson masses are not randomly distributed, but instead follow a hierarchical structure within a high-dimensional functional space. The found analytic solution depends on only two input parameters, representing the simplest mathematical model that could provide a foundation for developing a future theoretical framework to address the SM.

Figures

Figures reproduced from arXiv: 2509.07713 by the authors.

Figure 1
Figure 1. The summary of random variations of the SM parameters around their nominal values (open symbols). The [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

    work page

  2. [2]

    Navas, et al., Review of particle physics , Phys

    Particle Data Group Collaboration, S. Navas, et al., Review of particle physics , Phys. Rev. D 110 (3) (2024) 030001. http://dx.doi.org/10.1103/PhysRevD.110.030001 doi:10.1103/PhysRevD.110.030001

    doi:10.1103/physrevd.110.030001 2024

  3. [3]

    Weinberg, Dreams of a Final Theory , Pantheon Books, 1994

    S. Weinberg, Dreams of a Final Theory , Pantheon Books, 1994

    work page 1994

  4. [4]

    H. B. Nielsen, C. Surlykke, S. E. Rugh, Seeking inspiration from the standard model in order to go beyond it , in: 4th Hellenic School on Elementary Particle Physics , 1994, pp. 476--501. http://arxiv.org/abs/hep-th/9407012 arXiv:hep-th/9407012

    work page Pith review arXiv 1994

  5. [5]

    C. D. Froggatt, H. B. Nielsen, http://dx.doi.org/10.1080/0142241032000156559 Trying to understand the standard model parameters , Surveys in High Energy Physics 18 (1–4) (2003) 55–75. http://dx.doi.org/10.1080/0142241032000156559 doi:10.1080/0142241032000156559 . ://dx.doi.org/10.1080/0142241032000156559

    work page doi:10.1080/0142241032000156559 2003

  6. [6]

    S. V. Chekanov, H. Kjellerstrand, Discovering the underlying analytic structure within Standard Model constants using artificial intelligence (2025). http://arxiv.org/abs/2507.00225 arXiv:2507.00225

    arXiv 2025

  7. [7]

    J. R. Koza, Genetic Programming, MIT Press, 1992

    work page 1992

  8. [8]

    Weinberg, A model of leptons https://link.aps.org/doi/10.1103/PhysRevLett.19.1264, Phys

    S. Weinberg, A model of leptons https://link.aps.org/doi/10.1103/PhysRevLett.19.1264, Phys. Rev. Lett. 19 (1967) 1264--1266. http://dx.doi.org/10.1103/PhysRevLett.19.1264 doi:10.1103/PhysRevLett.19.1264 . ://link.aps.org/doi/10.1103/PhysRevLett.19.1264

    doi:10.1103/physrevlett.19.1264 1967

Show all 11 references
  1. [9]

    Raby, Grand Unified Theories , in: 2nd World Summit: Physics Beyond the Standard Model , 2006

    S. Raby, Grand Unified Theories , in: 2nd World Summit: Physics Beyond the Standard Model , 2006. http://arxiv.org/abs/hep-ph/0608183 arXiv:hep-ph/0608183

    2006 arXiv

  2. [10]

    N.-F. Zhou, H. Kjellerstrand, J. Fruhman, Picat programming language, https://picat-lang.org, [Online; accessed ] (2013)

    2013

  3. [11]

    S. V. Chekanov, H. Kjellerstrand, GC4PhysicalConstants - Genetic computing for physical constants , https://github.com/chekanov/GC4PhysicalConstants, GitHub Repository (2025)

    2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.