Pith. sign in

REVIEW 4 minor 43 references

Rare multi-nucleon decays with the full data sets of the Majorana Demonstrator

T0 review · 0 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Using the full Majorana Demonstrator dataset, this paper sets the most stringent partial-lifetime limits to date for tri-nucleon decays in germanium, with 1.83 x 10^26 years for the ppp and ppn visible modes.

desk verdict Solid, conservative limits on rare multi-nucleon decays from the full Majorana dataset; the results are new and the analysis holds up, though the technique is incremental. read the letter →

arxiv 2412.16047 v4 pith:IGNNCSMR submitted 2024-12-20 nucl-ex

I.J. Arnquist , F.T. Avignone III , A.S. Barabash , E. Blalock , B. Bos , M. Busch , Y.-D. Chan , J.R. Chapman
show 34 more authors
C.D. Christofferson P.-H. Chu C. Cuesta J.A. Detwiler Yu. Efremenko H. Ejiri S.R. Elliott N. Fuad G.K. Giovanetti M.P. Green J. Gruszko I.S. Guinn V.E. Guiseppe R. Henning E.W. Hoppe R.T. Kouzes A. Li R. Massarczyk S.J. Meijer L.S. Paudel W. Pettus A.W.P. Poon D.C. Radford A.L. Reine K. Rielage D.C. Schaper S.J. Schleich D. Tedeschi R.L. Varner S. Vasilyev S.L. Watkins J.F. Wilkerson C. Wiseman C.-H. Yu
This is my paper · ORCID
classification nucl-ex PACS 23.40-s23.40.Bw14.60.Pq27.50.+j
keywords tri-nucleondecaybaryonnumberviolationMajoranaDemonstratorgermanium-76double-betapulse-shapediscriminationpartiallifetimelimitrarenucleardecays
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

The paper searches for baryon-number-violating decays in germanium isotopes using the complete dataset of the Majorana Demonstrator, an ultra-low-background experiment originally built for neutrinoless double-$\beta$ decay. It sets new lower limits on partial lifetimes for three-nucleon (ΔB=3) and two-nucleon (ΔB=2) decays, the most stringent being $1.83 \times 10^{26}$ years for $^{76}$Ge(ppp)$\rightarrow^{73}$Cu e$^+\pi^+\pi^+$ and $^{76}$Ge(ppn)$\rightarrow^{73}$Zn e$^+\pi^+$. These numbers matter because baryon-number violation is one of the Sakharov conditions for the matter-antimatter asymmetry; tri-nucleon decay is a specific Standard Model-allowed ΔB=3 process. The limits are derived from a two-step signature: a high-energy saturated event from the decay, followed by the $\beta$ decay of the unstable daughter isotope. No candidate events beyond expected background were found, so the results are lower bounds on the decay lifetimes.

What carries the argument

The search uses a two-step 'saturated event + daughter $\beta$ decay' signature. For semi-inclusive visible modes, the first event is a saturated energy deposit (above the ~10 MeV dynamic range) from the positron and pions produced in the multi-nucleon decay, which occurs inside a germanium crystal and is not tagged by the muon veto; the second is the $\beta$ decay of the unstable daughter within 5 half-lives and with energy between 100 keV and the Q value. For fully inclusive modes, both steps are $\beta$ decays of the daughter and granddaughter (e.g., $^{73}$Cu$\rightarrow^{73}$Zn$\rightarrow^{73}$Ga) with energies above 2 MeV, and pulse-shape discrimination (AvsE, DCR, LQ) is applied to suppress $\gamma$, $\alpha$, and surface backgrounds. Each step's efficiency ($\epsilon_0$, $\epsilon_\tau$, $\epsilon_{E1}$, $\epsilon_{E2}$, $\epsilon_{PSD}$) is computed from Monte Carlo simulation and calibration data, and combined with the isotope exposure $NT$ to obtain the effective exposure $NT\epsilon_{\mathrm{Tot}}$; the partial-lifetime limit is $\tau > NT\epsilon_{\mathrm{Tot}}/S$, with $S$ the Feldman-Cousins 90% upper limit on the observed signal count.

What would settle it

Measure the saturated-event efficiency with tagged high-energy events in the actual detector; if it falls below the assumed >99%, the limits must be reduced proportionally. In a larger dataset, test whether candidate events' time differences follow the daughter half-lives (e.g., 4.2 s for $^{73}$Cu) or stay flat in time; a flat time distribution would confirm the one candidate here is background.

Watch

Extended reading notes

Core claim

Using the full 64.5 kg yr enriched and 27.4 kg yr natural germanium exposure of the Majorana Demonstrator, the collaboration establishes the most stringent partial-lifetime limits to date for tri-nucleon decays in $^{76}$Ge with visible products: $\tau > 1.83 \times 10^{26}$ yr (90% CL) for $^{76}$Ge(ppp)$\rightarrow^{73}$Cu e$^+\pi^+\pi^+$ and $^{76}$Ge(ppn)$\rightarrow^{73}$Zn e$^+\pi^+$. For the fully inclusive tri-proton channel $^{76}$Ge(ppp)$\rightarrow^{73}$Cu+X the limit is $\tau > 2.1 \times 10^{25}$ yr. The analysis updates the earlier Demonstrator search by applying additional pulse-shape discrimination cuts (AvsE and LQ), which removed the two previously reported candidate sequences. One new candidate sequence in the semi-inclusive search is consistent with the expected random-coincidence background of about 0.017 counts. The paper also reports updated limits for other ΔB=2 and ΔB=3 modes across $^{74}$Ge, $^{73}$Ge, $^{72}$Ge, and $^{70}$Ge.

Load-bearing premise

The quoted limits scale linearly with Monte Carlo-derived detection efficiencies, especially the saturated-event efficiency assumed to be above 99%, so a simulation error in those efficiencies changes all quoted lifetimes by the same factor.

Editorial extensions

If this is right

  • The new limits for $^{76}$Ge(ppp) and $^{76}$Ge(ppn) visible tri-nucleon decays are the most stringent for these channels, improving on the previous Demonstrator result.
  • The fully inclusive tri-proton limit of $2.1 \times 10^{25}$ yr is weaker than GERDA's $1.2 \times 10^{26}$ yr because the Demonstrator sees background events from cosmogenic $^{73m}$Ge decays, which reduces sensitivity to that mode.
  • With no signal candidates, the observation is consistent with background; the single candidate in the semi-inclusive search is likely a random coincidence (expected 0.017 counts).
  • The limits constrain ΔB=3 baryon-number-violating processes in germanium, complementing proton-decay and dinucleon-decay bounds from other experiments.
  • Future experiments like LEGEND-1000 with ~$10^4$ kg yr exposure are expected to improve these limits by two orders of magnitude, reaching ~$10^{28}$ yr.

Reading between the lines

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

  • If the new limits are correct, they begin to probe the effective scale of the dimension-15 operator responsible for ΔB=3 transitions; a rough estimate would place that scale far above the TeV, implying that any observable signal in near-future experiments would require a specific UV completion rather than generic new physics.
  • The saturated-event trigger technique is transferable: any large-volume, low-background detector with a dynamic energy range can search for multi-nucleon decays by looking for a high-energy deposit followed by a characteristic beta-decay chain, so similar searches could be performed with other isotopes in existing double-beta experiments.
  • The fact that the fully inclusive channel is background-limited by cosmogenic $^{73m}$Ge in the Demonstrator suggests that next-generation germanium detectors should minimize surface exposure during fabrication or add a tag for the $^{73m}$Ge two-step waveform to recover sensitivity in that channel.
  • A testable prediction: if the one semi-inclusive candidate were signal, its time difference would need to match a daughter half-life (e.g., ~487 s for $^{74}$Ga); the observed 1712 s makes the signal hypothesis unlikely, and additional events in the LEGEND dataset should show a flat time distribution if the background interpretation is correct.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The Majorana Demonstrator reports limits on multi-nucleon decays in germanium isotopes using its full data set (64.5 kg yr enriched, 27.4 kg yr natural). For the two headline semi-inclusive visible channels, 76Ge(ppp)→73Cu e+π+π+ and 76Ge(ppn)→73Zn e+π+, zero candidate events are observed, leading to 90% confidence level partial-lifetime limits of 1.83×10^26 yr each. The fully inclusive tri-proton decay of 76Ge is bounded at 2.1×10^25 yr, and additional ΔB=2 and ΔB=3 channels are tabulated. The analysis uses a saturated-event trigger followed by a daughter-decay search, with efficiencies from MaGe simulations and calibration data. One candidate event in semi-inclusive channels is conservatively treated as a potential signal in each possible channel.

Significance. The paper presents the most stringent existing limits on the two specific semi-inclusive tri-nucleon decay modes of 76Ge, improving on the previous Demonstrator result with the full exposure. The analysis is straightforward and internally consistent: the limits follow from zero observed candidates, near-unity total efficiencies (~0.96) for the headline modes, and Feldman-Cousins statistics. The single observed candidate is handled conservatively, and the systematic uncertainties are itemized. The paper is also notable for its explicit disclosure of limitations, including the non-uniqueness of 73Cu production for ΔB=3 and the reduced sensitivity in the fully inclusive channel due to cosmogenic 73mGe backgrounds. The result is a solid experimental constraint that will be useful for baryon-number-violation studies.

minor comments (4)
  1. [Eq. (3) and Table I] The saturated-trigger efficiency ϵ0 is quoted as a single value (0.998 or 0.999) across all three detector types (PPC, ICPC, BEGe). Since the saturation response may depend on detector geometry and electronics, the authors should either provide per-detector ϵ0 values or justify the uniformity with calibration data.
  2. [Systematic uncertainties paragraph] The systematic uncertainties (exposure <2%, DC <0.1%, PSD <5%, simulation 2%, statistical <1%) are listed but not propagated into the quoted limits; the authors state they are ignored because the limit is statistically limited. A sentence quantifying the combined systematic (e.g., <6%) and its negligible impact on the 90% CL limit would make the treatment explicit.
  3. [Fully inclusive mode description] The paper does not provide a quantitative background estimate for the fully inclusive sequential beta-decay search; the one observed sequence is attributed to 208Tl, but a background-rate estimate for this search would strengthen the presentation.
  4. [Eq. (1) and abstract] The symbol τ is defined as the partial lifetime limit in Eq. (1), but the text does not explicitly state the relation to the half-life (T1/2 = τ ln 2). A brief clarification would avoid ambiguity for readers comparing with half-life limits in other experiments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: limits follow from observed counts, exposure, and independently determined efficiencies.

full rationale

The central limits are derived from zero observed candidate events, the active isotope exposures, and efficiency factors that come from Geant4-based Monte Carlo simulations and calibration data in prior collaboration papers [14, 35, 40]. These efficiencies are not fitted to the target decay limits; they are independently determined from detector-response simulation and calibration sources. The paper's Eq. (1), tau > NT*epsilonTot/S, is the standard partial-lifetime-limit formula, and Eqs. (3) and (4) merely multiply independent efficiency components. Footnote [41] explicitly describes how previous candidate sequences from Ref. [14] were re-evaluated under new pulse-shape cuts and treated as observed events, which is a data-reanalysis step, not a parameter fit masquerading as a prediction. The self-citations to Refs. [14] and [35] provide detector parameters, simulation frameworks, and PSD efficiencies; they do not inject the claimed result into the derivation. The comparison with GERDA [15] and the acknowledged limitation that observing 73Cu does not uniquely imply Delta B = 3 are external benchmarks and honest caveats rather than circular reasoning. No equation in the paper reduces a headline limit to an input by construction, and no load-bearing claim is justified solely by a self-citation.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

This is an experimental search with no fitted free parameters. The efficiencies and exposures are measured quantities with uncertainties from calibration and Monte Carlo, not parameters tuned to produce the result. The analysis assumes the theoretical decay modes of Babu et al. and uses published nuclear data; these are listed as axioms. No new particles, forces, or conserved quantities are introduced.

assumptions (3)
  • domain assumption The Z6-symmetric Standard Model extension permits Delta B = 3 tri-nucleon decay modes ppp -> e+ pi+ pi+, ppn -> e+ pi+, pnn -> e+ pi0, and nnn -> neutrino pi0 with the assumed phase-space distributions.
    Motivates the search channels; taken from Babu et al. Ref. [8], not derived or tested in this paper.
  • domain assumption The daughter and granddaughter isotopes and their half-lives and Q values (Table II, ENSDF) are correct.
    The time windows and energy cuts are set from these values; a wrong half-life would change epsilonTau and the candidate selection windows.
  • domain assumption The detection efficiency of the saturated trigger plus daughter sequence is correctly captured by Geant4-based MaGe Monte Carlo simulations and calibration data.
    The central limits scale linearly with the reported efficiencies, so modeling errors would directly shift the limits.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rare multi-nucleon decays with the full data sets of the Majorana Demonstrator." pith.science (2026). https://pith.science/paper/IGNNCSMR

@misc{pith2026241216047,
  author       = {Pith},
  title        = {Pith review of: Rare multi-nucleon decays with the full data sets of the Majorana Demonstrator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGNNCSMR}},
  note         = {Machine review of arXiv:2412.16047}
}
abstract

The Majorana Demonstrator was an ultra-low-background experiment designed for neutrinoless double-beta decay ($0\nu\beta\beta$) investigation in $^{76}$Ge. Located at the Sanford Underground Research Facility in Lead, South Dakota, the Demonstrator utilized modular high-purity Ge detector arrays within shielded vacuum cryostats, operating deep underground. The arrays, with a capacity of up to 40.4 kg (27.2 kg enriched to $\sim 88\%$ in $^{76}$Ge), have accumulated the full data set, totaling 64.5 kg yr of enriched active exposure and 27.4 kg yr of exposure for natural detectors. Our updated search improves previously explored three-nucleon decay modes in Ge isotopes, setting new partial lifetime limits of $1.83\times10^{26}$ years (90\% confidence level) for $^{76}$Ge($ppp$) $\rightarrow$ $^{73}$Cu e$^+\pi^+\pi^+$ and $^{76}$Ge($ppn$) $\rightarrow$ $^{73}$Zn e$^+\pi^+$. The partial lifetime limit for the fully inclusive tri-proton decay mode of $^{76}$Ge is found to be $2.1\times10^{25}$ yr. Furthermore, we have updated limits for corresponding multi-nucleon decays.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 39 canonical work pages

  1. [1]

    A. D. Sakharov. Violation of CP Invariance, C asymme- try, and baryon asymmetry of the universe, Pisma Zh. Eksp. T eor. Fiz., 5:32–35, 1967. [Sov. Phys. Usp. 34, 392 (1991)]

    work page 1991

  2. [2]

    K. S. Babu et al. , Baryon Number Violation, arXiv:1311.5285

    arXiv

  3. [3]

    P. S. B. Dev et al., Searches for Baryon Number Violation in Neutrino Experiments: A White Paper, J. Phys. G: Nucl. Part. Phys., 51, 033001 (2024)

    work page 2024

  4. [4]

    Pavel Fileviez Perez et al., On Baryon and Lepton Num- ber Violation, arXiv:2208.00010

    arXiv

  5. [5]

    Abe et al., Search for proton decay via p → e+π0 and p → µ+π0 in 0.31 megaton ·years exposure of the Super- Kamiokande water Cherenkov detector, Phys

    K. Abe et al., Search for proton decay via p → e+π0 and p → µ+π0 in 0.31 megaton ·years exposure of the Super- Kamiokande water Cherenkov detector, Phys. Rev. D, 95(1):012004, 2017

    work page 2017

  6. [6]

    Takhistov et al

    V. Takhistov et al. , Search for Nucleon and Dinucleon Decays with an Invisible Particle and a Charged Lepton in the Final State at the Super-Kamiokande Experiment, Phys. Rev. Lett., 115(12):121803, 2015

    work page 2015

  7. [7]

    Gustafson et al., Search for dinucleon decay into pions at Super-Kamiokande, Phys

    J. Gustafson et al., Search for dinucleon decay into pions at Super-Kamiokande, Phys. Rev. D, 91(7):072009, 2015

    work page 2015

  8. [8]

    K. S. Babu, Ilia Gogoladze, and Kai Wang. Gauged baryon parity and nucleon stability, Phys. Lett. B, 570:32–38, 2003

    work page 2003

Show all 43 references
  1. [9]

    Bakker, A.I

    B.L.G. Bakker, A.I. Veselov, and M.A. Zubkov. A hidden symmetry in the standard model, Physics Letters B, 583(3–4):379–382, March 2004

    2004

  2. [10]

    Bernabei et al

    R. Bernabei et al. , Search for rare processes with DAMA/LXe experiment at Gran Sasso, Eur. Phys. J. A, 27(S1):35–41, 2006

    2006

  3. [11]

    J. B. Albert et al., Search for nucleon decays with EXO- 200, Phys. Rev. D, 97(7):072007, 2018

    2018

  4. [12]

    Hazama, H

    R. Hazama, H. Ejiri, K. Fushimi, and H. Ohsumi. Limits on single- and multinucleon decays in I-127 by inclusive measurements of nuclear gamma and x rays,Phys. Rev. C, 49:2407–2412, 1994

    1994

  5. [13]

    Inclusive Nu- cleon Decay Searches as a Frontier of Baryon Number Violation, Phys

    Julian Heeck and Volodymyr Takhistov. Inclusive Nu- cleon Decay Searches as a Frontier of Baryon Number Violation, Phys. Rev. D, 101(1):015005, 2020

    2020

  6. [14]

    S. I. Alvis et al. , Search for trinucleon decay in the Ma- jorana Demonstrator, Phys. Rev. D, 99(7):072004, 2019

    2019

  7. [15]

    Agostini et al

    M. Agostini et al. , Search for tri-nucleon decays of 76Ge in GERDA, Eur. Phys. J. C, 83(9):778, 2023

    2023

  8. [16]

    I. J. Arnquist et al. , Signatures of muonic activation in the Majorana Demonstrator, Phys. Rev. C, 105(1):014617, 2022

    2022

  9. [17]

    K. S. Babu et al. , |∆B| = 2: A State of the Field, and Looking Forward–A brief status report of theoretical and experimental physics opportunities, arXiv:2010.02299

    2010 arXiv

  10. [18]

    Abgrall et al., The Majorana DemonstratorNeu- trinoless Double-Beta Decay Experiment, Adv

    N. Abgrall et al., The Majorana DemonstratorNeu- trinoless Double-Beta Decay Experiment, Adv. High Energy Phys., 2014:365432, 2014

    2014

  11. [19]

    Abgrall et al

    N. Abgrall et al. , The Majorana Demonstrator experiment’s construction, commissioning, and perfor- mance, arXiv:2501.02060

    arXiv

  12. [20]

    The Sanford Underground Research Facility, arXiv.2203.08293

    Jaret Heise. The Sanford Underground Research Facility, arXiv.2203.08293

    arXiv

  13. [21]

    P. S. Barbeau, J. I. Collar, and O. Tench. Large-Mass Ultra-Low Noise Germanium Detectors: Performance and Applications in Neutrino and Astroparticle Physics, J. Cosmol. Astropart. Phys., 09:009, 2007

    2007

  14. [22]

    Abgrall et al., Muon Flux Measurements at the Davis Campus of the Sanford Underground Research Facility with the Majorana DemonstratorVeto System, As- tropart

    N. Abgrall et al., Muon Flux Measurements at the Davis Campus of the Sanford Underground Research Facility with the Majorana DemonstratorVeto System, As- tropart. Phys., 93:70–75, 2017

    2017

  15. [23]

    E. W. Hoppe, C. E. Aalseth, O. T. Farmer, T. W. Hoss- bach, M. Liezers, H. S. Miley, N. R. Overman, and J. H. 7 Reeves. Reduction of radioactive backgrounds in elec- troformed copper for ultra-sensitive radiation detectors, Nucl. Instrum. Methods Phys. Res. A, 764:116– 121, 2014

    2014

  16. [24]

    Abgrall et al

    N. Abgrall et al. , The Majorana Demonstrator ra- dioassay program, Nucl. Instrum. Methods Phys. Res. A, 828:22–36, 2016

    2016

  17. [25]

    Abgrall et al., The Majorana Demonstratorread- out electronics system, J

    N. Abgrall et al., The Majorana Demonstratorread- out electronics system, J. Inst., 17(05):T05003, 2022

    2022

  18. [26]

    S. I. Alvis et al. , Multisite event discrimination for the Majorana Demonstrator, Phys. Rev. C , 99(6):065501, 2019

    2019

  19. [27]

    I. J. Arnquist et al. , α-event characterization and rejec- tion in point-contact HPGe detectors, Eur. Phys. J. C, 82(3):226, 2022

    2022

  20. [28]

    Abgrall et al

    N. Abgrall et al. , ADC Nonlinearity Correction for the Majorana Demonstrator, IEEE T rans. Nucl. Sci., 68(3):359–367, 2021

    2021

  21. [29]

    I. J. Arnquist et al. , Charge trapping correction and energy performance of the Majorana Demonstrator, Phys. Rev. C, 107(4):045503, 2023

    2023

  22. [30]

    Abgrall et al

    N. Abgrall et al. , The Majorana Demonstratorcali- bration system, Nucl. Instrum. Methods Phys. Res. A, 872:16–22, 2017

    2017

  23. [31]

    I. J. Arnquist et al. , Energy calibration of germanium detectors for the Majorana Demonstrator, J Inst., 18(09):P09023, 2023

    2023

  24. [32]

    P. N. Luke, F. S. Goulding, N. W. Madden, and R.H. Pehl, Low capacitance large volume shaped-field ger- manium detector, IEEE T ransactions on Nuclear Science, 36(1):926–930, 1989

    1989

  25. [33]

    R. J. Cooper, D. C. Radford, P. A. Hausladen, and K. Lagergren, A novel HPGe detector for gamma- ray tracking and imaging, Nucl. Instrum. Methods Phys. Res. A, 665:25–32, 2011

    2011

  26. [34]

    Canberra Industries Inc. (now Mirion Technologies), 800 Research Parkway Meriden, CT 06450, informa- tion available at https://www.mirion.com/products/ bege-broad-energy-germanium-detectors , 2023, ac- cessed on October 27, 2023

    2023

  27. [35]

    I. J. Arnquist et al. , Final Result of the Majorana Demonstrator’s Search for Neutrinoless Double- β De- cay in 76Ge, Phys. Rev. Lett., 130(6):062501, 2023

    2023

  28. [36]

    The semi-inclusive visible mode is called the decay- specific mode in Ref. [14]

  29. [37]

    The fully inclusive mode is called the invisible mode in Ref. [14]

  30. [38]

    Feldman and Robert D

    Gary J. Feldman and Robert D. Cousins, Unified ap- proach to the classical statistical analysis of small signals, Phys. Rev. D, 57:3873–3889, 1998

    1998

  31. [39]

    nndc.bnl.gov, 2023, accessed on October 27, 2023

    ENSDF Database, version available at http://www. nndc.bnl.gov, 2023, accessed on October 27, 2023

    2023

  32. [40]

    , MaGe-a Geant4-based Monte Carlo Application Framework for Low-background Ger- manium Experiments

    Melissa Boswell et al. , MaGe-a Geant4-based Monte Carlo Application Framework for Low-background Ger- manium Experiments. IEEE T rans. Nucl. Sci., 58:1212–1220, 2011

    2011

  33. [41]

    There were two sequences (Sequence 1 and 2) as shown in Table II of Ref. [14]. In this study, Sequence 1 was excluded due to AvsE and LQ cuts, while Sequence 2 was excluded based on the AvsE cut. The new case in- troduced in this study is part of the updated dataset

  34. [42]

    Abgrall et al

    N. Abgrall et al. , The Large Enriched Germanium Ex- periment for Neutrinoless ββ Decay: LEGEND-1000 Pre- conceptual Design Report, 7 2021. arxiv:2107.11462

    2021 arXiv

  35. [43]

    I. J. Arnquist et al. , Majorana DemonstratorData Release for AI/ML Applications, arxiv:2308.10856, zen- odo.8257027

    arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.