Partition function zeros for the Blume-Capel model on a complete graph

Bertrand Berche; Mariana Krasnytska; Ralph Kenna; Yulian Honchar; Yurij Holovatch

REVIEW 3 major objections 4 minor 23 references

Partition function zeros for the Blume-Capel model on a complete graph

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For the Blume-Capel model on a complete graph, partition-function zeros show that finite-size systems up to N=1500 still display effective critical exponents far from the thermodynamic-limit mean-field values.

desk verdict A useful finite-size zeros map for the complete-graph Blume-Capel model, but the headline claim rests on power-law fits that need corrections-to-scaling and error bars before I'd trust the exponents. read the letter →

arxiv 2501.04452 v1 pith:4TTOKQW3 submitted 2025-01-08 cond-mat.stat-mech

Yulian Honchar , Mariana Krasnytska , Bertrand Berche , Yurij Holovatch , Ralph Kenna This is my paper · ORCID

classification cond-mat.stat-mech MSC 82B2082B2782B26 PACS 05.70.Jk05.50.+q

keywords Blume-CapelmodelpartitionfunctionzerosFisherLee-Yangcrystalfieldcompletegraphfinite-sizescalingtricriticalpoint

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 examines the Blume-Capel model on a complete graph, where every spin interacts with every other spin, and asks how the finite-size system's critical behaviour is seen through the zeros of its partition function. It establishes that, even for graphs of 1500 sites, the scaling of the first Fisher zero gives effective exponents $T_1 \sim N^{-0.35}$ or $T_1 \sim N^{-0.41}$ near the tricritical point, whereas the exact thermodynamic limit predicts $T_1 \sim N^{-2/3}$. At the ordinary critical point the effective exponent is about 0.44 rather than the mean-field value 0.5. Crystal-field zeros along the critical line are even more strongly skewed toward tricritical behaviour, with exponents between 0.62 and 0.69. If the paper is right, finite-size data at accessible sizes cannot be read directly as mean-field asymptotics; corrections or larger sizes are needed.

What carries the argument

The load-bearing object is the exact integral representation of the partition function, Eq. (4), which expresses $Z_N$ as a single integral over a collective variable $x$ for any finite $N$. On top of this, the paper uses partition-function zeros in three complex planes: Fisher zeros (complex temperature), crystal-field zeros (complex $\Delta$), and Lee-Yang zeros (complex magnetic field). These zeros are found by solving the simultaneous equations $\mathrm{Re}\,Z=0$ and $\mathrm{Im}\,Z=0$ (for Lee-Yang zeros, $\mathrm{Re}\,Z=0$ alone). The argument is carried by scaling relations of the form $T_j \sim (j/N)^{1/(2-\alpha)}$, $H_j \sim (j/N)^{\beta\delta/(2-\alpha)}$, and the impact-angle relation (12), which connect the first zero's distance from the real axis to the critical exponents. By fitting the size dependence of the first zero over $N=20$ to $1500$, the paper extracts effective gap exponents and compares them with the mean-field expectations $1/2$, $2/3$, $3/4$ and $5/6$.

What would settle it

A concrete check: compute the first Fisher zero at the tricritical point ($T_t=1/3$, $\Delta_t=2\ln 2/3$) for $N=10^4$ to $10^6$ using the exact integral representation and fit $T_1(N)$ with the asymptotic form $a N^{-2/3}$ plus a logarithmic-correction term. If the effective exponent moves toward $2/3$ once those corrections are included, the paper's claim of non-asymptotic criticality at $N=1500$ is a finite-size fitting artifact; if the exponent remains near $0.4$ even at $N=10^6$, the claim is supported.

Watch

Extended reading notes

Core claim

The central claim is that finite-size criticality in this exactly solvable mean-field model is not asymptotic for the sizes studied. The authors analyse the exact integral representation of the partition function and locate its zeros in the complex temperature, crystal-field, and magnetic-field planes. Near the tricritical point at $(T_t,\Delta_t)=(1/3,2\ln 2/3)$, the first Fisher zero scales as $T_1 \sim N^{-0.35}$ for all sizes up to $N=1500$, or $N^{-0.41}$ using only the three largest sizes, compared with the asymptotic mean-field exponent $2/3$. Near the Ising critical point $(T=2/3,\Delta=0)$ the effective exponent is about 0.44, not 0.5. Crystal-field zeros behave well only near the tricritical point and scale with effective exponents close to $2/3$ even when the thermodynamic point is critical ($1/2$). Lee-Yang zeros in the complex magnetic field obey the Lee-Yang circle theorem and their gap exponent approaches $3/4$ at the critical point and $5/6$ at the tricritical point, but along most of the critical line the fitted exponents are skewed toward the tricritical value.

Load-bearing premise

The load-bearing premise is that a pure power law $T_1 \sim N^{-g}$, fitted over the available range of $N$, captures the asymptotic exponent; if the data instead contain logarithmically suppressed corrections, the fitted effective exponents could be artifacts of omitting them.

Editorial extensions

If this is right

Finite-size scaling analyses that use only a few hundred or thousand sites for this model will report effective exponents that are not the true mean-field values; the gap is especially large at the tricritical point.
Crystal-field zeros are a reliable probe near the tricritical point but a poor one near the Ising critical point, because the scaling field is perpendicular to the tricritical point rather than to the critical point.
The point $(T=1/2,\Delta=\ln 2/2)$, which lies on the pseudo-transition line $\Delta=T\ln 2$, shows unusually fast convergence to the critical exponent and may be a useful reference point in finite-size studies.
Lee-Yang zeros confirm the Lee-Yang circle theorem on the complete graph: all zeros lie on the imaginary magnetic-field axis at the critical line and the tricritical point.
Effective exponents extracted from the first Fisher zero depend on which system sizes are included (0.35 versus 0.41 at the tricritical point), so reported exponents should carry this dependence.

Reading between the lines

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

An implication the authors leave implicit is that the fitted pure power laws may be reconciled with the true mean-field asymptotics by including logarithmic corrections, which are known to appear in mean-field models above the upper critical dimension; if so, the effective exponents would be a crossover phenomenon rather than evidence of a new universality class.
A testable extension would be to repeat the zero analysis at substantially larger $N$ (for example $10^4$ to $10^5$) and check whether the effective Fisher-zero exponent increases monotonically toward $2/3$; the paper does not report such data.
The observation that crystal-field zeros are skewed toward tricritical behaviour even along the critical line suggests that fits of thermodynamic quantities near tricritical points in other spin-1 models may similarly be contaminated by tricritical scaling, a caution that could be tested on related models.
One could also fit the first Fisher zero with explicit correction terms, such as $T_1 = a N^{-2/3}(1 + b N^{-\omega})$, to see whether the effective exponent 0.35 to 0.41 at $N \le 1500$ is a leading-order artefact or a genuine slow crossover.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know: this paper maps out finite-size effective exponents from Fisher, Lee-Yang, and crystal-field zeros for the Blume-Capel model on a complete graph, and the qualitative message—that effective behavior is far from mean-field even at N=1500—is plausible and worth taking seriously. The exact integral representation of the partition function is clean, the numerical zero-finding is systematic, and the comparison of the three zero types at the same points in the phase diagram is genuinely useful. The observation about the pseudo-transition point at T=1/2, Delta=(ln2)/2, where the Lee-Yang scaling is unexpectedly good, is a nice detail.

That said, the central quantitative claim is not as solid as the text implies. The effective exponents are extracted from pure power-law fits over N=50–1500, with no corrections-to-scaling, no logarithmic terms, and no error bars. For a mean-field/tricritical model with known log corrections, that's a real gap. The fitted value g=0.41 at the tricritical point could move toward the asymptotic 2/3 once corrections are accounted for; the paper does not test that. I don't think the claim collapses—the systematic pattern across different zero types is suggestive—but it needs a stability analysis.

Two smaller problems. The introduction promises an "expanded function approximation" that never appears anywhere in the paper; either it's in a supplementary file or it should be cut. And the conclusion says the equivalence between Fisher and crystal-field zeros was "confirmed analytically," but reading Sections 5.1 and 5.2, I see the equivalence assumed from the scaling-field argument, not derived or verified analytically. That overstatement should be fixed. No code or data files are shipped, which makes independent checks harder.

None of this is fatal. The paper's core contribution—a careful finite-size study of zeros in three complex planes for the complete-graph Blume-Capel model—is real and within the scope of a specialist journal. My own verdict is close to the reader's: conditional. If the authors add corrections-to-scaling fits, report uncertainty, and either deliver or remove the promised expansion, the results would be a solid reference for people doing zeros-based finite-size studies.

I would send this to peer review. A good referee can push for the fixes without needing a re-derivation of the whole method.

Referee Report

3 major / 4 minor

Summary. The paper studies finite-size effects in the Blume-Capel model on a complete graph by locating zeros of the exact integral representation of the partition function, Eq. (4), in the complex temperature, crystal-field, and magnetic-field planes for system sizes up to N=1500. The thermodynamic-limit mean-field exponents are recalled, and the finite-size scaling of the first Fisher, crystal-field, and Lee-Yang zeros is compared with those asymptotic exponents. The central claim is that even at the largest accessible sizes the effective critical behaviour read from Fisher zeros is not asymptotic, e.g. T_1 ~ N^{-0.35} or N^{-0.41} at the tricritical point instead of N^{-2/3}. The paper also reports angle estimates, crystal-field-zero scaling along the critical line, and running Lee-Yang gap exponents.

Significance. If the central claim is correct, the paper provides a useful example in which partition-function-zero scaling at accessible finite sizes is governed by effective exponents that differ substantially from the known mean-field asymptotics, which matters for interpreting finite-size data in mean-field and tricritical systems. The work has clear strengths: the starting point is an exact integral representation, the thermodynamic-limit phase diagram is derived analytically, the standard zero-scaling relations are applied, and three different complex-field planes are compared systematically. However, the central quantitative conclusion rests on pure-power-law fits without corrections to scaling, logarithmic terms, or error bars, so the distinction between 'non-asymptotic effective behaviour' and 'large corrections to scaling' is not yet established.

major comments (3)

[§5.1.3, Eq. (4), Fig. 9] The main claim that criticality is not asymptotic at the tricritical point is based on fitting Im T_1(N) to a pure power law, giving N^{-0.35} or N^{-0.41} versus the expected N^{-2/3}. No corrections to scaling, no logarithmic factors, and no error bars are reported. At a tricritical point the quartic Landau coefficient vanishes, so logarithmic and higher-order corrections are expected; Ref. [8] is precisely about such logarithmic corrections to mean-field scaling. The authors should test fits of the form T_1 = A N^{-g}(1 + B N^{-theta}) or T_1 = A N^{-g}(ln N)^p over the available N range, and report the resulting g-values with uncertainties. If g moves toward 2/3 under such fits, the conclusion reduces to 'corrections are large' rather than 'the asymptotic exponent is not approached.'
[§5.1.1, Fig. 6] The analogous pure-power-law analysis at the Ising critical point gives g=0.44 or 0.46 versus the expected 0.5, but the fitted range is limited (N=50 to 500, or only the three largest sizes) and no error bars or corrections-to-scaling fits are shown. The statement that the scaling 'approaches the theoretical value, even though slowly' is plausible, but the numerical support is not quantitative. Please provide stability checks under different N-windows and fits with corrections, and report confidence intervals for the effective exponents.
[§5.3.1, Figs. 18-19] The running Lee-Yang exponent g_h is computed by repeated fits that add one system size at a time, but no uncertainties are quoted, and the conclusion that the exponents 'finally converge' toward 3/4 or 5/6 is drawn from trajectories without error bars. The qualitative trend is visible, but a quantitative statement about convergence requires either error bars or a stated criterion for convergence. This is important because the conclusions section asserts that Lee-Yang scaling 'matched theoretical predictions.'

minor comments (4)

[Introduction, Section 5] The Introduction promises an 'expanded function approximation' that 'simplifies the calculations without sacrificing accuracy,' but this method is never defined or used anywhere in the paper. The authors should either deliver it or remove the promise from the Introduction.
[§5.2.1] The claim that crystal-field zeros at the Ising critical point are not well defined because the scaling field is perpendicular to the tricritical point is stated very briefly; a short explanation of why this makes the zeros poorly defined would improve readability.
[Conclusions] The conclusion states that 'the equivalence between Fisher and crystal field zeros was confirmed analytically,' but the body of the paper only establishes that both obey the same expected scaling relation, Eq. (13) and Eq. (20). The wording is stronger than the presented derivation and should be adjusted.
[Notation in §5.1.3, Fig. 9] The text writes T_1 ~ N^{0.35} in one place and T_1 ~ (1/N)^{0.35} in another. Since the quantity is the imaginary part of the first zero decreasing with N, the notation should be made uniform to avoid sign ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: finite-size zero scalings are measured from the exact partition function and compared with independently known mean-field exponents; self-citations supply methodology, not the result.

full rationale

The paper's central claim is that the first Fisher-zero coordinate scales as N^{-0.44} at the Ising critical point and N^{-0.35}/N^{-0.41} at the tricritical point, versus the asymptotic mean-field values N^{-1/2} and N^{-2/3}. These asymptotic exponents are not fitted parameters; they are taken from the standard mean-field solution of the Blume-Capel model and from the known scaling relation (13), which is cited to general results [5,8,10,19] rather than derived from the present fits. The finite-size exponents are outputs of power-law fits to zeros obtained by numerically solving the exact integral representation (4), so the measured exponents are not imposed by the asymptotic values being tested. The self-citations in the paper ([8,10,14,15,22]) are used for methodological background, for prior applications of partition-function-zero techniques, and for heuristic expectations about crystal-field zeros; none of them is invoked as a uniqueness theorem or as the source of the numerically fitted exponent values. The discussion of the special point T=1/2, Delta=ln2/2 is a post-hoc observation that this point lies on the pseudo-transition line, but it does not define the scaling exponents through that line. Concerns about neglecting corrections to scaling, logarithmic terms, or error bars are legitimate statistical robustness questions, but they are not circularity: the comparison between a fitted effective exponent and an independently known asymptotic exponent remains an external benchmark. No equation is defined in terms of the quantity it purports to predict, and no fitted parameter is renamed as a prediction. Therefore the derivation chain is self-contained with respect to the central finite-size claim, and no circular step can be exhibited from the text.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The model itself has no invented entities. The central claim rests on the exact integral representation, standard zero-scaling relations, known mean-field exponents, and the paper's own power-law fitting assumption. The fitted effective exponents are outcomes, listed here for completeness because the headline claim is about their values.

free parameters (4)

Effective Fisher-zero exponent near Ising point = 0.44 (all sizes), 0.46 (largest only)
Fitted from log-log slope of first Fisher zero vs N in Fig. 6; compared to asymptotic 0.5.
Effective Fisher-zero exponent near tricritical point = 0.35 (all sizes), 0.41 (largest only)
Fitted from Fig. 9; compared to asymptotic 2/3.
Effective crystal-field-zero exponent = 0.62 (T=0.4), 0.69 (T=0.35)
Fitted from Fig. 13; expected 0.5 at critical, 2/3 at tricritical.
Effective Lee-Yang gap exponent = Roughly 0.75 to 0.85 depending on point and fit window
Fitted from Figs. 18 and 19; expected 0.75 (critical) and 0.833 (tricritical).

assumptions (4)

standard math Stratonovich-Hubbard transformation yields exact integral representation of the partition function (Eq. 4).
Used in Section 2 to reduce the N-spin sum to a single integral; standard and exact.
domain assumption The scaling relations (12) and (13) connect partition function zero coordinates to critical exponents (known from Lee-Yang and Fisher theory).
Quoted from Refs. [4,5,8,10,19] in Section 4; assumed valid for the complete graph.
domain assumption Mean-field critical exponents for the Blume-Capel model (Table 3) are the asymptotic values.
Taken from Refs. [20,21,22]; used to set expected gap exponents and angles.
ad hoc to paper The finite-size scaling of the first zero follows a pure power law T1 ~ N^{-g} with no corrections-to-scaling over the fitted range.
The paper fits straight lines on log-log plots without testing logarithmic or higher-order corrections; this is the paper's central modeling assumption.

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Cite this review

Pith. "Pith review of Partition function zeros for the Blume-Capel model on a complete graph." pith.science (2026). https://pith.science/paper/4TTOKQW3

@misc{pith2026250104452,
  author       = {Pith},
  title        = {Pith review of: Partition function zeros for the Blume-Capel model on a complete graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TTOKQW3}},
  note         = {Machine review of arXiv:2501.04452}
}

read the original abstract

In this paper we study finite-size effects in the Blume-Capel model through the analysis of the zeros of the partition function. We consider a complete graph and make use of the behaviour of the partition function zeros to elucidate the crossover from effective to asymptotic properties. While in the thermodynamic limit the exact solution yields the asymptotic mean-field behaviour, for finite system sizes an effective critical behaviour is observed. We show that even for large systems, the criticality is not asymptotic. We also present insights into how partition function zeros in different complex fields (temperature, magnetic field, crystal field) give different precision and provide us with different parts of the larger picture. This includes the differences between criticality and tricriticality as seen through the lens of Fisher, Lee-Yang, and Crystal Field zeros.

Figures

Figures reproduced from arXiv: 2501.04452 by the authors.

**Figure 2.** Behaviour of function 𝐹(𝑇,Δ, 𝑥) (6) vs 𝑥 at (a) 𝑇 = 1/4, (b) 𝑇 = 𝑇𝑡 = 1/3, (c) 𝑇 = 1/2 and different values of Δ. In all figures the blue and the red curves correspond to Δ = 0.1 and Δ = 1. The green curve corresponds to the first-order phase transition at 𝑇 = 1/4, Δ = 0.4762658366 (a), tricritical at 𝑇 = 𝑇𝑡 = 1/3, Δ = Δ𝑡 = (2ln2)/3 = 0.4620981204 (b) and the second order at 𝑇 = 1/2, Δ = 0.3465735903 (c) transitions… view at source ↗

**Figure 3.** (a) Behaviour of the function 𝐹(𝑇,Δ, 𝑥) (6) vs 𝑥 at the tricritical point (𝑇,Δ)= (1/3,(2ln(2))/3) (red curve) and at several points along the first-order phase transition line. (𝑇,Δ) = (0.3, 0.4667635075), (0.25, 0.4762658366), (0.2, 0.4867012711), (0.15, 0.4951587768), orange, yellow, green, blue lines, correspondingly. (b) Behaviour of 𝐹(𝑇,Δ, 𝑥) (6) as a function of 𝑥 at the tricritical point (𝑇𝑡 ,Δ𝑡)= (1/3,(2ln(2… view at source ↗

Figures from the paper (16 more)

**Figure 4.** Figure 4: Phase diagram of the Blume-Capel model that follows from the analysis of function [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Solutions of the equations Re 𝑍𝑁 (𝑇𝑅 +𝑖𝑇𝐼 ,Δ = 0, 𝐻 = 0) = 0 (purple curves) and Im𝑍𝑁 (𝑇𝑅 + 𝑖𝑇𝐼 ,Δ = 0, 𝐻 = 0) = 0 (green curves) in the complex temperature plane for 𝑁 = 500 in the vicinity of 𝑇𝑐 (”Ising” point). Fisher zeros are obtained by seeking the intersections …

**Figure 6.** Figure 6: Coordinate of the first Fisher zero 𝑇1 as a function of system size 𝑁. The expected scaling (yellow line) is 𝑇1 ∼ (1/𝑁) 1/2 . For the graph sizes up to 𝑁 = 500 we obtain 𝑇1 ∼ (1/𝑁) 0.44 or 𝑇1 ∼ (1/𝑁) 0.46 if we only take into account three largest system sizes, which s…

**Figure 7.** Figure 7: Fisher zeros for 𝑁 = 500 along the critical line at Δ = 0.1 (left) and near 𝑇𝑡 (right). As we approach the tricritical point, the lines Re 𝑍𝑁 (𝑇𝑅 + 𝑖𝑇𝐼 ,Δ, 𝐻 = 0) = 0 and Im𝑍𝑁 (𝑇𝑅 + 𝑖𝑇𝐼 ,Δ, 𝐻 = 0) = 0 become more dense, and the amplitudes of the functions are more subs…

**Figure 8.** Figure 8: Fisher zeros for 𝑁 = 500 at the 𝑇𝑡 = 1/3. Only a few first zeros can be seen, while the rest hides behind the limits of our computing capabilities. always at a higher temperature than the critical point of the infinite system. We also want to observe whether the impact…

**Figure 9.** Figure 9: Coordinate of the first Fisher zero 𝑇1 as a function of system size 𝑁 near 𝑇𝑡 = 1/3 at Δ𝑡 = 2/3ln2. The mean-field expectation is 𝑇1 ∼ (1/𝑁) 2/3 , while for the graph sizes up to 𝑁 = 1500 we obtain 𝑇1 ∼ (1/𝑁) 0.35 (green line) or 𝑇1 ∼ (1/𝑁) 0.41 if we only take into ac…

**Figure 10.** Figure 10: Fisher zeros for 𝑁 = 200 at different points along the critical line (Δ = 0.01,0.1,Δ𝑡 −0.05) and at the tricritical point Δ𝑡 = 2/3ln2. The fitted angle values read 𝜑 = 37.27,37.38,34.48,32.51 correspondingly. Note that angles 𝜑 should be equal (𝜑𝑀𝐹 = 45◦ ) for the inf…

**Figure 11.** Figure 11: Angles 𝜑1𝑐 and 𝜑12 between the Fisher zeros and real temperature axis for different system sizes at Δ = 0 and along the critical line at Δ = ln2/2. In the thermodynamic limit both angles should be 𝜑𝑀𝐹 = 45◦ , and we can already see that for graphs up to the size 𝑁 = 5…

**Figure 12.** Figure 12: Solutions of equations Re 𝑍 = 0 and Im𝑍 = 0 in complex crystal field plane for 𝑁 = 200 at 𝑇𝑐 = 2/3. Crystal field zeros are difficult to spot due to the high density of zero lines. This happens due to the fact that the critical line here is not perpendicular to the sc…

**Figure 13.** Figure 13: Coordinate of the first crystal field zero [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

**Figure 14.** Figure 14: Crystal field zeros for different system sizes at different points along the critical line [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

**Figure 15.** Figure 15: Solutions of equations Re 𝑍 = 0 and Im𝑍 = 0 in complex crystal field plane for 𝑁 = 200 at 𝑇𝑡 . The crystal field zeros are clearly visible. 5.3.1 Scaling exponent analysis for Lee-Yang zeros We are interested in the finite-size scaling of the first Lee-Yang zero 𝐻1 al…

**Figure 16.** Figure 16: Angles 𝜑1𝑐 and 𝜑12 between the crystal field zeros and real crystal field axis for different system sizes at 𝑇𝑡 . Both angles come within an error bar to the expected value 𝜑 𝑡𝑟𝑖 𝑀𝐹 = 60◦ before graphs reach size 𝑁 = 500. follow how the scaling exponent 𝑔ℎ changes wit…

**Figure 17.** Figure 17: Function Re 𝑍(𝐻𝑅 = 0, 𝐻𝐼) at 𝑁 = 900. Solutions of the equation Re 𝑍(𝐻𝑅 = 0, 𝐻𝐼) = 0 are the Lee-Yang zeros. The first solution 𝐻1 is the first zero whose scaling we will follow through. Interestingly, the point 𝑇 = 1/2,Δ = ln2/2 lies exactly on the line Δ = 𝑇 ln2, de…

**Figure 18.** Figure 18: Scaling exponent 𝑔ℎ that governs the FSS of the first Lee-Yang zeros 𝐻1 at different points along the critical line, as well as at thee tricritical line. The approach of taking into account all of the previous graph sizes up to size 𝑁 when fitting the FSS is used here…

**Figure 19.** Figure 19: Scaling exponent 𝑔ℎ that governs the FSS of the first Lee-Yang zeros 𝐻1 at different points along the critical line, as well as at the tricritical line. The approach of taking into account only five largest graph sizes up to size 𝑁 when fitting the FSS is used here. S…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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