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Investigating Kozai-Lidov Oscillations and Disc Tearing in Be Star Discs

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper identifies the binary mass ratio and disc viscosity as the controls for whether misaligned Be star discs undergo Kozai-Lidov oscillations and disc tearing, and shows these states produce distinct observable signatures including…

desk verdict A useful parameter-space extension with the first radiative-transfer observables for KL oscillations, but the viscosity mapping uncertainty and thin q=0.1 sampling mean the claimed boundaries should be treated as provisional. read the letter →

arxiv 2412.04299 v1 pith:CWUZYAHL submitted 2024-12-05 astro-ph.SR astro-ph.EP

M.W. Suffak , C.E. Jones , A.C. Carciofi This is my paper · ORCID
classification astro-ph.SRastro-ph.EP
keywords BestarscircumstellardiscsKozai-Lidovoscillationsdisctearingsmoothedparticlehydrodynamicsradiativetransfertriple-peakedH-alphastellarinterferometry
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 extends three-dimensional smoothed-particle-hydrodynamics simulations of Be star discs in misaligned binary systems to ask what controls two dramatic behaviours: Kozai-Lidov (KL) oscillations, where the disc's eccentricity and inclination repeatedly trade off while it dissipates, and disc tearing, where the disc breaks into separate precessing rings. The paper finds that lowering the binary mass ratio from 1 to 0.5 preserves both phenomena with a slightly longer timescale, while a mass ratio of 0.1 suppresses them for the parameters examined, and that raising the Shakura-Sunyaev viscosity parameter from 0.1 to 0.5 or 1.0 damps them as well. Using the HDUST radiative-transfer code, the paper produces the first triple-peaked H$\alpha$ line profile from a Be disc model, tied to the asymmetric disc during a KL oscillation, and shows that polarization position angle, line peaks, and photometry oscillate with the disc's changing inclination. If these predictions hold, they give observers a concrete route to recognize KL oscillations and torn discs in real Be binaries, and they identify which binaries should be dynamically quiet.

What carries the argument

The machinery is the three-dimensional smoothed-particle-hydrodynamics (SPH) code used for the disc models, with the Shakura-Sunyaev viscosity parameter $\alpha_{\rm ss}$ converted into the SPH artificial viscosity through $\alpha_{\rm sph} = 10\,\alpha_{\rm ss} H/h$ (with $\beta_{\rm sph}=0$), paired with the HDUST non-local-thermodynamic-equilibrium Monte Carlo radiative-transfer code that converts the particle distributions into synthetic spectra, polarization, images, and interferometric visibilities. A second piece of machinery is the comparison between two viscosity prescriptions: constant $\alpha_{\rm ss}$ and constant $\alpha_{\rm sph}$. Because the simulations have $H/h < 1$, the constant-$\alpha_{\rm sph}$ runs correspond to an effective $\alpha_{\rm ss}$ above the nominal 0.1, which inflates the viscous torque and is why those runs fail to reproduce disc tearing; the constant-$\alpha_{\rm ss}$ runs are therefore the ones used to map the parameter boundaries.

What would settle it

Run the constant-$\alpha_{\rm sph}$ equal-mass, 40-degree misalignment model with particle resolution high enough that $H/h \approx 1$ throughout the disc; if disc tearing then appears, the suppression of tearing in constant-$\alpha_{\rm sph}$ models is an artifact of the viscosity conversion rather than a physical effect of the higher viscosity.

Watch

Extended reading notes

Core claim

The central claim is that in a Be star with a misaligned binary companion, the binary mass ratio $q$ and the disc viscosity parameter $\alpha_{\rm ss}$ decide whether the disc tears apart or undergoes Kozai-Lidov oscillations. For $q=0.5$ the paper finds the same evolution as the equal-mass case, only delayed, whereas for $q=0.1$ neither phenomenon occurs for the examined parameters, and the disc simply grows larger and precesses; this is consistent with the analytical tearing radius moving to the disc's outer edge. Raising $\alpha_{\rm ss}$ from 0.1 to 0.5 or 1.0 damps the oscillations and suppresses tearing because the faster viscous communication keeps the disc intact. The paper then reports the first radiative-transfer-generated triple-peaked H$\alpha$ line profile in a Be disc model, produced when the KL oscillation makes the disc strongly asymmetric, and shows that the observables—equivalent width, V/R ratio, peak separation, $V$-band magnitude, and polarization degree and position angle—oscillate in step with the changing disc inclination. Finally, it predicts that a gap in the disc, from either tearing or KL oscillations, creates hump-like bumps in the squared-visibility versus baseline curve and can increase the differential phase across H$\alpha$ and Br$\gamma$ lines by up to an order of magnitude, making the dynamical state recognizable with long-baseline interferometers.

Load-bearing premise

The main load-bearing assumption is that the formula converting the Shakura-Sunyaev viscosity into the simulation's artificial viscosity ($\alpha_{\rm sph} = 10\,\alpha_{\rm ss} H/h$ with $\beta_{\rm sph}=0$) is faithful, since in these simulations $H/h < 1$ makes the effective viscosity higher than the nominal $\alpha_{\rm ss}$, and an error in that mapping would shift the mass-ratio and viscosity boundaries at which tearing and KL oscillations occur.

Editorial extensions

If this is right

  • Be stars in misaligned binaries with a low-mass companion ($q \sim 0.1$) are predicted not to show KL oscillations or disc tearing, so any observed such variability in those systems would point to a different mechanism or different parameters.
  • Discs with $\alpha_{\rm ss} \gtrsim 0.5$ should dissipate smoothly without tearing or KL oscillations, which can be tested by comparing stars with known high-viscosity discs.
  • Triple-peaked H$\alpha$ profiles can be produced by an asymmetric disc during a KL oscillation, and the line-formation mapping identifies the slow-moving, radially extended side and fast-moving, small side of the disc as the sources of the separate peaks.
  • A long-baseline interferometer whose projected baseline crosses a torn or eccentric gap should see a hump in the squared visibility versus baseline curve and an enhanced differential phase across H$\alpha$ and Br$\gamma$, a signature that could be searched for in Pleione.
  • The quantitative criteria (tearing radius and KL timescale) used to interpret the simulations can be applied to observed binaries to estimate whether their discs are in the tearing/KL regime or the quiet regime.

Reading between the lines

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

  • The boundary between $q=0.1$ and $q=0.5$ suggests that many Be/X-ray binaries with low-mass neutron-star companions may lie below the tearing threshold, so superorbital variability from tearing should be rarer than the equal-mass case would suggest (an inference, not a paper claim).
  • Because the triple-peaked profile is attributed to disc asymmetry rather than to KL oscillations specifically, the same line-formation mapping may help interpret triple peaks seen in other Be stars with spiral waves or outbursts, if their discs have similar geometry (editorial extension).
  • The $H/h < 1$ issue implies that past constant-$\alpha_{\rm sph}$ simulations of Be discs may have effectively used a higher viscosity than intended; re-scaling their results with the effective $\alpha_{\rm ss}$ could shift which systems are expected to tear (editorial inference).
  • The predicted relationship between visibility hump location and gap radius could be inverted: a single observation with two position angles might estimate the gap's size and orientation, turning interferometry into a diagnostic of the tearing state (editorial suggestion).
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper extends the authors' earlier SPH study of misaligned Be star binary discs (Paper I) by running eight new simulations that vary the binary mass ratio (q = 0.1, 0.5, 1.0) and the Shakura–Sunyaev viscosity parameter (alpha_ss = 0.1, 0.5, 1.0), and by post-processing selected runs with the radiative transfer code HDUST. The central dynamical claims are that q = 0.5 reproduces the previously seen disc-tearing (for 40 deg misalignment) and Kozai–Lidov (KL) oscillations (for 60 deg) with somewhat longer timescales, while q = 0.1 shows neither phenomenon in the parameters examined; that increasing alpha_ss to 0.5 or 1.0 damps these oscillations; and that a constant-SPH-viscosity prescription yields effectively higher alpha_ss values, explaining why tearing is suppressed in those comparison runs. On the observational side, the paper predicts that KL oscillations produce periodic changes in H-alpha equivalent width, V/R ratio, peak separation, polarization degree and position angle, including a triple-peaked H-alpha profile that the authors state is the first produced by radiative transfer modelling of a Be star disc. It also presents interferometric visibility and differential-phase signatures of gaps caused by KL oscillations and disc tearing, and suggests Pleione as a candidate for detecting such signatures.

Significance. If the dynamical claims hold, the paper offers a useful map of how misaligned Be discs transition between tearing, KL oscillation, and quiescent precession as functions of mass ratio and viscosity, which is directly relevant to Be/X-ray binaries and the growing population of Be stars with faint companions. The strengths of the work are that it uses a well-established SPH code, cross-checks its results against independent analytic scalings (Doğan et al. tearing radius, Eq. 6; Larwood et al. precession, Eq. 7; Lidov conservation, Eq. 8; Kiseleva et al. KL timescale, Eqs 9–10), and forward-models observables with HDUST rather than tuning the models to match data. The radiative-transfer and interferometric predictions, especially the triple-peaked H-alpha profile and the visibility-hump signature of a torn disc, are novel, falsifiable predictions that should guide future observations of systems like Pleione. The main reservations are that the viscosity conversion underlying the claimed q/alpha_ss boundaries is not quantitatively validated in the warped and tearing regions, and that the q = 0.1 null result rests on only one simulation per misalignment angle.

major comments (3)
  1. [§4.2, Eq. (2), Fig. 6] The mapping between the chosen alpha_ss and the SPH artificial viscosity parameter alpha_sph in Eq. (2) is load-bearing for the paper's central conclusion that q = 0.1 does not produce KL oscillations or disc tearing, and that alpha_ss = 0.5 or 1.0 damps them. The manuscript itself notes in §4.2 that 'areas in the simulation where the disc does not follow this scale height prescription may have inflated artificial viscosity values,' and Fig. 6 demonstrates H/h < 1 for a constant-alpha_sph run, inflating the realized alpha_ss. However, no analogous diagnostic is provided for the constant-alpha_ss runs on which the main scientific claims rest, even though those runs are precisely the ones that become warped, tilted, and torn. As a result, the effective viscosity in the main runs may be substantially larger than the nominal alpha_ss = 0.1, which could suppress tearing/KL in the q = 0.1 runs for reasons unrelated to mass ratio. I request that the authors report the radial and temporal distribution of the realized effective alpha_ss (e.g., from alpha_sph * h/H) for the constant-alpha_ss models, and, ideally, test whether the q = 0.1 null persists when either the resolution is increased (so that H/h approaches unity) or the nominal alpha_ss is lowered. Without such a test, the q and alpha_ss boundaries are not yet quantitatively secure.
  2. [§3, Table 2, Figs 2–3] The null result for q = 0.1 is based on exactly one simulation for each of the two misalignment angles (40 deg and 60 deg), with no variation of random seed or independent realisation. Given the stochastic nature of SPH particle injection and the sensitivity of tearing and KL oscillations to the detailed disc structure at the end of the growth phase, a single realisation per parameter set is insufficient to establish a null result. The authors should either present multiple realisations with different noise seeds, or demonstrate robustness by varying some other parameter (e.g., lower alpha_ss, different mass-injection profile) while keeping q = 0.1. This is important because the paper's abstract and conclusions state that q = 0.1 does not cause KL oscillations or disc tearing, which is a strong claim for the current evidence.
  3. [§3, Eq. (10), Figs 2–3] The statement that applying the factor 2.7 from Martin & Franchini (2019) to the predictions of Eq. (10) 'results in the exact initial KL periods found in our simulations' is an overstatement. That factor is derived for a test particle with an initial eccentricity of 0.2, whereas the disc here is extended, has a changing surface-density profile, and is dissipating; moreover, the measured periods are read from the first eccentricity peak, which involves some judgement given the oscillations. The agreement is encouraging and worth reporting, but it should be presented as an approximate consistency check (within the factor-of-2.7 approximation), not as an 'exact' reproduction. This is a minor point for the overall conclusions, but it appears in the main body as a quantitative validation of the KL interpretation.
minor comments (4)
  1. [§4.2, Fig. 6] The sentence 'we find we can recover the same qualitative disc evolution for a range of constant artificial viscosity as when alpha_ss is held constant' is immediately followed by the statement that disc tearing does not occur in the constant-alpha_sph = 1 models where it previously did. These two sentences are in tension; please rephrase to clarify that the qualitative agreement holds for some diagnostics (disc mass, inclination, precession) but not for tearing/KL, which depend sensitively on viscosity.
  2. [§6, Figs 12–13] The interferometric predictions assume a distance of 100 pc and a specific orientation, but the text does not state whether the uncertainties in the HDUST grid resolution (50x50x50) or in the disc density have been checked for their effect on the visibility curves. A brief comment on numerical convergence of the image/visibility computation would strengthen the quantitative claims about the size of the visibility humps.
  3. [§8, Data Availability] The statement 'No new data was generated' is confusing given that eight new SPH simulations and their HDUST observables were produced. I suggest rewording it to 'No observational data were generated; the simulation outputs are available from the authors upon request,' which would be consistent with the models actually being new.
  4. [§1, Introduction] The sentence 'Recently, simulations of Be stars in misaligned binary systems have revealed that misalignment between the disc and binary orbit can cause the disc to undergo Kozai-Lidov oscillations or disc-tearing' could benefit from a citation to the early papers that established these phenomena in Be-disc context (e.g., Martin et al. 2014; Suffak et al. 2022), rather than only to Paper I and the later sections.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new SPH/HDUST results are forward-modeled and cross-checked against external analytic criteria.

full rationale

The derivation chain is largely self-contained. The new claims about mass ratio and viscosity (q=0.5 reproduces tearing/KL with delay, q=0.1 does not, higher alpha_ss damps the phenomena) come from new SPH runs with independently varied parameters, not from fitting the conclusions into the inputs. Where the paper invokes external criteria, these are genuine independent checks: Eq (6) from Doğan et al. (2015) is used to interpret the q=0.1 null result, Eqs (7)-(10) are analytic precession/KL timescales from Larwood et al. and Martin et al., and the 2.7 KL-period correction is an externally published result (Martin & Franchini 2019), not a parameter fitted in this work. The HDUST observables and the triple-peaked H-alpha profile are forward models with no free parameters tuned to reproduce the line; the profile is an emergent output of the simulation and radiative transfer, and the paper presents line-formation loci to explain its origin. Equation (2), converting alpha_ss to alpha_sph, is a modeling assumption from Okazaki et al. (2002), not a claimed derivation; the paper explicitly tests its limitations in Section 4.2 with constant-alpha_sph runs and acknowledges that warped/tearing regions may have inflated artificial viscosity. That is a robustness caveat, not a circular step. Self-citations (Paper I base models and the Suffak et al. 2024 HDUST interface) are used as tools and reference states, but the central results rest on the new computations reported here. No equation in the paper reduces to its own input by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central claims depend on a grid of input parameters (mass ratio, viscosity, misalignment, injection settings) and on standard astrophysical assumptions (Shakura-Sunyaev viscosity, isothermal thin-disc scale height, VDD transport, circular orbit, KL benchmarks). No new physical entities are introduced; all new structures (gap, tearing, triple peak) are emergent properties of the simulations.

free parameters (7)
  • Shakura-Sunyaev viscosity alpha_ss = 0.1, 0.5, 1.0
    Free input scanned in Models 3,4,7,8; central result that high viscosity damps tearing/KL is defined by these choices.
  • binary mass ratio q = 1.0, 0.5, 0.1
    Free input scanned in Models 1,2,5,6 and from Paper I; q=0.1 null result depends on this grid.
  • misalignment angle beta = 40 deg, 60 deg
    Selected to match Paper I base models that tear (40 deg) and undergo KL (60 deg); not swept.
  • orbital period P_orb = 30 days
    Fixed base value; controls companion torque strength and all timescales reported in P_orb units.
  • mass injection rate Mdot_inj = 1e-8 Msun/yr
    Fixed input; sets disc mass and the 'anchoring' of the inner disc discussed in Section 7.1.
  • disc temperature T_d = 12000 K (0.6 T_eff)
    Fixed isothermal disc temperature from Carciofi & Bjorkman (2006); enters sound speed and scale height.
  • particle injection rate = 5000 particles per timestep, 5e5 per orbital period
    Numerical resolution choice; fixes smoothing length h in Eq 2 and thus the effective alpha_ss, which affects whether tearing occurs (Section 4.2).
assumptions (7)
  • domain assumption Shakura-Sunyaev viscosity: nu = alpha_ss c_s H (Eq 1) represents turbulent transport in the disc.
    This prescription underlies all simulations; the alpha_ss damping conclusions depend on it (Section 2.1).
  • domain assumption The disc is isothermal with scale height H = c_s / Omega.
    Used to define H in the viscosity mapping and the tearing radius estimates (Section 2.1).
  • ad hoc to paper The conversion alpha_sph = 10 alpha_ss H / h with beta_sph = 0 (Eq 2) faithfully maps SPH artificial viscosity to Shakura-Sunyaev viscosity.
    Adopted from Okazaki et al. (2002); Section 4.2 shows the mapping is resolution dependent and can suppress tearing, so the q/alpha_ss boundaries depend on it.
  • domain assumption The viscous decretion disc (VDD) model of Lee et al. (1991) governs disc growth and dissipation.
    Framework adopted in the introduction and used to interpret disc mass and dissipation.
  • domain assumption The binary orbit is circular (e_s = 0, presented implicitly).
    The orbital period is fixed but no eccentricity is stated; Eq 9 includes (1 - e_s^2)^{3/2}, so the KL timescale comparison assumes a circular companion orbit.
  • standard math KL theory for test particles (Eqs 8-10) is a valid benchmark for global disc oscillations.
    Used in Section 3 to identify KL oscillations and to compare measured periods to analytic predictions.
  • ad hoc to paper Mass injection into the equatorial plane at 1.04 R_p is a stable boundary condition.
    All discs are built by this injection; Section 7.1 argues injection anchors the disc, so its rate and radius affect oscillation behaviour.

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Cite this review

Pith. "Pith review of Investigating Kozai-Lidov Oscillations and Disc Tearing in Be Star Discs." pith.science (2026). https://pith.science/paper/CWUZYAHL

@misc{pith2026241204299,
  author       = {Pith},
  title        = {Pith review of: Investigating Kozai-Lidov Oscillations and Disc Tearing in Be Star Discs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWUZYAHL}},
  note         = {Machine review of arXiv:2412.04299}
}
read the original abstract

Recent simulations of Be stars in misaligned binary systems have revealed that misalignment between the disc and binary orbit can cause the disc to undergo Kozai-Lidov (KL) oscillations or disc-tearing. We build on our previous suite of three-dimensional smoothed particle hydrodynamics simulations of equal-mass systems by simulating eight new misaligned Be star binary systems, with mass-ratios of 0.1 and 0.5, or equal-mass systems with varying viscosity. We find the same phenomena occur as previously for mass ratios of 0.5, while the mass ratio of 0.1 does not cause KL oscillations or disc-tearing for the parameters examined. With increased viscosity in our equal-mass simulations, we show that these phenomena and other oscillations are damped out and do not occur. We also briefly compare two viscosity prescriptions and find they can produce the same qualitative disc evolution. Next, we use the radiative transfer code HDUST to predict observable trends of a KL oscillation, and show how the observables oscillate in sync with disc inclination and cause large changes in the polarization position angle. Our models generate highly complex line profiles, including triple-peak profiles that are known to occur in Be stars. The mapping between the SPH simulations and these triple-peak features gives us hints as to where they originate. Finally, we construct interferometric predictions of how a gap in the disc, produced by KL oscillations or disc-tearing, perturbs the visibility versus baseline curve at multiple wavelengths, and can cause large changes to the differential phase profile across an emission line.

Figures

Figures reproduced from arXiv: 2412.04299 by the authors.

Figure 1
Figure 1. Schematic of our simulation setup with a misaligned binary com￾panion. The primary and secondary stars are shown as blue spheres, while the disc is shown in light grey in the 𝑥 𝑦 plane. Labelled are the 𝑦 and 𝑧 axes, the misalignment angle 𝛽, and azimuthal angles 𝜙 of 90◦ and 270◦ , which are aligned with the positive and negative 𝑦-axis, respectively [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Top to bottom, total disc mass, disc inclination with respect to the primary equatorial plane, disc inclination with respect to the binary orbital plane, disc eccentricity, and the longitude of the ascending node of the disc, for models with a 40◦ misalignment angle and varying mass ratio as indicated by the legend. The 𝑥-axis is in units of binary orbital periods. approximation of the tearing radius, 𝑅break, by (Do… view at source ↗
Figure 4
Figure 4. Same format as [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (20 more)
Figure 6
Figure 6. Figure 6: Shakura-Sunyaev viscosity parameter, calculated using Equation 2, versus radius for each particle in a simulation with a constant 𝛼sph = 1 as indicated by the solid line. The points representing each particle are coloured by their smoothing length. This is for an equal…
Figure 7
Figure 7. Figure 7: Disc inclination with respect to the binary orbit, and disc eccen￾tricity versus time, for models with an equal-mass binary companion, and a constant 𝛼ss= 0.1 (blue solid line) or a constant 𝛼sph = 1 (yellow dashed line). The binary orbit is inclined by 60◦ . The dotte…
Figure 8
Figure 8. Figure 8: Same format as [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: Top to bottom: images of the disc in H𝛼, created by hdust; the associated H𝛼 line profiles; H𝛼 EW; H𝛼 V/R ratio; H𝛼 peak separation, with the absolute values of the blue and red peak positions in blue and red respectively; 𝑉-band magnitude; polarization degree; and pol…
Figure 10
Figure 10. Figure 10: Same format as [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: The various emitting regions of different portions of the triple-peaked H𝛼 line produced during KL oscillations are shown. The center panel shows the normalized emission line, while the four outer panels show images in H𝛼 of the disc and star, at the four coloured ban…
Figure 12
Figure 12. Figure 12: Top: hdust images in H𝛼 for our 60◦ base sph model (a) before and (b) during KL oscillations. The baseline PAs of 0 ◦ and 90◦ are labelled on each image. Middle: Squared visibility versus baseline, and bottom: differential phase versus wavelength, before and during KL…
Figure 13
Figure 13. Figure 13: Same format as [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: Same format as [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]
Figure 14
Figure 14. Figure 14: Continued MNRAS 000, 1–14 () [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
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Figure 14. Figure 14: Continued MNRAS 000, 1–14 () [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
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Figure 14. Figure 14: Continued MNRAS 000, 1–14 () [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
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Figure 15. Figure 15: Same format as [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
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Figure 15. Figure 15: Continued MNRAS 000, 1–14 () [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
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Figure 15. Figure 15: Continued MNRAS 000, 1–14 () [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
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Figure 15. Figure 15: Continued MNRAS 000, 1–14 () [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
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Figure 16. Figure 16: Continued MNRAS 000, 1–14 () [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
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Figure 16. Figure 16: Continued MNRAS 000, 1–14 () [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]

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    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

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