The dynamical evolution of the stellar clumps in the Sparkler galaxy

Ad\'ela\"ide Claeyssens; Antonio Ragagnin; Carlo Nipoti; Eric Giunchi; Federico Marinacci; Francesco Calura; Raffaele Pascale

REVIEW 3 major objections 4 minor 108 references

The dynamical evolution of the stellar clumps in the Sparkler galaxy

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper argues that the stellar clumps around the z≈1.4 Sparkler galaxy that survive to z=0 remain 20–50 pc in half-mass radius, about ten times the size of local globular clusters, and are therefore too large in size to be…

desk verdict A careful, useful dynamical-evolution study of the Sparkler clumps, but the headline size conclusion needs a bound-mass re-analysis before it is solid. read the letter →

arxiv 2507.13904 v1 pith:TCKWTM4X submitted 2025-07-18 astro-ph.GA

Eric Giunchi , Federico Marinacci , Carlo Nipoti , Adélaïde Claeyssens , Raffaele Pascale , Francesco Calura , Antonio Ragagnin This is my paper · ORCID

classification astro-ph.GA

keywords globularclusterformationSparklergalaxystellarclumpsdynamicalfrictiontidalstrippingN-bodysimulationsgravitationallensingJWST

topics Dark Matter

open problems Dark Matter

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 asks whether the bright stellar clumps seen around the $z\approx1.4$ Sparkler galaxy—objects with globular-cluster-like ages and metallicities but masses and sizes about ten times larger than local globular clusters—can evolve into globular clusters by $z=0$. Using N-body simulations of dynamical friction and tidal stripping over 9.23 Gyr, the authors find that clumps more massive than about $10^7\,\mathrm{M_\odot}$ sink into the galactic centre, while the clumps that survive beyond 1 kpc lose enough mass, through tidal shocks from a stellar disk, to match massive globular clusters in mass. Their sizes, however, stay between 20 and 50 pc, about ten times larger than present-day globular clusters. The paper's central conclusion is that the surviving Sparkler clumps are too large in size to be globular-cluster progenitors, and that most of the clumps are instead destined for the bulge or a nuclear star cluster.

What carries the argument

Two complementary N-body setups carry the argument. In the first, each clump is a single particle moving in a live halo of $10^7$ dark-matter particles built to match a Sparkler-like NFW/Hernquist potential, so dynamical friction arises self-consistently; the clump system is initialised in three geometries—face-on disk, edge-on disk, and spherical—to bracket the unknown morphology. In the second, clumps are resolved as Plummer spheres of $10^5$ particles orbiting in a static external potential that includes the same halo and, in one set of runs, an exponential stellar disk (represented by three Miyamoto-Nagai disks fixed at $z=0$ mass and size) whose crossings produce tidal shocks. The two channels are joined by fitting the simulated mass loss as a function of initial mass and applying it before or after the dynamical-friction selection, bracketing the order in which the processes act.

What would settle it

Re-estimate each clump's lensing magnification and intrinsic size from a deeper lens model or from resolved stellar kinematics: if the true half-mass radii come out near a few parsecs rather than 30–50 pc, the size-based conclusion fails. An independent check is to search local galaxies of roughly $10^{10}$ solar masses for surviving extended clusters with radii of 20–50 pc and masses above $10^6$ solar masses; their absence would support the paper's prediction.

Watch

Extended reading notes

Core claim

For a Sparkler-like galaxy with a dark-matter halo reaching $M_{200}\simeq5\times10^{11}\,\mathrm{M_\odot}$ by $z=0$ (the high end of the inferred range), the simulations predict that only about 40–60% of the ten extraplanar clumps survive outside 1 kpc. Clumps with stellar masses above roughly $10^7\,\mathrm{M_\odot}$ are dragged by dynamical friction into the central regions on timescales shorter than the 9.23 Gyr available. Without tidal stripping the survivor mass distribution peaks near $5\times10^6\,\mathrm{M_\odot}$, implying unusually over-massive clumps at $z=0$; when mass loss from disk tidal shocks is included and corrected for, the peak shifts to about $2\times10^6\,\mathrm{M_\odot}$, consistent with the most massive globular clusters known. Yet the same simulations show that the surviving clumps keep half-mass radii of 20–50 pc down to $z=0$. The paper concludes that most of the Sparkler clumps end up as bulge fossil fragments or contributors to a nuclear star cluster, while those that remain outside are too large in size to be the progenitors of today's globular clusters.

Load-bearing premise

The conclusion that survivors are too large to become globular clusters rests on the measured 30–52 pc effective radii being real physical sizes; if JWST resolution or lensing magnification inflates them, the clumps would be denser, tidal stripping would remove less mass, and the over-massive survivors would persist.

Editorial extensions

If this is right

Clumps more massive than about $10^7\,\mathrm{M_\odot}$ should sink into the Sparkler's central regions by $z=0$, where they can feed bulge growth or merge into a nuclear star cluster.
Only about 40–60% of the observed extraplanar clumps are expected to survive beyond 1 kpc, so the clump population around the galaxy thins substantially with time.
Correcting survivors for tidal stripping shifts their peak mass from about $5\times10^6$ to $2\times10^6\,\mathrm{M_\odot}$, placing the descendants at the high-mass end of the globular-cluster mass function.
Even after tidal stripping, survivor half-mass radii remain 20–50 pc, about ten times larger than local globular clusters, so mass-compatible descendants are still not size-compatible globular clusters.
If the lensing magnification were underestimated by about a factor of 10, the clump masses would become consistent with massive globular clusters while their sizes would remain up to five times too large.

Reading between the lines

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

A direct corollary of the paper's bracketing is that the order in which tidal stripping and dynamical friction act changes the fraction of survivors (from about 0.42 to 0.60) but leaves the final mass distribution nearly unchanged, which may make the relative timing hard to constrain from masses alone.
If the intrinsic sizes really are 20–50 pc, one would expect to find a population of extended, massive clusters—objects unlike ordinary globular clusters—in local galaxies of roughly $10^{10}\,\mathrm{M_\odot}$; searching nearby analogues for such extended clusters is a testable consequence the paper does not pursue.
The paper assumes no dark-matter subhalo around the clumps; if they were embedded in mini-halos, its own dynamical-friction results imply they would sink even faster, which would suppress the over-massive survivors rather than solve the size problem.
A revision of the lens model that shrinks the inferred radii to a few parsecs would not by itself rescue the globular-cluster scenario: the paper's argument shows the clumps would then be dense enough to resist tidal stripping and would survive as over-massive outliers.

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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

First thing to know: this is the first time anyone has put the Sparkler clumps through actual dynamical evolution, and the headline result is conditional but plausible — most of the clumps either sink to the centre by z=0 or survive as objects that are still too big to be globular-cluster progenitors. The paper is worth reading and worth refereeing.

What it does well: the authors use standard tools (live halo for dynamical friction, resolved Plummer clumps in an external potential for tidal stripping, Chandrasekhar drag for orbit calibration) and apply them carefully to one specific lensed galaxy. They test three spatial configurations because the morphology is unconstrained, and they are upfront about the assumptions. The survival fractions (~40–60%) and the prediction that clumps above ~1e7 Msun sink are sensible. The comparison with bulge fossil fragments and NSCs is useful.

The soft spot, and it is a real one: the claim that surviving clumps stay 20–50 pc relies on Plummer fits to the total density within 4rs,i, with no selection of bound particles. In the disk-shock runs the low-mass clumps keep only 15–40% of their mass inside that radius and have obvious tidal tails (Fig. 11). A single Plummer fitted to the total profile can return a scale radius set by the debris rather than by the compact bound remnant. If the bound remnants have half-mass radii below ~10 pc and still hold ~1e5 Msun, the abstract's 'too large' conclusion does not follow. This is checkable from their snapshots, and the authors should do that before publication.

The second issue is that the size–mass relation used to initialise clumps is fitted to only seven resolved clumps, with intercept q = 0.19 ± 0.65. The paper does not propagate that uncertainty. A lower intercept would mean more compact initial conditions and, presumably, more compact survivors. The qualitative conclusion about sizes is therefore more fragile than the paper lets on.

The mass-distribution shift to ~2e6 Msun in Section 6.2 uses a semi-analytic combination with no propagated uncertainties. That is a lesser concern; the qualitative shift is believable.

Bottom line: this is a solid case study for anyone interested in high-z star clusters or GC formation. It deserves peer review. I would ask the referee to focus on the bound-mass re-analysis of the final sizes and on the sensitivity to the size–mass intercept. If those hold up, the paper is publishable as a valuable addition to the Sparkler literature.

Referee Report

3 major / 4 minor

Summary. The paper studies whether the stellar clumps observed around the z=1.4 lensed Sparkler galaxy can evolve into globular clusters by z=0. The authors build initial conditions from the observed clump masses, effective radii, and distances, then run two complementary suites of N-body simulations with the AREPO code: (i) dynamical-friction runs in which point-mass clumps orbit inside a live NFW/Hernquist halo, and (ii) tidal-stripping runs in which resolved Plummer clumps orbit in static spherical or disk+halo potentials. They find that dynamical friction sends most clumps more massive than about 10^7 M_sun to the central regions, that tidal shocks from a disk can disrupt or heavily strip low-mass clumps, and that the surviving clumps retain half-mass radii of roughly 20-50 pc. Combining the two processes with a semi-analytic mass-loss model shifts the survivor mass peak to about 2e6 M_sun, consistent with very massive globular clusters, but the authors conclude that the surviving clumps are too large in size to be globular-cluster progenitors.

Significance. If the size conclusion survives detailed scrutiny, the paper is significant: it directly tests a JWST-motivated hypothesis that the Sparkler clumps are proto-globular clusters, using explicit N-body integrations rather than purely analytic estimates. The paper also has genuine strengths: the resolved clumps are followed with 10^5 particles, three spatial configurations are explored, the eccentric-orbit case is treated in an appendix, and the assumptions that maximize tidal effects are stated transparently. The headline result is falsifiable and observationally meaningful. However, the central claim currently rests on a size measurement that may be contaminated by unbound tidal debris, and the quantitative mass claim depends on an extrapolation of a maximum-stripping calibration; both points need to be addressed before the conclusion can be accepted.

major comments (3)

[Sec. 6.1, Fig. 12, and Appendix A] The final scale radii rs,f are obtained by fitting a single Plummer profile to the total density profile of all particles within 4 rs,i, with no selection of gravitationally bound particles. The disk-shock runs (Sec. 5.2, Fig. 11) generate extended tidal tails, and for low-mass clumps the retained mass within 4 rs,i can be only a small fraction of the initial mass (bottom panel of Fig. 10). In such a two-component system, a single Plummer fit measures the envelope containing both the bound remnant and the unbound debris, so the fitted rs,f is not necessarily the size of the object that would survive as a globular cluster. Because the abstract's central claim that 'the remaining clumps are too large in size to be progenitors of GCs' is based directly on Fig. 12, the authors should repeat the analysis on the bound remnant, using an iterative energy-based unbinding procedure, and report the bound mass and bound half-mass radius for each run in Secs. 5.1 and 5.2. Without this re-analysis, the size conclusion is not established for the low-mass survivors.
[Sec. 6.2, Eq. (17), and Fig. 13] The tidal mass-loss relation of Eq. (17) is calibrated on resolved-clump runs at r0 = 1 kpc and inclination theta = pi/6, parameters chosen explicitly to maximize disk shocks, and is then applied without any radial dependence to all surviving clumps in the spherical dynamical-friction runs, which end at distances ranging from about 1 to 10 kpc (Figs. 6 and 8). The predicted peak near 2e6 M_sun is therefore conditional on a maximum-stripping assumption, and the statement that the corrected mass distribution is 'compatible with massive GCs' is a lower limit on the true final masses. The authors note the maximization at the end of Sec. 6.2, but they do not quantify how the peak and the fraction of over-massive survivors would change if the disk-shock model were applied only to clumps with pericentres close to 1 kpc, or if a distance-dependent calibration were used. This quantification is needed to support the mass claim in the abstract.
[Sec. 2.1, Eq. (1), and Fig. 12] The size-mass relation used to set the initial Plummer scale radii has a poorly constrained intercept, q = 0.19 +/- 0.65, and is fitted to only seven resolved clumps. The simulations then extrapolate this relation to masses as low as 1e6 M_sun and as high as 6e7 M_sun, extending beyond the observed range 10^6.3-10^7.4 M_sun. Because the final sizes in Fig. 12 are directly tied to these initial radii, the authors should propagate the 1-sigma (or 2-sigma) uncertainties in m and q into the final-size predictions, or restrict the size conclusion to the mass range directly sampled by the resolved clumps.

minor comments (4)

[Sec. 4.3 and Figs. 6-7] The text says 'For each spatial configuration we ran 15 simulations', but Sec. 4.4.1 and the captions of Figs. 6 and 7 refer to 'the 30 simulations'. This inconsistency should be corrected, since it affects the interpretation of the quoted survival fractions and their uncertainties.
[Fig. 12 caption] The caption refers to 'the size-mass relation of Eq. 2.1' but the correct reference is Eq. (1).
[Table 5] The third row is labelled 'MN2', duplicating the second row; it should presumably be labelled 'MN3'.
[Fig. 10 caption] The bottom-panel caption says 'same as the left panel', but it should say 'same as the top panel'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are outputs of direct N-body integrations with observationally motivated initial conditions; the semi-analytic tidal-loss model is a transparent fit to simulation outputs, not a hidden reuse of the target conclusion.

full rationale

Walked the derivation chain: observed masses, sizes, and distances enter only as initial conditions (Table 1, Sect. 2); the dynamical-friction runs (Sect. 4) and resolved tidal-stripping runs (Sect. 5) are genuine AREPO N-body integrations whose outputs (survival fraction, final masses, final Plummer scale radii) are not imposed by construction. The combination step (Sect. 6.2, Eq. 17) fits a mass-loss law to the disk-shock simulations and then applies it to the dynamical-friction survivor distribution; this is a transparent emulator, explicitly labeled as a best fit with uncertainties, and the comparison to local GC masses and sizes is external, so the shifted peak near 2e6 Msun and the 'too large' size statement are not statistically forced by the target data. The main robustness caveat is that final sizes come from Plummer fits to all particles within 4 rs,i (Appendix A), so unbound tidal debris may inflate the fitted radius; this is a methodological concern about the bound remnant, not a definitional circularity, and the paper itself flags the complementary magnification uncertainty in Sect. 6.1. Self-citations (Claeyssens et al. 2023/2025, Adamo et al. 2023, Calura et al. 2022/2024) supply external observational data and prior literature comparisons; none carries a load-bearing uniqueness argument or an unverified ansatz. Verdict: no significant circularity.

Assumptions & free parameters 12 free parameters · 7 assumptions · 0 invented entities

The dynamical predictions rest on the standard N-body treatment of the halo, a set of fitted astrophysical relations (size-mass, mass function, spatial distribution, mass-loss model), and explicitly chosen bracketing assumptions (maximum halo mass, maximum disk, no DM in clumps, no major mergers). The central size conclusion depends mainly on the observed sizes being intrinsic, which is flagged in the paper.

free parameters (12)

Size-mass relation slope m = 0.21 ± 0.10
Fitted to seven resolved Sparkler clumps (Sect. 2.1) and used to set initial Plummer radii for all tidal stripping runs.
Size-mass relation intercept q = 0.19 ± 0.65
Same fit; poorly constrained, so initial radii have large uncertainty that is not propagated to final size or mass predictions.
Initial mass-distribution mean = 6.86 dex
Gaussian fit to the ten clump masses (Sect. 2.2); used to sample initial clump masses in the N-body runs.
Initial mass-distribution sigma = 0.31 dex
Same fit as above; sets the spread of sampled clump masses.
Distance distribution scale radius Re = 2.52 kpc (face-on), 4.06 kpc (edge-on), 2.41 kpc (spherical)
Fitted to observed projected distances under three morphology assumptions (Sect. 2.3.1).
Spherical Sersic index n = 0.70
Same distance fit for the spherical configuration.
Coulomb logarithm ln Lambda = 10
Assumed for dynamical friction timescale and Chandrasekhar formula (Sect. 4).
Tidal mass-loss model slope m = -0.35 ± 0.02
Fit to seven resolved-clump simulations with disk shocks (Eq. 17); used to shift mass distributions.
Tidal mass-loss model intercept q = 2.06 ± 0.15
Same fit; sets minimum disruption mass around 8e5 Msun.
Survival distance threshold = 1 kpc
Chosen by hand to separate survived from accreted clumps (Sect. 4.4).
Halo mass M200(z=0) = 5.43e11 Msun
Taken from the highest-mass end of the stellar-to-halo mass relation; chosen to maximize dynamical effects (Sect. 3).
Disk mass = 8.79e9 Msun
Set to z=0 stellar mass from the universemachine relation to maximize tidal shocks (Sect. 5.2).

assumptions (7)

domain assumption The Sparkler halo is adequately described by an NFW/Hernquist profile and does not undergo major mergers from z=1.4 to z=0.
Sect. 3: the halo is modeled as isolated and stationary; mass accretion is only pseudo-evolution. If a major merger occurred, orbits and stripping would change.
domain assumption The clumps contain no dark matter and their stellar mass equals their dynamical mass.
Sect. 4.2 and 6: used for dynamical friction estimates; if clumps had DM halos they would sink faster.
domain assumption The observed effective radii are intrinsic 2D half-mass radii.
Sect. 6.1: if sizes are overestimated by resolution or lensing, the central conclusion about size would reverse.
ad hoc to paper Clump internal density profiles are Plummer spheres with scale radius from the fitted size-mass relation.
Sect. 5.1: used to initialize resolved clumps; a different profile (e.g., King) would change tidal mass loss.
domain assumption Stellar mass loss by winds and supernovae is negligible for clump ages greater than 1 Gyr.
Sect. 4.3, citing Calura et al. 2014.
ad hoc to paper Orbits are circular or quasi-circular in the fiducial runs; eccentricities are treated only in an appendix.
Sect. 4.2 and 5.2; radial orbits would increase stripping by up to about 10 percent.
ad hoc to paper Disk properties (mass, scale length, scale height) are fixed at their z=0 values to maximize tidal effects.
Sect. 5.2.

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Cite this review

Pith. "Pith review of The dynamical evolution of the stellar clumps in the Sparkler galaxy." pith.science (2026). https://pith.science/paper/TCKWTM4X

@misc{pith2026250713904,
  author       = {Pith},
  title        = {Pith review of: The dynamical evolution of the stellar clumps in the Sparkler galaxy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TCKWTM4X}},
  note         = {Machine review of arXiv:2507.13904}
}

abstract

Recent JWST observations detected stellar clumps around the z=1.4 gravitationally lensed Sparkler galaxy (of stellar mass $M_*\sim 10^9\,\mathrm{M_\odot}$), with ages and metallicities compatible with globular clusters (GCs). However, most of their masses ($>10^6\,\mathrm{M_\odot}$) and sizes (>30 pc) are about 10 times those of GCs in the local Universe. To assess whether these clumps can evolve into GCs, we performed N-body simulations of their dynamical evolution from z=1.4 to z=0 (9.23 Gyr), under the effect of dynamical friction and tidal stripping. Dynamical friction is studied performing multiple runs of a clump system in a Sparkler-like spherical halo of mass $M_{200}\simeq 5\times 10^{11}\,\mathrm{M_\odot}$ (from the stellar-to-halo mass relation). For the tidal stripping, we simulated resolved clumps, orbiting in an external, static gravitational potential including the same halo as in the dynamical friction simulations and also a Sparkler-like stellar disk. Dynamical friction causes the clumps with mass $>10^7\,\mathrm{M_\odot}$ to sink into the galaxy central regions, possibly contributing to the bulge growth. In absence of tidal stripping, the mass distribution of the surviving clumps (40%) peaks at $5\times 10^6\,\mathrm{M_\odot}$, implying the presence of uncommonly over-massive clumps at z=0. Tidal shocks by the stellar disk strip considerable mass from low-mass clumps, even though their sizes remain larger than those of present-day GCs. When the surviving clumps are corrected for tidal stripping, their mass distribution peak shifts to $2\times 10^6\,\mathrm{M_\odot}$, compatible with massive GCs. Our simulations suggest that a fraction of the Sparkler clumps is expected to fall into the central regions, where they might become bulge fossil fragments or contribute to form a nuclear star cluster. The remaining clumps are too large in size to be progenitors of GCs.

Figures

Figures reproduced from arXiv: 2507.13904 by the authors.

**Figure 1.** Size-mass relation for the extraplanar clumps of the Sparkler galaxy (Sect. 2). Clumps are shown as black empty circles, while the best-fitting line (obtained as described in Sect. 2.1) is plotted as the red solid line. The red shaded areas (from opaque to transparent red) show the 1σ, 2σ, 3σ intervals of the fit, respectively [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Top panel: density profile of the highest-mass dark matter halo of the Sparkler galaxy, as predicted by the models by Correa et al. (2015). Black lines indicate the z = 0 profile while red lines the z = 1.378 one. The dotted lines mark r−2, the dashed lines mark r200. The grey shaded area marks the region within 1 and 10 kpc, which is the interval under study. Bottom panel: difference between the halo density profil… view at source ↗

**Figure 3.** Dynamical-friction timescales for clumps of mass 106 , 107 and 108 M⊙ (dashed, dotted and solid lines, respectively) as a function of the distance from the centre of the halo. The red-dashed area covers the range of dynamical friction timescales relative to the interval of clump masses and distances of interest. The grey horizontal line corresponds to the look-back time at z ≈ 1.4 tlb = 9.23 Gyr, which divides the p… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Left panel: evolution over 9.23 Gyr of the distances from the centre of the point-mass clumps orbiting in a Sparkler-like potential with circular orbit initial conditions. On the x−axis, the mass of the clump is reported, and the black arrows point from the initial to …

**Figure 5.** Figure 5: Initial spatial distribution of the 10 clumps in representative simulations of the three different spatial configurations described in Sect. 2.3.1. The clump initial positions are determined as described in Sect. 4.3. From left to right: face-on disk, edge-on disk and …

**Figure 6.** Figure 6: Fraction of survived clumps of each of the 30 simulations ran in the face-on (top panel, in red), edge-on (middle panel, in green) and spherical (bottom panel, in blue) configurations. A clump is flagged as survived if its distance from the centre of the halo at the en…

**Figure 9.** Figure 9 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: Top panel: retained mass (relative to the initial one) as a function of time, for clumps of different masses in circular orbits at 1 kpc from the centre of the system. The residual mass is computed within 4rs , where rs is the initial Plummer scale radius. In this ca…

**Figure 11.** Figure 11: Comparison between the initial conditions and the snapshot at 0.51 Gyr for the tidal-shock simulation described in Sect. 5.2 in the case of mass M∗ = 106 M⊙. The top and bottom panels show the projections on the x−y and x−z planes, respectively. The clump stellar part…

**Figure 12.** Figure 12: Comparison between the final (rs,f) and initial (rs,i) scale radii of the clumps simulated in Sect. 5, when modelled as Plummer sphere, as in Eq. 15). The initial scale radii are computed from the size-mass relation of Eq. 2.1. The final scale radii are obtained fitti…

**Figure 13.** Figure 13: Left panel: fraction of the clump mass lost after 9.23 Gyr log(|∆M∗,fin|/M∗,ini) as a function of the initial mass log(M∗,ini). Here ∆M∗,fin = M∗,fin(4rs) − M∗,ini, where M∗,fin(4rs) is the final mass within four initial Plummer scale radii, computed as in Sect. 5.2. …

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

[2] [2]

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 format....

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