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Lensing Without Borders: Measurements of galaxy-galaxy lensing and projected galaxy clustering in DESI DR1

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

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Pith's one-line read This paper establishes that DESI DR1 galaxy-galaxy lensing is free of significant lens-sample systematics and is validated for cosmological use once the HSC-Y3 photometric redshift shifts and the covariance uncertainty are applied.

desk verdict A thorough, honest validation of DESI DR1 galaxy-galaxy lensing that deserves referee time, but the source-redshift trend explanation would be stronger with a leave-HSC-out test and propagated shift uncertainties. read the letter →

arxiv 2506.21677 v1 pith:TQENNRJD submitted 2025-06-26 astro-ph.CO

S. Heydenreich , A. Leauthaud , C. Blake , Z. Sun , J. U. Lange , T. Zhang , M. DeMartino , A. J. Ross
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J. Aguilar S. Ahlen D. Bianchi D. Brooks F. J. Castander T. Claybaugh A. Cuceu A. de la Macorra J. DeRose Arjun Dey Biprateep Dey P. Doel N. Emas S. Ferraro A. Font-Ribera J. E. Forero-Romero C. Garcia-Quintero E. Gaztañaga S. Gontcho A Gontcho G. Gutierrez B. Hadzhiyska K. Honscheid D. Huterer M. Ishak N. Jeffrey S. Joudaki E. Jullo S. Juneau D. Kirkby T. Kisner A. Kremin A. Krolewski O. Lahav C. Lamman M. Landriau L. Le Guillou M. Manera A. Meisner R. Miquel S. Nadathur N. Palanque-Delabrouille W. J. Percival A. Porredon F. Prada I. Pérez-Ràfols G. Rossi R. Ruggeri E. Sanchez C. Saulder D. Schlegel A. Semenaite J. Silber D. Sprayberry G. Tarlé B. A. Weaver S. Yuan P. Zarrouk R. Zhou H. Zou
This is my paper
classification astro-ph.CO
keywords galaxy-galaxylensingDESIDR1excesssurfacemassdensityphotometricredshiftcalibrationsourcetrendprojectedclusteringweaksystematicscovarianceestimation
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 reports galaxy-galaxy lensing measurements — the excess surface mass density $\Delta\Sigma$ and tangential shear $\gamma_t$ — from the first DESI data release, cross-correlating the Bright Galaxy Sample and Luminous Red Galaxies with source galaxies from HSC, KiDS, DES, and SDSS. Its aim is to show that these measurements are clean enough to feed cosmological analyses. The authors find no significant trends of the lensing amplitude with DESI lens properties such as fiber completeness, stellar density, imaging depth, or extinction, but they do find a significant rise of the lensing amplitude with source redshift. They argue that this trend is explained by photometric redshift shifts in the HSC-Y3 source sample reported by Li et al. (2023), after which all source-redshift trends become consistent with mock expectations. The remaining excess scatter between surveys is confined to scales below $1\,h^{-1}\mathrm{Mpc}$ and loses significance when the uncertainty in the analytical covariance is marginalized over, so the paper concludes the measurements are validated for cosmology with those two caveats carried forward.

What carries the argument

The load-bearing element is the matched-filter lensing amplitude $A_{\Delta\Sigma}$, a scalar summary of each $\Delta\Sigma$ datavector obtained with weights $\mathbf{w}\propto \mathbf{C}^{-1}\Delta\Sigma_{\mathrm{ref}}$, where the reference profile comes from mock galaxy catalogs calibrated to clustering and $\mathbf{C}$ is the analytical covariance. Because multiplicative systematics such as shear calibration errors and photo-z biases rescale the lensing signal, any discrepancy between surveys shows up as a difference in this one number. The source-redshift test then plots $A_{\Delta\Sigma}$ against the effective source redshift, exploiting the fact that the true $\Delta\Sigma$ is a physical property of the lens and should not depend on the sources. The proposed explanation mechanism is the set of HSC-Y3 photometric redshift shifts $\Delta z_3=0.115$, $\Delta z_4=0.192$, which the paper applies to the source distributions and shows to remove the observed trends.

What would settle it

A direct clustering-redshift or spectroscopic calibration of the HSC-Y3 source redshift distributions that rules out shifts of $\Delta z_3=0.115$ and $\Delta z_4=0.192$ would falsify the claim that the trend is a photo-z bias; equivalently, re-running the source-redshift test on an independent lens sample and finding the slope persists after applying the shifts would contradict the validation.

Watch

Extended reading notes

Core claim

The central claim is that the DESI DR1 galaxy-galaxy lensing signal is free of significant lens-sample systematics and that its one notable anomaly — an increasing lensing amplitude with source redshift — is a source-side calibration effect. Using a matched-filter amplitude $A_{\Delta\Sigma}$ for each DESI lens bin, the paper shows the amplitude does not vary with potential contaminants of the DESI lens sample. A significant slope with source redshift is detected, strongest in the first LRG bin; the paper rules out under-estimated intrinsic alignments as the cause and shows that shifting the HSC-Y3 source redshift distributions by $\Delta z_3=0.115$ and $\Delta z_4=0.192$ brings all measured trends into agreement with the mock-based predictions. Excess scatter between surveys is driven by small scales, $r_p \leq 1\,h^{-1}\mathrm{Mpc}$, and disappears once a roughly 10% uncertainty in the analytical covariance is marginalized over. The paper's conclusion is that the measurements are sufficiently validated for subsequent cosmological analyses, provided the HSC-Y3 redshift shifts and the covariance uncertainty are taken into account.

Load-bearing premise

The central claim stands on the assumption that the HSC-Y3 source-galaxy redshift distributions are biased by exactly the shifts reported in Li et al. (2023), so that applying those shifts removes the source-redshift trend; if the true cause is a different combination of systematics, the validation conclusion loses its main support.

Editorial extensions

If this is right

  • The released DESI DR1 $\Delta\Sigma$, $\gamma_t$, and projected clustering $w_p$ measurements can be used in combined 3x2pt cosmological analyses without adding an unknown DESI lens-systematics term, provided the HSC-Y3 redshift shifts are applied.
  • A future analysis that does not apply the HSC-Y3 shifts should expect a spurious positive slope of the lensing amplitude with source redshift in high-redshift lens bins, most prominently the first LRG bin.
  • Scale cuts at $r_p > 1\,h^{-1}\mathrm{Mpc}$ are the safe route for precision cosmology, because the small-scale excess scatter has no identified cause and its significance depends on the covariance model.
  • The absence of lens-homogeneity trends supports the DESI DR1 fiber-incompleteness and imaging-systematics weights as adequate for lensing and clustering studies.
  • The data-level tendency for KiDS amplitudes to sit below DES and HSC gives a direct observational counterpart to the lower $S_8$ values reported by KiDS cosmological analyses.

Reading between the lines

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

  • If the Li et al. shifts are correct, cosmic-shear analyses of HSC-Y3 that omit them would systematically underestimate the lensing signal for high-redshift source bins, which would bias their inferred clustering amplitude upward; testing whether the $S_8$ tension is reduced when the shifts are included would provide an independent check.
  • The same redshift bias is plausibly present, at smaller amplitude, in HSC-Y1 data; the paper only raises this as a suspicion, so a dedicated re-analysis of HSC-Y1 GGL with free photo-z shifts would be a direct extension.
  • A joint fit that lets the HSC-Y3 shifts, the covariance amplitude, and survey-specific calibration terms float simultaneously would convert the paper's assumption-then-check procedure into a proper parameter constraint.
  • For uses of the public data vectors, the conservative default is to restrict to $r_p > 1\,h^{-1}\mathrm{Mpc}$ unless the covariance uncertainty is explicitly marginalized, since the small-scale scatter remains unexplained.
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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. This paper presents galaxy-galaxy lensing (GGL) measurements of DESI DR1 Bright Galaxy Sample and Luminous Red Galaxy lenses, cross-correlated with source galaxies from HSC-Y3, KiDS-1000, DES-Y3, and SDSS. The excess surface mass density and tangential shear are measured with a unified pipeline, corrected for multiplicative shear bias, photo-z dilution, and lens magnification bias, and analyzed via matched-filter lensing amplitudes. The authors perform blinded lens-homogeneity tests (splits by NTILE, stellar density, depth, seeing, EBV) and source-redshift tests, compare outlier rates against random realizations, estimate an amplitude systematic with a Gaussian mixture likelihood, and measure projected clustering wp with PIP weights and angular upweighting. The main findings are: no significant trends with lens-sample properties; a significant trend of lensing amplitude with source redshift, primarily in the first LRG bin, that is reported to disappear after applying Li et al. (2023) HSC-Y3 redshift shifts; and excess scatter between lensing amplitudes on small scales that is not robust to marginalization over covariance uncertainty. The measurements and covariance estimates are intended for public release and as inputs to future cosmological analyses.

Significance. If the validation claims hold, this is a valuable public dataset and a strong data-level consistency check among the major weak-lensing surveys, directly relevant to the 'lensing is low' effect and to DESI 3x2pt analyses. The paper has notable strengths: a blinded analysis with pre-registered unblinding criteria; a mathematically proven statement (Appendix E) that matched-filter amplitude comparisons are unbiased for any template shape, with only sensitivity depending on the template; extensive systematics tests against AbacusSummit simulations and L+24 mocks; and explicit, honest caveats about jackknife covariance limitations and the covariance-marginalization sensitivity of the small-scale scatter. The central interpretation, however, rests on applying external HSC photo-z point estimates without propagating their uncertainty, and the evidence for that interpretation would be materially strengthened by a leave-HSC-out test. These are fixable within the scope of the manuscript, so the result is promising but not yet fully quantitatively supported.

major comments (3)
  1. [§8.1.2, Fig. 9, §8.3] The central validation statement that the source-redshift trend is "most likely" explained by HSC-Y3 photo-z shifts applies the Li et al. (2023) point estimates Δz3=0.115 and Δz4=0.192 as exact numbers. Their uncertainties are not propagated into Fig. 9, and no goodness-of-fit statistic is given for the statement that the shifted slope is "completely consistent" with the L+24 mock prediction. Please propagate the Li et al. posterior (or at minimum show a band on the shifted amplitudes derived from the quoted shift errors) and report the residual slope and its uncertainty after shifting. Without this, the agreement in Fig. 9 is not quantitatively established.
  2. [§8.1.2, Fig. 5] A leave-HSC-out diagnostic is needed. The trend in Fig. 5 is fitted jointly over HSC, KiDS, and DES points, and only the HSC bins 3 and 4 are shifted; the paper does not report the slope of AΔΣ versus source redshift obtained from KiDS and DES alone. Since §8.1.3 states that the first LRG-bin trend is dominated by the HSC third and fourth source bins, the HSC-shift explanation would be strongly supported by showing that the residual slope is consistent with zero after removing HSC or after shifting with uncertainties. If a significant positive slope remains in KiDS+DES, the conclusion in §8.3 that GGL is "sufficiently validated" would need to be weakened.
  3. [§7.5, §8.2, Abstract] The abstract states that the measurements "find excess scatter... driven primarily by small-scale measurements", but §8.2 reports that this excess disappears once the uncertainty in the covariance estimate is marginalized over, and §7.5 gives only ≳2σ evidence even before that marginalization. The abstract and §9 should be reworded to present the small-scale scatter as a model-dependent systematic-uncertainty indicator rather than a robust detection, or the analysis should include a covariance-inference treatment that quantifies the evidence while marginalizing over covariance uncertainty.
minor comments (4)
  1. [Throughout] There are numerous typographical errors that should be corrected: "the the first data release" in the Abstract and Section 1, "in the the first year" in the Abstract, "respecitvely" in Section 4.1, "rougly" in Section 2.1, "galaxes" in Section 3.2, "sigificant" in the Appendix, "neglible" in Section 4.2.4, "compomnent" in Section 4.5, and "trent" in Section 9.
  2. [§5.5, §5.8.1, Abstract] The abstract's unqualified statement that no significant trends with lens-systematic properties are found is stronger than the text's own caution: Section 5.5 states that the jackknife covariances used for the lens homogeneity tests "are unable to be validated... to the standards required for cosmological analysis", and Section 5.8.1 again urges caution. Please soften the abstract/conclusion wording to reflect this stated limitation.
  3. [5.2] The EBV test was added post-unblinding and this is disclosed, but the unblinding criteria in Section 5.7 do not mention that this additional test was not part of the pre-registered set. Please explicitly label the EBV test as a post-unblinding robustness check in the outlier-accounting discussion, since otherwise the outlier counting in Section 7.4 could be misread as including a test that was not part of the blinded protocol.
  4. [Appendix E] The proof that an arbitrary template yields unbiased comparisons assumes that the covariance matrix C is correct. If C is misestimated, the bias factor α in Eq. (E9) can differ between surveys with different covariance structures, potentially creating apparent amplitude differences. Please state this assumption explicitly and, where possible, show that the conclusions are robust to the 10% covariance calibration uncertainty quoted in Section 4.5.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the validation argument rests on external benchmarks and imported HSC photo-z shifts, not on fitted inputs.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The matched-filter lensing amplitudes are proven in Appendix E to give unbiased comparisons for any template shape, so the A_DeltaSigma comparisons are not defined to reproduce the AbacusSummit/L+24 reference by construction. The lens homogeneity tests compare DESI DR1 measurements split by systematic proxies such as NTILE, STARDENS, and imaging depth against jackknife and analytical covariances; no fitted parameter is relabeled as a prediction. The source-redshift trend is a measured finding, and the HSC-Y3 n(z) shifts (Delta z3 = 0.115, Delta z4 = 0.192) are imported from the external Li et al. (2023) cosmic shear analysis rather than fitted to the DESI data to force the trend to vanish. Figure 12 is explicitly diagnostic, computing the shifts that would set amplitudes to unity, but those values are not used as the correction; the correction comes from Li et al. (2023). Self-citations to L+24 and Y+24 are used as pre-existing simulation and covariance inputs that are externally benchmarked against AbacusSummit, Buzzard, and jackknife estimates, and they are not invoked as uniqueness theorems. The paper also self-reports its own limitations, including that the jackknife covariances for lens homogeneity tests are not validated to cosmological standards and that the significance of small-scale excess scatter disappears when covariance uncertainty is marginalized; these are statistical robustness caveats, not circular steps. The main conditional conclusion is honestly framed: GGL is validated for cosmology only if the HSC-Y3 shifts and covariance uncertainty are taken into account. Because the central validation claim does not reduce to a fitted parameter, a self-citation chain, or a definitional identity, no circularity is present.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claims rest on externally calibrated inputs (HSC photo-z shifts, mock contamination estimates, survey shear calibrations, analytical covariance) rather than on parameters fitted within this paper. The listed free parameters are analysis choices or imported corrections that the validation conclusions depend on; none are derived from first principles here.

free parameters (4)
  • HSC-Y3 redshift shifts = Δz3=0.115, Δz4=0.192
    Imported from Li et al. (2023), applied post-unblinding in Section 8.1.2 and Fig. 9; the central claim that the source-redshift trend is explained depends on these values.
  • BGS absolute magnitude cut thresholds = MR < -19.5, -20.5, -21 for bins [0.1,0.2], [0.2,0.3], [0.3,0.4]
    Hand-chosen following Y+24 to maintain constant comoving density (Section 2.1); defines the BGS subsample used in all measurements.
  • Source-lens contamination threshold = 3% contamination from L+24 mocks
    Analysis choice in Section 4.4 selecting conservative source-lens bin pairs; affects the data vector and all amplitude tests.
  • Scale cuts = 0.5 arcmin minimum separation; maximum at 50% pair completeness
    Section 4.4: angle and pair-count cuts to limit blending and unreliable covariance; changes the radial range of the measurements.
assumptions (6)
  • domain assumption The HSC-Y3 photometric redshift distributions are biased as quantified by Li et al. (2023), so applying those shifts makes all source-redshift trends consistent with mock expectations.
    Assumed in Section 8.1.2 and Fig. 9 to reach the paper's main validation conclusion; if the shifts are incorrect, the residual source-redshift trend is unexplained.
  • domain assumption The L+24 mock estimates of intrinsic alignment, boost factor, source magnification, and reduced-shear contamination are accurate enough to set the conservative source-lens cuts and the expected slope baseline.
    Section 4.4 uses these to exclude source-lens pairs; Section 8.1.1 uses the mock slope beta=0.05 as the expectation. Underestimated IA would bias the source-redshift test.
  • domain assumption The analytical covariance, with boundary corrections calibrated on lognormal simulations, accurately represents the true noise of the measurements.
    Section 4.5; excess scatter and outlier significances depend on this covariance. The authors note the small-scale scatter loses significance if covariance uncertainty is marginalized (Section 8.2).
  • domain assumption The AbacusSummit HOD reference datavector is a valid matched-filter template; Appendix E proves amplitude comparisons remain unbiased for any template, with only sensitivity affected.
    Section 5.1 uses this template to compress excess surface density into lensing amplitudes.
  • standard math The Planck fiducial cosmology is adequate for distance and critical surface density conversions.
    Section 1 and Appendix G show alternative cosmologies do not significantly change results.
  • domain assumption The conservative source-lens cuts (less than 3% predicted contamination) remove detrimental intrinsic alignment and boost factor contamination.
    Section 4.4 and Table 2; the conclusion of no significant intrinsic alignment contamination relies on this.

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Cite this review

Pith. "Pith review of Lensing Without Borders: Measurements of galaxy-galaxy lensing and projected galaxy clustering in DESI DR1." pith.science (2026). https://pith.science/paper/TQENNRJD

@misc{pith2026250621677,
  author       = {Pith},
  title        = {Pith review of: Lensing Without Borders: Measurements of galaxy-galaxy lensing and projected galaxy clustering in DESI DR1},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQENNRJD}},
  note         = {Machine review of arXiv:2506.21677}
}
abstract

We present Galaxy-Galaxy Lensing measurements obtained by cross-correlating spectroscopically observed galaxies from the first data release of the Dark Energy Spectroscopic Instrument (DESI) with source galaxies from the Hyper Suprime-Cam Subaru Strategic Survey, the Kilo-Degree Survey, the Sloan Digital Sky Survey, and the Dark Energy Survey. Specifically, we measure the excess surface mass density $\Delta\Sigma$ and tangential shear $\gamma_\mathrm{t}$ for the Bright Galaxy Sample and Luminous Red Galaxies measured within the first year of observations with DESI. To ensure robustness, we test the measurements for systematic biases, finding no significant trends related to the properties of the \acrshort{desi} lens galaxies. We identify a significant trend with the average redshift of source galaxies, however, this trend vanishes once we apply shifts to the Hyper Suprime-Cam Subaru Strategic Survey redshift distributions that are also favored by their fiducial cosmology analysis. Additionally, we compare the observed scatter in the measurements with the theoretical covariance and find excess scatter, driven primarily by small-scale measurements of $r\leq 1 \, \mathrm{Mpc}/h$; measurements on larger scales are consistent at the $2\,\sigma$ level. We further present the projected clustering measurements $w_p$ of the galaxy samples in the the first data release of DESI. These measurements, which will be made publicly available, serve as a foundation for forthcoming cosmological analyses.

Figures

Figures reproduced from arXiv: 2506.21677 by the authors.

Figure 1
Figure 1. The observed number density of DESI DR1 BGS, overlaid with the footprints of the four imaging surveys. One can see that DESI DR1 has very varying completeness over its footprint, in particular it is much more complete in the KiDS and HSC regions than in DES. We note that the footprint of the DESI DR1 LRG looks very similar to the BGS [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Absolute magnitudes of DESI DR1 BGS as a function of redshift. The galaxies that pass the magnitude cut described in Sect. 2.1 are the ones above the dashed lines. In total, 61% of all galaxies pass the magnitude cut. high number density of galaxies. L+24 show that this effect strongly biases the lensing measurements, but can be com￾pletely removed by applying individual-inverse-probability (IIP) weights to the lens… view at source ↗
Figure 3
Figure 3. Photo-z distribution of source bins compared to DESI DR1 lens galaxies. The different panels show the source redshift distributions for the different imaging surveys, respectively. In every panel, the dashed lines denote the redshift boundaries of the BGS and LRG lens samples, where the BGS extends from 0.1 ≤ z < 0.4, and the LRGs extend from 0.4 ≤ z < 1.1. Each lens sample is subsequently split into three redshift … view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Lensing signal for the different source samples. The grey shaded regions are excluded due to our scale cuts outlined in Sect. 4.4. The dotted line corresponds to the boundary between small and large scales. The dashed lines corresponds to the reference datavector extra…
Figure 5
Figure 5. Figure 5: Lensing amplitude as a function of source redshift. The red line represents the best-fit slope β; the grey band denotes its 1σ uncertainty. The black line represents the slope we find when feeding the datavectors contaminated by all uncorrected lensing systematics (int…
Figure 6
Figure 6. Figure 6: Estimating significance of A∆Σ slopes. Top panel: The number of outliers from the 108 measurements of degeneracies with potential system￾atics. We plot the number of outliers in the data (blue), the average number of outliers in the randoms (black), and the probability…
Figure 7
Figure 7. Figure 7: Estimate of the systematic uncertainty between different lensing surveys, compared to the systematic uncertainties of the lensing surveys themselves. The downwards-facing triangles denote the statistical uncertainties of the measurements determined from our covariance …
Figure 8
Figure 8. Figure 8: The projected clustering measurements wp for the DESI DR1 BGS and LRG. One can see that, apart from some noise upticks at small scales in the regions with small NTILE values, the projected clustering is broadly consistent between different NTILE values. 8. DISCUSSION W…
Figure 9
Figure 9. Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: The lensing amplitude A∆Σ as a function of the parameter NTILE. For the BGS galaxies, we split the samples into 4 bins with NTILE=1,2,3,4, respectively. For the LRG galaxies, we split into NTILE=1, NTILE=2, NTILE ∈ {3, 4}, NTILE ∈ 5, 6, 7. APPENDIX SECONDARY FIGURES A…
Figure 11
Figure 11. Figure 11: Lensing B-modes for the different source surveys. The grey shaded regions are excluded due to our scale cuts outlined in Sect. 4.4. The dashed line corresponds to the boundary between small and large scales. We also denote χ 2 and p-values for each source survey. We d…
Figure 12
Figure 12. Figure 12: Shifts in mean redshift that would set all lensing amplitudes to the mean lensing amplitude. We calculate the shifts by determining the linear offset of the source redshift distribution that would lead to a Σcrit that yields a lensing amplitude of unity. entirely accu…
Figure 13
Figure 13. Figure 13: Same as [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: Validation plots for the fits to secondary quantity, especially ∆χ 2 . Left: BGS, right: LRG and the first LRG bins, but the significance does not cross the traditional p < 0.05 threshold. The lens homogeneity tests do not show any sigificant trends when the HSC-Y3 da…
Figure 15
Figure 15. Figure 15: Magnification bias measured with different methods and comparison with simulations Category Bin 1 Bin 2 Bin 3 # cut α # cut α # cut α FMC 2 0.00 0 -0.00 1 0.00 g-r > -1 0 0.00 0 0.00 0 0.00 g-r < 4 0 0.00 0 0.00 0 0.00 r-z > -1 0 0.00 0 0.00 0 0.00 r-z < 4 0 0.00 0 0.…
Figure 16
Figure 16. Figure 16: Same as [PITH_FULL_IMAGE:figures/full_fig_p029_16.png]
Figure 17
Figure 17. Figure 17: The boost factor for the measurements presented in [PITH_FULL_IMAGE:figures/full_fig_p030_17.png]
Figure 18
Figure 18. Figure 18: Difference between the ∆Σ measurements of the alternative cosmologies and the fiducial cosmology. The dashed lines indicate the covariance estimates. IMPACT OF ALTERNATIVE COSMOLOGIES To calculate ∆Σ, we have to assume a cosmological model to calculate the angular dia…

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