Fact-checked by Grok 2 weeks ago

Baade's Window

Baade's Window is a compact region of the sky, spanning approximately 1° in diameter, located in the constellation Sagittarius at galactic coordinates roughly l = 1°, b = −4°, where interstellar dust extinction is unusually low, enabling a relatively unobscured optical view into the dense central bulge of the Milky Way galaxy, about 26,000 light-years distant.^[1]^[2] Named after German-American astronomer Walter Baade, who identified it in the 1940s while seeking a clear path to the galactic center using the 100-inch Hooker Telescope at Mount Wilson Observatory, Baade's Window became instrumental in early efforts to map the structure and scale of our galaxy.^[3] Baade exploited this "window" to observe variable stars, particularly RR Lyrae types, in the bulge, which allowed him to refine distance estimates to the galactic nucleus and contribute to understanding stellar populations.^[1] The region's significance lies in its role as one of the few dust-free sightlines toward the galactic bulge, facilitating detailed studies of its ancient stellar content despite the overall opacity of the Milky Way's disk.^[2]^[3] Through Baade's Window, astronomers have resolved millions of stars, including metal-rich giants and variable stars, revealing the bulge's chemical composition—typically enhanced in heavy elements compared to the solar neighborhood—and supporting evidence for a barred structure in the galaxy's core aligned toward the Sun.^[1]^[3] Prominent features visible include the globular clusters NGC 6522 (magnitude 8.6, approximately 25,000 light-years away and potentially 12 billion years old) and NGC 6528 (magnitude 9.6, about 25,800 light-years distant), alongside dark nebulae such as Barnard 298.^[2] In modern astronomy, Baade's Window serves as a primary field for microlensing surveys, such as the Optical Gravitational Lensing Experiment (OGLE) and the Massive Compact Halo Object (MACHO) project, which have detected dozens of gravitational lensing events by monitoring stars in this dense region to probe dark matter and the bulge's mass distribution.^[3] These observations continue to yield insights into the galaxy's formation, stellar evolution, and interstellar medium, underscoring the window's enduring value despite advancements in infrared and radio telescopes that bypass dust in other directions.^[2]

Discovery and History

Discovery

Walter Baade, a German-born astronomer at Mount Wilson Observatory, identified Baade's Window in the mid-1940s while conducting deep-sky observations with the 100-inch Hooker Telescope.^[4] These efforts were facilitated by World War II-era blackouts in the Los Angeles area, which drastically reduced light pollution and permitted extended exposure times on photographic plates to capture faint stars near the galactic center.^[4] Baade's work during this period capitalized on the unusually dark skies, allowing him to push the limits of ground-based imaging in the direction of Sagittarius. Through these observations, Baade recognized a relatively dust-free aperture in the dense interstellar medium, offering an unobscured line of sight to remote stars within the Milky Way's bulge approximately 8 kiloparsecs away.^[5] This region, centered near globular clusters NGC 6522 and NGC 6528, stood out amid the typical heavy extinction that obscures views toward the galactic core. Baade's detailed plates revealed a rich field of resolved stars, marking the first clear glimpse into the inner galaxy's stellar content. The feature was subsequently named Baade's Window in honor of its discoverer and described as a "window" providing access to the otherwise hidden galactic bulge, with initial findings published in 1946.^[5] This identification laid foundational groundwork for probing the structure and composition of the Milky Way's central regions.

Early Observations

Following its identification as a relatively dust-free line of sight toward the galactic bulge, Walter Baade initiated systematic photographic observations of Baade's Window in the mid-1940s using the 100-inch Hooker telescope at Mount Wilson Observatory. These early plates resolved the region into a dense field of individual stars, revealing the high stellar density of the inner Milky Way and enabling the detection of variable stars, including RR Lyrae stars characteristic of Population II systems. Baade's work highlighted the window's utility for probing the structure of the galactic bulge, where globular clusters such as NGC 6522 could be studied in detail for the first time.^[6] In the late 1940s, Baade exposed 12 one-hour photographic plates of the window, assisted by night assistants Gerrit Oom and others, specifically to search for RR Lyrae variables and assess the bulge's stellar content. These observations confirmed the presence of numerous RR Lyrae stars, whose periods and luminosities served as distance indicators, contributing to an initial estimate of the galactic center's distance at approximately 27,000 light-years. By the 1950s, with access to the 200-inch Hale Telescope at Palomar Observatory, Baade extended these efforts, obtaining deeper plates that further delineated the bulge's stellar density and identified additional Cepheid variables, aiding in the calibration of absolute magnitudes for Population II objects.^[6] These investigations played a pivotal role in revising the cosmic distance scale. Baade compared the apparent magnitudes of RR Lyrae stars in the bulge with those in nearby globular clusters to establish their absolute magnitudes, which informed the period-luminosity relation for Cepheids and revealed the distinction between classical (Population I) and Population II Cepheids. This led to a doubling of extragalactic distances, as announced in Baade's 1952 report to the International Astronomical Union. Seminal publications include Baade's 1944 Astrophysical Journal papers introducing stellar populations and his subsequent works through the 1950s on variable stars and galactic structure.^[7] Early analyses faced challenges from residual interstellar extinction, necessitating corrections for color excesses in variable star photometry, and from contamination by foreground disk stars, which obscured pure bulge samples and complicated density estimates. Despite these obstacles, Baade's photographic surveys established Baade's Window as a cornerstone for mid-20th-century studies of the galactic bulge.^[8]

Physical Characteristics

Location and Extent

Baade's Window is situated in the constellation Sagittarius at equatorial coordinates of right ascension 18ʰ 03ᵐ 24ˢ and declination −30° 01′ 30″ (J2000 epoch).^[9] In Galactic coordinates, its center lies at longitude l = 1.02^\circ and latitude b = -3.92^\circ.^[10] This positioning places it in a direction that aligns closely with the inner Galactic bulge, approximately 4° from the nominal Galactic center. The region spans an angular diameter of about 1°, forming a relatively clear "peephole" through the interstellar medium, centered near the bright star Gamma Sagittarii (also known as Alnasl).^[6] Observations through this window reveal stars at a typical distance of roughly 8 kpc from Earth, corresponding to the far side of the Galactic bulge.^[11] The low levels of obscuring material in this area enable direct views of the bulge's stellar content, unlike adjacent sightlines where extinction is significantly higher. The boundaries of Baade's Window are demarcated by a sharp increase in interstellar dust opacity in the surrounding fields, which progressively obscures views toward the Galactic center beyond the central ~1° extent.^[10] This localized transparency arises from a relative paucity of dust along the line of sight, allowing the window to serve as a key vantage point for studying the bulge's structure.

Interstellar Dust and Extinction

Baade's Window is distinguished by its low column density of interstellar dust along the line of sight to the Galactic bulge, which results in a visual extinction of A_V ranging from 1.26 to 2.79 mag. This value is markedly lower than the typical A_V exceeding 20 mag found in most directions toward the bulge, where dense dust lanes severely obscure the view.^[12]^[13] The corresponding reddening in this region is E(B-V) ≈ 0.5 mag, reflecting the reduced amount of dust scattering and absorption affecting shorter wavelengths. This low extinction enables deeper observations of bulge stars compared to heavily obscured sightlines, where cumulative dust layers amplify attenuation. The interstellar dust in the line of sight primarily consists of silicate and carbon grains, typical of the Galactic diffuse interstellar medium, with silicates contributing to the 9.7 μm absorption feature and carbon-based particles (such as graphite or amorphous carbon) responsible for the 2175 Å extinction bump. These components arise from a combination of stellar ejecta and grain growth processes in the interstellar medium. The relative clarity of Baade's Window stems from its alignment with a gap in the dust lane within the Galactic plane, allowing a clearer path through the otherwise dense foreground material near Sagittarius.^[13] In comparison, adjacent fields known as "mini-windows" exhibit higher extinction, with near-infrared A_K values around 0.28 mag versus 0.18 mag in Baade's Window, underscoring the localized nature of these low-dust regions.^[13]

Scientific Significance

Stellar Populations

Baade's observations through the low-extinction region known as Baade's Window enabled the first clear distinction of stellar populations in the Milky Way's Galactic bulge, where he identified Population II stars as older, metal-poor systems contrasting with the younger, metal-rich Population I stars dominant in the disk. This classification, initially inspired by resolved imaging of external galaxies like M31 but applied to our Galaxy via the Window, revealed the bulge as hosting a predominantly ancient stellar component with subdued colors and fainter luminosities compared to disk populations.^[14] The dominant stellar types observed in Baade's Window include RR Lyrae variables, red giants, and horizontal branch stars, which serve as key tracers of the old bulge population. RR Lyrae stars, pulsating horizontal branch objects with periods of 0.2–1 day, are particularly prominent and indicate a metal-poor, evolved stellar cohort. Red giants, often late-type M giants, form the bulk of the visible population due to their high luminosity, while horizontal branch stars provide insights into helium-burning phases in low-mass stars. Spectroscopic studies of these stars reveal a metallicity distribution with evidence of multiple populations, peaking near [Fe/H] ≈ -0.3 for the metal-poor component and around +0.3 for the metal-rich one, reflecting a bimodal structure rather than a unimodal spread. This distribution underscores the bulge's complex formation history, with contributions from both early, metal-poor accretion and later enrichment. Age estimates for bulge stars in Baade's Window place the dominant population at 10–12 Gyr, consistent with rapid formation shortly after the Big Bang and minimal recent star formation.^[15] Observations in Baade's Window have been instrumental in calibrating the period-luminosity relation for RR Lyrae stars, serving as a standard distance indicator for old populations across galaxies. By measuring the apparent magnitudes and periods of these variables through the Window, astronomers derived the distance to the Galactic center at approximately 8 kpc, establishing a benchmark for bulge structure and extragalactic distances.

Galactic Bulge Structure

Baade's Window provides a clear line of sight into the Galactic bulge, revealing its bar-like morphology through star counts of red clump giants. Observations in this region, combined with near-infrared photometry across multiple bulge fields, confirm an X-shaped structure extending along the bar's major axis, with the two arms separated by approximately 1.5 kpc near the Galactic plane. This configuration arises from the buckling instability of a stellar bar, as evidenced by the double-peaked luminosity function of red clump stars viewed edge-on, which distinguishes the near and far arms of the X-shape. The window's position at galactic longitude l \approx 1^\circ and latitude b \approx -4^\circ places it approximately 8 kpc from the Sun, aligning closely with the distance to the Galactic center and enabling precise mapping of bulge kinematics. Radial velocity distributions measured for late-type giants in Baade's Window exhibit cylindrical rotation with a pattern speed of about 40 km s^{-1} kpc^{-1}, supporting a rotating bar model that extends to the bulge's core. These velocities, peaking at around -100 km s^{-1} for metal-rich stars, indicate coherent orbital streaming along the bar, consistent with dynamical simulations of an elongated, triaxial structure. Formation models for the bulge, informed by observations through Baade's Window, favor secular evolution from an inner disk via bar instabilities over dominant merger remnants, as the observed boxy/X-shaped morphology and kinematics align with N-body simulations of bar buckling without requiring a classical spheroid. While minor merger contributions cannot be ruled out, the lack of a pressure-supported, isotropic velocity dispersion in the bulge—unlike merger-built systems—points to disk-driven secular processes as the primary mechanism, with the bar forming around 8-10 Gyr ago. Kinematic data from the window further constrain the bar angle to approximately 20-30 degrees relative to the Sun-Galactic center line, reinforcing this evolutionary pathway.^[16] The star formation history of the bulge, traced via color-magnitude diagrams and chemical abundances in Baade's Window, indicates a major burst approximately 10 Gyr ago that formed the bulk of its stellar mass, followed by quiescent evolution with only minor recent activity contributing less than 5% of stars younger than 3 Gyr. This rapid early enrichment, evidenced by enhanced alpha-element abundances in metal-poor stars, suggests a short formation timescale of 1-2 Gyr, consistent with secular bar-driven infall rather than prolonged disk-like star formation. Low levels of ongoing activity are inferred from a small population of intermediate-age asymptotic giant branch stars, but the dominant old population underscores the bulge's ancient assembly.^[17]^[18] Baade's Window lies in near line-of-sight proximity to the supermassive black hole Sgr A* at the Galactic center, approximately 590 pc projected distance away, but the surrounding interstellar dust obscures direct optical views except through this low-extinction aperture. This positioning allows infrared probes of the central region's influence on bulge dynamics, though the window's offset prevents resolved imaging of the black hole itself.

Observational Studies

Ground-Based Surveys

Ground-based surveys of Baade's Window have played a crucial role in probing the Galactic bulge since the 1980s, leveraging optical and near-infrared telescopes to overcome partial interstellar extinction and measure stellar kinematics, abundances, and distributions. These efforts, conducted from terrestrial observatories, have provided large-scale datasets essential for understanding bulge structure and evolution, despite limitations from Earth's atmosphere. The Bulge Radial Velocity Assay (BRAVA), initiated in the 2000s, targeted approximately 10,000 M-type red giant stars selected from the 2MASS catalog across multiple bulge fields, including Baade's Window, using the CTIO 4m Blanco telescope with the Hydra multi-fiber spectrograph. This survey measured radial velocities to map kinematic properties, revealing cylindrical rotation consistent with a bar-like structure and confirming streaming motions along the bar axis, with peak velocities of about 100 km/s at longitude l ≈ 10°. These findings supported the dominance of a bar population in the bulge without evidence for significant classical bulge components or unknown streams.^[19] In the 2010s, the Apache Point Observatory Galactic Evolution Experiment (APOGEE) extended near-infrared spectroscopy to Baade's Window using the 2.5m Sloan telescope, observing over 400 unique stars with high-resolution (R ≈ 22,500) H-band spectra via the APOGEE spectrograph. This yielded precise metallicities, α-element abundances, and age estimates for these giants, deriving a metallicity distribution function (MDF) from the ASPCAP pipeline that highlighted a bimodal structure with peaks at [Fe/H] ≈ -0.3 and +0.3 dex. Age determinations, based on [C/N] ratios, indicated a predominantly old population (~10 Gyr) with a subset of intermediate-age stars (~3-4 Gyr), providing evidence for multiple stellar components in the bulge.^[15] Complementing spectroscopic efforts, the VISTA Variables in the Vía Láctea (VVV) survey, also from the 2010s, conducted deep near-infrared imaging of the bulge using the 4.1m VISTA telescope at ESO Paranal, cataloging millions of stars—including targeted fields in Baade's Window—as a follow-up to 2MASS with J, H, and Ks bands reaching Ks ≈ 18 mag. In Baade's Window, VVV data enabled high-resolution (2′–6′) extinction maps via red clump star colors, revealing variations up to ΔA_{Ks} ≈ 0.1 mag on small scales and confirming patchy interstellar dust with a mean E(J - Ks) ≈ 0.7 mag, which informs corrections for deeper bulge studies. The survey's color-magnitude diagrams further delineated stellar populations, supporting the identification of multiple bulge components through density profiles of red clump giants.^[20] Key findings from these surveys include a bimodal metallicity distribution in the bulge giants, as robustly measured by APOGEE, and kinematic evidence for a barred structure with distinct components—a boxy/peanut bar and possible underlying spheroid—evident in BRAVA velocities and VVV spatial distributions. These results underscore Baade's Window as a low-extinction probe for bulge demographics, revealing a complex formation history involving both secular bar evolution and early mergers.^[15] Despite advances in adaptive optics and infrared detectors, ground-based observations face persistent challenges, including atmospheric seeing that limits resolution to ~0.5–1 arcsec and light pollution from urban sites, which restricts faint-star detection and increases extinction uncertainties in crowded fields like Baade's Window. These factors necessitate careful data processing, such as multi-epoch observations in VVV to mitigate variability effects.

Space-Based Observations

Space-based observations of Baade's Window have revolutionized our understanding of the Galactic bulge by providing high-resolution imaging and astrometry free from atmospheric distortion, enabling access to ultraviolet and full infrared wavelengths that are heavily absorbed by Earth's atmosphere.^[21] This allows for the resolution of faint, crowded stellar fields and precise measurements of stellar motions and compositions that ground-based telescopes struggle to achieve.^[22] The Hubble Space Telescope (HST), operational since the 1990s, delivered deep photometry in Baade's Window using instruments like the Wide Field Planetary Camera 2 and Near-Infrared Camera and Multi-Object Spectrometer (NICMOS). These observations resolved individual stars down to I ≈ 24.3 mag (corresponding to masses ~0.3 M⊙), deriving a luminosity function similar to the solar neighborhood and a mass function with a power-law slope of -2.2 for masses above ~0.7 M⊙, flattening at lower masses.^[23] Color-magnitude diagrams (CMDs) from HST data revealed the star formation history, showing that over 80% of stars formed more than 8 Gyr ago, with 10-25% of metal-rich stars younger than 5 Gyr and an age-metallicity relation of dZ/dt ≈ 0.005 Gyr⁻¹.^[24] NICMOS near-infrared imaging provided evidence for continuous star formation shaping the central stellar cusp, best fit by models with ongoing activity rather than ancient bursts.^[25] HST also measured proper motions across 35 fields near Baade's Window, sampling a 5° × 2.5° area to trace bulge kinematics. Additionally, HST resolved source stars in microlensing events toward the bulge, placing them on CMDs calibrated with Baade's Window fields to confirm low-mass exoplanet candidates. The Gaia mission, from the 2010s onward, supplied astrometry for over 100,000 stars in Baade's Window and surrounding bulge fields in data releases such as DR2 and DR3, enabling mapping of the 3D structure and velocities with microarcsecond precision. These proper motions distinguished metal-rich and metal-poor populations, revealing vertex deviations of l_v ≈ -40° for metal-rich stars and l_v ≈ 10° for metal-poor ones, and supported models of the bulge's bar-like kinematics integrated with spectroscopic surveys.^[26] In the 2020s, the James Webb Space Telescope (JWST) has targeted the Galactic bulge, including fields overlapping Baade's Window, using NIRCam for deep imaging and NIRSpec for spectroscopy. NIRCam observations probed the low-mass end of the initial mass function down to the hydrogen-burning limit (~0.08 M⊙), detecting faint main-sequence stars and candidate brown dwarfs with effective temperatures ~1000 K, and a mass function slope of α = 0.88 ± 0.36 terminating around 0.15 M⊙.^[27] These data revealed young stellar populations and chemical abundances, complementing HST's historical insights with enhanced sensitivity to dust-obscured, low-mass objects.^[27] The Euclid space telescope, launched in 2023, has further advanced microlensing studies through its Galactic Bulge Survey, imaging nine fields in Baade's Window to monitor dense stellar regions for gravitational lensing events. As of 2025, analyses of these observations, combined with historical data from OGLE and others, have cataloged thousands of microlensing events, aiding in the characterization of lens properties and constraints on dark matter contributions in the bulge.^[28]

References

[1]
Baade's Window
### Definition and Key Facts about Baade's Window
[2]
Baade's Window | COSMOS
- **Definition**: Baade’s Window is a region providing a relatively unobscured view toward the Galactic center, one of the few lines-of-sight not heavily obscured by dust.
[3]
February 1, 1996 - Lensing through Baade's Window - APOD
Feb 1, 1996 · In the 1940s, however, astronomer Walter Baade identified a "window" near the center of our Galaxy where there is comparatively little opaque ...
[4]
Wilhelm Heinrich Walter Baade - Department of Physics & Astronomy
Baade and Rudolph Minkowski identified and took spectrograms of optical counterparts of many of the first-discovered radio sources, including Cygnus A and ...
[5]
A Search For the Nucleus of Our Galaxy
- **Title**: A Search For the Nucleus of Our Galaxy
[6]
What Can You See Through Baade's Window? - Sky & Telescope
Jun 10, 2015 · Named for Mt. Wilson astronomer Walter Baade who first recognized its significance, this porthole into deep space spans about 1° in diameter ...Missing: definition | Show results with:definition
[7]
[PDF] 1949–1999 The Early Years of Stellar Evolution, Cosmology, and ...
The principal scientific address was by Walter Baade (1948) titled “A Program of Extragalactic Research for the 200-inch Telescope.” This was a prescient ...<|control11|><|separator|>
[8]
(PDF) The Zero Point of Extinction toward Baade's Window from RR ...
Aug 9, 2025 · PDF | We measure the zero point of the Stanek (1996) extinction map by comparing the observed (V-K) colors of 20 RR Lyrae stars (type ab) found
[9]
The Optical Gravitational Lensing Experiment. The General Catalog ...
In this paper we present stars from the center of the Baade's Window field designated as BWC. The J2000 coordinates of the BWC field are: ~ = 18h03m24s ...
[10]
Coordinates and 2MASS and OGLE identifications for all stars in ...
The region towards the Galactic Bulge situated at Galactic coordinates (l,b) = (1, −4) known as Baade's window is a much studied area as it provides a low ...
[11]
Two estimates of the distance to the Galactic Centre - Oxford Academic
Using the same calibration, Stanek et al. (2000) gave R0 = 8.67 ± 0.4 kpc, also from red clump stars in Baade's window. These estimates are high ...
[12]
[astro-ph/9512137] Extinction Map of Baade's Window - arXiv
An extinction map of (40')^2 region of Baade's Window, with resolution of \sim30\;arcsec. Such a map should be useful for studies of this frequently observed ...Missing: dust BV) composition
[13]
Low-extinction windows in the inner Galactic Bulge
Baade (1963) identified the windows Sgr I, Sgr II, and the. NGC 6522 field; this latter has been widely referred to as Baade's Window in subsequent studies.
[14]
Modeling the infrared extinction toward the galactic center
The best-fit model estimates the total visual extinction toward the GC to be AV ~ 38–42 mag. In view that the starlight from the GC passes through different ...Missing: 25 | Show results with:25<|control11|><|separator|>
[15]
https://arxiv.org/abs/1702.01547
[16]
[1702.01547] Baade's window with APOGEE: Metallicities, ages and ...
Feb 6, 2017 · Baade's window (BW) is one of the most observed Galactic bulge fields in terms of chemical abundances. Due to its low and homogeneous ...
[17]
[PDF] Theoretical Models of the Galactic Bulge - arXiv
Apr 20, 2015 · Qin et al. (2015) also found that stars in the X-shaped bulge do not necessarily stream along simple banana orbits. Clearly, more studies ...
[18]
formation history of the Galactic bulge
Models with Ceff= 10, τ1=τ2= 0.5 Gyr form 50 per cent of the stars in ∼0.75 Gyr, compared with a longer duration of ∼1.9 Gyr when the efficiency is lowered to C ...
[19]
[PDF] The formation history of the Galactic bulge - arXiv
20% in stars younger than 10 Gyr imposes a further constraint on ... straints on the duration of the star formation burst which gave way to the Galactic bulge.
[20]
Baade's Window and NGC 6522
The Galactic centre is itself obscured by dust clouds at visible wavelengths, but in the 1940s, Walter Baade, working at Mt Wilson in the USA, identified a ...Missing: original | Show results with:original
[21]
[PDF] THE BULGE RADIAL VELOCITY ASSAY (BRAVA). II. COMPLETE ...
Feb 6, 2012 · M giants in Baade's Window. ... We select red giants from the K, J − K RGB that are approximately at the distance to the Galactic center, R0 =.
[22]
I. The 84-million star colour–magnitude diagram of the Galactic bulge
Milky Way demographics with the VVV survey - I. The 84-million star colour–magnitude diagram of the Galactic bulge | Astronomy & Astrophysics (A&A)
[23]
Why Have a Telescope in Space? - NASA Science
Jan 24, 2025 · This is good for life on our planet, since it keeps away some dangerous radiation, but it also blinds us to some of the light emitted by cosmic ...
[24]
Space Telescopes - Las Cumbres Observatory
Space telescopes have the advantage of being above the blurring effects of the Earth's atmosphere. In addition, there are many wavelengths from the ...Missing: Baade's Window distortion
[25]
[astro-ph/9801321] The Luminosity Function and Mass ... - arXiv
Jan 30, 1998 · Abstract: We present deep photometry obtained with the Hubble Space Telescope (HST) in a field in Baade's Window in the Galactic bulge.
[26]
[1801.01426] Star formation history of the Galactic bulge from deep ...
Jan 4, 2018 · Our metallicity distribution function accurately reproduces that observed by several spectroscopic surveys of Baade's window, with the bulk of ...
[27]
An Extended Star Formation History for the Galactic Center ... - arXiv
Sep 28, 2003 · We present Hubble Space Telescope (HST) Near-Infrared Camera and Multiobject Spectrometer (NICMOS) observations as evidence that continuous star ...
[28]
[2011.13905] Mapping the tilt of the Milky Way bulge velocity ... - arXiv
Nov 27, 2020 · In the Baade's window, the metal-rich stars display a larger vertex deviation (l_{v} = -40^{\circ}) than the metal-poor stars (l_{v} = 10^{\circ} ...
[29]
JWST Imaging of the Closest Globular Clusters -- VI. The Lowest-Mass Objects in M 4 and the Galactic Bulge
### Summary of arXiv:2507.13564