<figure><img src="image_22.png" /><figcaption>**Fig. 21.** Model compact flux density estimates from the Bayesian Image reconstruction methods for the April 21 HOPS data. The error bars represent the 68% confidence intervals around the median (square points).</figcaption></figure>

of uncertainty in what we should expect the compact flux density to be at EHT-scales.

To illustrate this, we investigate the compact flux density estimates produced by the various Bayesian methods that fit the complex visibilities (THEMIS) or the visibility amplitudes (Comrade). For the THEMIS raster and Hybrid Themage models, we quantify the compact flux density by summing the flux density from the raster and the raster plus crescent, respectively. For Comrade, the compact flux density is a model parameter which sets the total flux density of the raster as described in Sect. 5.1.5. The THEMIS xs-ringauss model constructs the compact flux density estimate by summing the flux density from the crescent and nuisance Gaussians.

We summarize the compact flux density estimates across all observing bands on April 21 from the THEMIS raster, Hybrid Themage, Comrade raster, and THEMIS xs-ringauss models in Fig. 21, using the HOPS data. There are two clear modes: a “low-estimate” for compact flux density around 0.6 Jy, and a “high-estimate” for compact flux density around 1.0 Jy. The THEMIS raster model is the only model that consistently produces compact flux density estimates below 0.7 Jy for all bands. In comparison, the THEMIS xs-ringauss and Comrade raster produce compact flux density estimates that are greater than 0.9 Jy. The Comrade raster prefers larger compact flux density because of the larger FOV and the choice of prior distribution. We found that the compact flux density strongly depends on the Comrade prior assumptions because of the lack of intermediate baselines. The Hybrid Themage model prefers lower compact flux densities in band 3 and band 4, but a higher compact flux density in band 1, with uncertainty that spans 0.5 Jy. The error bars in Fig. 21 are taken from the 68% confidence intervals and that, generally, the methods that produce low compact flux density estimates do not overlap with the high compact flux density estimates.

There is no consensus among the Bayesian methods about what the compact flux density should be in the 2018 EHT data. The chain parameters for the Hybrid Themage reconstruction of the April 21 band 1 data give us clues about how we might achieve such high values for compact flux density. In Fig. 22, we show the total compact flux density, the raster FOV, and the crescent flux density. Recall that for the Hybrid Themage, we construct the compact flux density by adding the crescent flux density and the flux density from all the raster points. We see

<figure><img src="image_23.png" /><figcaption>**Fig. 22.** Triangle plot comparing the compact flux density (CF), raster FOV in the x and y directions, and crescent flux ($I_X$) parameters for the Hybrid Themage reconstruction of April 21 band 1. The compact flux density and raster FOV parameters exhibit clearly bi-modal distributions in this parameter space. The color regions represent the 99%, 90%, and 50% quantile regions.</figcaption></figure>

clear bi-modal distributions in the compact flux density, $FOV_y$, and to a lesser degree, in the crescent flux density. This suggests that, at least for this reconstruction, adding the entire raster flux density may not be appropriate for computing the compact flux density; the raster is elongated in one direction and places some flux outside of what we might consider the compact region. This is evident in the April 21 band 1 Hybrid Themage image in Fig. N.2, where the raster extends flux in the northwest direction.

The most crucial parameters of interest, such as the diameter and the position angle, do not seem to depend strongly on the recovered compact flux density. This is consistent with the findings from the RML and CLEAN methods, which are not able to independently constrain the ratio between the total flux density and the compact flux density. The measured diameters and position angles are entirely consistent between models that produce different values of compact flux density (see Appendix M).

## 8. Conclusion

We present the results of the 2018 EHT observations of M 87* at 1.3 mm in order to further investigate the nature of the black hole shadow, and the year to year variation in the image structure. During the 2018 observations, the GLT participated for the first time as part of the VLBI array and provided additional baselines to improve the (u, v) coverage. Following the example of the 2017 data analysis, we use multiple independent calibration, imaging, and modeling methods to analyze and interpret the 2018 EHT data. We analyze data on two independent epochs (April 21 and April 25) across four independent frequency bands. The 2018 data show a clear null in the visibility amplitudes at 3.4 Gλ, and exhibit significant variation in the time-dependent closure phases compared to 2017. The differences in the closure phases indicate a significant change in the asymmetric structure from 2017 to 2018.
All of the imaging and modeling methods indicate the presence of a ring-like structure, and the extracted ring characteristics are consistent across all bands and days. Similar to the 2017 data, the current EHT data does not provide strong constraints on extended or diffuse emission outside the ring. All methods on April 21 data suggest a ring diameter of ~42 µas for the image domain analyses (Sect. 5.3) and ~45 µas for the direct modeling methods (Sect. 6.3). When combining all methods with equal weights, the median diameter of the ring in 2018 is 43.3±1.5$_{-3.1}^{+1.5}$ µas, in good agreement with the diameter measured in the analysis of the 2017 data.

The lack of variability in the ring diameter between 2017 and 2018 is consistent with the predictions from general relativity for strongly lensed emission around a black hole. In contrast with the 2017 data, we detect a significant shift in the position angle of the ring brightness asymmetry from ~180° as measured in 2017 to ~210° in the 2018 data. This shift in position angle is consistent with the expected variability from GRMHD simulations. In particular, when converting the brightness asymmetry position angle to a nominal black hole spin direction, the 2018 image is more consistent with the orientation of the large-scale jet seen at longer wavelengths.

**Acknowledgements.** The Event Horizon Telescope Collaboration thanks the following organizations and programs: the Academia Sinica; the Academy of Finland (projects 274477, 284495, 312496, 315721); the Agencia Nacional de Investigación y Desarrollo (ANID), Chile via NCN19_058 (TITANs), Fondecy 1221421 and BASAL FB210003; the Alexander von Humboldt Stiftung; an Alfred P. Sloan Research Fellowship; Allegro, the European ALMA Regional Centre node in the Netherlands, the NL astronomy research network NOVA and the astronomy institutes of the University of Amsterdam, Leiden University, and Radboud University; the ALMA North America Development Fund; the Astrophysics and High Energy Physics programme by MCIN (with funding from European Union NextGenerationEU, PRTR-C171i); the Black Hole Initiative, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation (although the opinions expressed in this work are those of the author(s) and do not necessarily reflect the views of these Foundations); the Brinson Foundation; "la Caixa" Foundation (ID 100010434) through fellowship codes LCF/BQ/DI22/11940027 and LCF/BQ/DI22/11940030; the Fondo CAS-ANID folio CAS220010; Chandra DD7-18089X and TM6-17006X; the China Scholarship Council; the China Postdoctoral Science Foundation fellowships (2020M671266, 2022M712084); Consejo Nacional de Humanidades, Ciencia y Tecnología (CONAHCYT, Mexico, projects U0004-246083, U0004-259839, F0003-272050, M0037-279006, F0003-281692, 104497, 275201, 263356); the Cofuturo Scholarship; the Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (grant P18-FR-1769), the Consejo Superior de Investigaciones Científicas (grant 2019AEP112); the Delaney Family via the Delaney Family John A. Wheeler Chair at Perimeter Institute; Dirección General de Asuntos del Personal Académico-Universidad Nacional Autónoma de México (DGAPANAM, projects IN112820 and IN108324); the Dutch Organization for Scientific Research (NWO) for the VICI award (grant 639.043.513), the grant OCENW.KLEIN.113, and the Dutch Black Hole Consortium (with project No. NWA 1292.19.202) of the research programme the National Science Agenda; the Dutch National Supercomputers, Cartesius and Snellius (NWO grant 2021.013); the EACOA Fellowship awarded by the East Asia Core Observatories Association, which consists of the Academia Sinica Institute of Astronomy and Astrophysics, the National Astronomical Observatory of Japan, Center for Astronomical Mega-Science, Chinese Academy of Sciences, and the Korea Astronomy and Space Science Institute; the European Research Council (ERC) Synergy Grant "BlackHoleCam: Imaging the Event Horizon of Black Holes" (grant 610058); the European Union Horizon 2020 research and innovation programme under grant agreements RadioNet (No. 730562) and M2FINDERS (No. 101018682); the Horizon ERC Grants 2021 programme under grant agreement No. 101040021; the Generalitat Valenciana (grants APOSTD/2018/177 and ASFAE/2022/018) and GenT Program (project CIGEGENT/2018/021); MICINN Research Project PID2019-108995GB-C22; the European Research Council for advanced grant "JETSET: Launching, propagation and emission of relativistic jets from binary mergers and across mass scales" (grant No. 884631); the FAPESP (Fundação de Amparo á Pesquisa do Estado de São Paulo) under grant 2021/01183-8; the Institute for Advanced Study; the Istituto Nazionale di Fisica Nucleare (INFN) sezione di Napoli, iniziative speci-

fiche TEONGRAV; the International Max Planck Research School for Astronomy and Astrophysics at the Universities of Bonn and Cologne; DFG research grant "Jet physics on horizon scales and beyond" (grant No. 443220636); Joint Columbia/Flatiron Postdoctoral Fellowship (research at the Flatiron Institute is supported by the Simons Foundation); the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT; grant JPMXP1020200109); the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for JSPS Research Fellowship (JP17J08829); the Joint Institute for Computational Fundamental Science, Japan; the Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS, grants QYZDJ-SSW-SLH057, QYZDJSSW-SYS008, ZDBS-LY-SLH011); the Leverhulme Trust Early Career Research Fellowship; the Max-Planck-Gesellschaft (MPG); the Max Planck Partner Group of the MPG and the CAS; the MEXT/JSPS KAKENHI (grants 18KK0909, JP21H01137, JP18H03721, JP18K13594, 18K03709, JP19K14761, 18H01245, 25120007, 19H01943, 21H01137, 21H04488, 22H00157, 23K03453); the MIT International Science and Technology Initiatives (MISTI) Funds; the Ministry of Science and Technology (MOST) of Taiwan (103-2119-M-001-010-MY2, 105-2112-M-001-025-MY3, 105-2119-M-001-042, 106-2112-M-001-011, 106-2119-M-001-013, 106-2119-M-001-027, 106-2923-M-001-005, 107-2119-M-001-017, 107-2119-M-001-020, 107-2119-M-001-041, 107-2119-M-110-005, 107-2923-M-001-009, 108-2112-M-001-048, 108-2112-M-001-051, 108-2923-M-001-002, 109-2112-M-001-025, 109-2124-M-001-005, 109-2923-M-001-001, 110-2112-M-001-033, 110-2124-M-001-007, 110-2923-M-001-001, and 112-2112-M-003-010-MY3); the National Science and Technology Council (NSTC) of Taiwan (111-2124-M-001-005, 112-2124-M-001-014); the Ministry of Education (MoE) of Taiwan Yushan Young Scholar Program; the Physics Division, National Center for Theoretical Sciences of Taiwan; the National Aeronautics and Space Administration (NASA, Fermi Guest Investigator grant 80NSSC23K1508, NASA Astrophysics Theory Program grant 80NSSC20K0527, NASA NuSTAR award 80NSSC20K0645); NASA Hubble Fellowship grants HST-HF2-51431.001-A, HST-HF2-51482.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS-26555; the National Institute of Natural Sciences (NINS) of Japan; the National Key Research and Development Program of China (grant 2016YFA0400704, 2017YFA0402703, 2016YFA0400702); the National Science and Technology Council (NSTC, grants NSTC 111-2112-M-001-041, NSTC 111-2124-M-001-005, NSTC 112-2124-M-001-014); the National Science Foundation (NSF, grants AST-0096454, AST-0352953, AST-0521233, AST-0705062, AST-0905844, AST-0922984, AST-1126433, AST-1140030, DGE-1144085, AST-1207704, AST-1207730, AST-1207752, MRI-1228509, OPP-1248097, AST-1310896, AST-1440254, AST-1555365, AST-1614868, AST-1615796, AST-1715061, AST-1716327, OISE-1743747, AST-1816420, AST-1935980, AST-2034306, AST-2307887); NSF Astronomy and Astrophysics Postdoctoral Fellowship (AST-1903847); the Natural Science Foundation of China (grants 11650110427, 10625314, 11721303, 11725312, 11873028, 11933007, 11991052, 11991053, 12192220, 12192223, 12273022); the Natural Sciences and Engineering Research Council of Canada (NSERC, including a Discovery Grant and the NSERC Alexander Graham Bell Canada Graduate Scholarships-Doctoral Program); the National Research Foundation of Korea (the Global PhD Fellowship Grant; grants NRF-2015H1A2A1033752, the Korea Research Fellowship Program: NRF-2015H1D3A1066561, Brain Pool Program: 2019H1D3A1A01102564, Basic Research Support Grant 2019R1F1A1059721, 2021R1A6A3A01086420, 2022R1C1C1005255, 2022R1F1A1075115); Netherlands Research School for Astronomy (NOVA) Virtual Institute of Accretion (VIA) postdoctoral fellowships; NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation; Onsala Space Observatory (OSO) national infrastructure, for the provisioning of its facilities/observational support (OSO receives funding through the Swedish Research Council under grant 2017-00648); the Perimeter Institute for Theoretical Physics (research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science); the Princeton Gravity Initiative; the Spanish Ministerio de Ciencia e Innovación (grants PGC2018-098915-B-C21, AYA2016-80889-P, PID2019-108995GB-C21, PID2020-117404GB-C21); the University of Pretoria for financial aid in the provision of the new Cluster Server nodes and SuperMicro (USA) for a SEEDING GRANT approved toward these nodes in 2020; the Shanghai Municipality orientation program of basic research for international scientists (grant no. 22JC1410600); the Shanghai Pilot Program for Basic Research, Chinese Academy of Science, Shanghai Branch (JCYJ-SHFY-2021-013); the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award for the Instituto de Astrofísica de Andalucía (SEV-2017-0709); the Spanish Ministry for Science and Innovation grant CEX2021-001131-S funded by MCIN/AEI/10.13039/5011000111033; the Spinoza Prize SPI 78-409; the South African Research Chairs Initiative, through the South African Radio Astronomy Observatory (SARAO, grant ID 77948), which is a facility
of the National Research Foundation (NRF), an agency of the Department of
Science and Innovation (DSI) of South Africa; the Swedish Research Council
(VR); the Taplin Fellowship; the Toray Science Foundation; the UK Science and
Technology Facilities Council (grant no. ST/X508329/1); the US Department
of Energy (USDOE) through the Los Alamos National Laboratory (operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of the USDOE, contract 89233218CNA000001); and the YCAA Prize Postdoc-
toral Fellowship. We thank the staff at the participating observatories, correla-
tion centers, and institutions for their enthusiastic support. This paper makes
use of the following ALMA data: ADS/JAO.ALMA#2016.1.01154.V. ALMA
is a partnership of the European Southern Observatory (ESO; Europe, repre-
senting its member states), NSF, and National Institutes of Natural Sciences
of Japan, together with National Research Council (Canada), Ministry of Sci-
ence and Technology (MOST; Taiwan), Academia Sinica Institute of Astron-
omy and Astrophysics (ASIA; Taiwan), and Korea Astronomy and Space Sci-
ence Institute (KASI; Republic of Korea), in cooperation with the Republic of
Chile. The Joint ALMA Observatory is operated by ESO, Associated Univer-
sities, Inc. (AUI)/NRAO, and the National Astronomical Observatory of Japan
(NAOJ). The NRAO is a facility of the NSF operated under cooperative agree-
ment by AUI. This research used resources of the Oak Ridge Leadership Com-
puting Facility at the Oak Ridge National Laboratory, which is supported by
the Office of Science of the US Department of Energy under contract No. DE-
AC05-00OR22725; the ASTROVIVES FEDER infrastructure, with project code
IDIFEDER-2021-086; the computing cluster of Shanghai VLBI correlator sup-
ported by the Special Fund for Astronomy from the Ministry of Finance in China;
We also thank the Center for Computational Astrophysics, National Astronom-
ical Observatory of Japan. This work was supported by FAPESP (Fundacao
de Amparo a Pesquisa do Estado de Sao Paulo) under grant 2021/01183-8.
APEX is a collaboration between the Max-Planck-Institut für Radioastronomie
(Germany), ESO, and the Onsala Space Observatory (Sweden). The SMA is a
joint project between the SAO and ASIAA and is funded by the Smithsonian
Institution and the Academia Sinica. The JCMT is operated by the East Asian
Observatory on behalf of the NAOJ, ASIAA, and KASI, as well as the Min-
istry of Finance of China, Chinese Academy of Sciences, and the National Key
Research and Development Program (No. 2017YFA0402700) of China and Nat-
ural Science Foundation of China grant 11873028. Additional funding support
for the JCMT is provided by the Science and Technologies Facility Council
(UK) and participating universities in the UK and Canada. The LMT is a project
operated by the Instituto Nacional de Astrofísica, Óptica, y Electrónica (Mex-
ico) and the University of Massachusetts at Amherst (USA). The IRAM 30-m
telescope on Pico Veleta, Spain is operated by IRAM and supported by CNRS
(Centre National de la Recherche Scientifique, France), MPG (Max-Planck-
Gesellschaft, Germany), and IGN (Instituto Geográfico Nacional, Spain). The
SMT is operated by the Arizona Radio Observatory, a part of the Steward Obser-
vatory of the University of Arizona, with financial support of operations from
the State of Arizona and financial support for instrumentation development from
the NSF. Support for SPT participation in the EHT is provided by the National
Science Foundation through award OPP-1852617 to the University of Chicago.
Partial support is also provided by the Kavli Institute of Cosmological Physics
at the University of Chicago. The SPT hydrogen maser was provided on loan
from the GLT, courtesy of ASIAA. This work used the Extreme Science and
Engineering Discovery Environment (XSEDE), supported by NSF grant ACI-
1548562, and CyVerse, supported by NSF grants DBI-0735191, DBI-1265383,
and DBI-1743442. XSEDE Stampede2 resource at TACC was allocated through
TG-AST170024 and TG-AST080026N. XSEDE JetStream resource at PTI and
TACC was allocated through AST170028. This research is part of the Fron-
tera computing project at the Texas Advanced Computing Center through the
Frontera Large-Scale Community Partnerships allocation AST20023. Frontera
is made possible by National Science Foundation award OAC-1818253. This
research was done using services provided by the OSG Consortium (Pordes et al.
2007; Sfiligoi et al. 2009), which is supported by the National Science Foun-
dation award Nos. 2030508 and 1836650. Additional work used ABACUS2.0,
which is part of the eScience center at Southern Denmark University, and the
Kultrun Astronomy Hybrid Cluster (projects Conicyt Programa de Astronomia
Fondo Quimal QUIMAL170001, Conicyt PIA ACT172033, Fondecyt Iniciacion
11170268, Quimal 220002). Simulations were also performed on the SuperMUC
cluster at the LRZ in Garching, on the LOEWE cluster in CSC in Frankfurt, on
the HazelHen cluster at the HLRS in Stuttgart, and on the Pi2.0 and Siyuan
Mark-I at Shanghai Jiao Tong University. The computer resources of the Finnish
IT Center for Science (CSC) and the Finnish Computing Competence Infras-
tructure (FCCI) project are acknowledged. This research was enabled in part
by support provided by Compute Ontario (http://computeontario.ca), Cal-
cul Quebec (http://www.calculquebec.ca), and Compute Canada (http:
//www.computecanada.ca). The EHTC has received generous donations of
FPGA chips from Xilinx Inc., under the Xilinx University Program. The EHTC
has benefited from technology shared under open-source license by the Collab-
oration for Astronomy Signal Processing and Electronics Research (CASPER).
The EHT project is grateful to T4Science and Microsemi for their assistance
with hydrogen masers. This research has made use of NASA's Astrophysics Data

System. We gratefully acknowledge the support provided by the extended staff
of the ALMA, from the inception of the ALMA Phasing Project through the
observational campaigns of 2017 and 2018. We would like to thank A. Deller
and W. Brisken for EHT-specific support with the use of DiFX. We thank Martin
Shepherd for the addition of extra features in the Difmap software that were used
for the CLEAN imaging results presented in this paper. We acknowledge the sig-
nificance that Maunakea, where the SMA and JCMT EHT stations are located,
has for the indigenous Hawaiian people.

References

Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, Phys. Rev. Lett., 116, 061102

Akiyama, K., Lu, R.-S., Fish, V. L., et al. 2015, ApJ, 807, 150

Akiyama, K., Ikeda, S., Pleau, M., et al. 2017a, AJ, 153, 159

Akiyama, K., Kuramochi, K., Ikeda, S., et al. 2017b, ApJ, 838, 1

An, T., Sohn, B. W., & Imai, H. 2018, Nat. Astron., 2, 118

Arras, P., Frank, P., Haim, P., et al. 2022, Nat. Astron., 6, 259

Asada, K., & Nakamura, M. 2012, ApJ, 745, L28

Bardeen, J. M. 1973, Les Astres Occlus, 215

Benkevitch, L., Akiyama, K., Lu, R., Doeleman, S., & Fish, V. 2016, ArXiv
e-prints [arXiv:1609.00055]

Bird, S., Harris, W. E., Blakeslee, J. P., & Flynn, C. 2010, A&A, 524, A71

Blackburn, L., Chan, C.-K., Crew, G. B., et al. 2019, ApJ, 882, 23

Blackburn, L., Pesce, D. W., Johnson, M. D., et al. 2020, ApJ, 894, 31

Blakeslee, J. P., Jordán, A., Mei, S., et al. 2009, ApJ, 694, 556

Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883

Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433

Blandford, R., Meier, D., & Readhead, A. 2019, ARA&A, 57, 467

Bower, G. C., Dexter, J., Markoff, S., et al. 2015, ApJ, 811, L6

Broderick, A. E., & Pesce, D. W. 2020, ApJ, 904, 126

Broderick, A. E., Gold, R., Karami, M., et al. 2020a, ApJ, 897, 139

Broderick, A. E., Pesce, D. W., Tiede, P., Pu, H.-Y., & Gold, R. 2020b, ApJ, 898, 9

Broderick, A. E., Pesce, D. W., Gold, R., et al. 2022a, ApJ, 935, 61

Broderick, A. E., Tiede, P., Pesce, D. W., & Gold, R. 2022b, ApJ, 927, 6

Byrd, R. H., Lu, P., Nocedal, J., & Zhu, C. 1995, SIAM J. Sci. Comput., 16, 1190

Cantiello, M., Blakeslee, J. P., Ferrarese, L., et al. 2018, ApJ, 856, 126

Carilli, C. L., & Thyagarajan, N. 2022, ApJ, 924, 125

CASA Team (Bean, B., et al.) 2022, PASP, 134, 114501

Chael, A. A., Johnson, M. D., Bouman, K. L., et al. 2018, ApJ, 857, 23

Chael, A., Bouman, K., Johnson, M., et al. 2019a, https://doi.org/10.5281/zenodo.2614016

Chael, A. A., Bouman, K. L., Johnson, M. D., et al. 2019b, Astrophysics Source
Code Library [record ascl:1904.004]

Chael, A., Johnson, M. D., & Lupsasca, A. 2021, ApJ, 918, 6

Chen, M.-T., Asada, K., Matsushita, S., et al. 2023, PASP, 135, 095001

Cho, I., Jung, T., Zhao, G.-Y., et al. 2017, PASI, 69, 87

Cho, I., Zhao, G.-Y., Kawashima, T., et al. 2022, ApJ, 926, 108

Clark, B. G. 1980, A&A, 89, 377

Cui, Y.-Z., Hada, K., Kino, M., et al. 2021, Res. Astron. Astrophys., 21, 205

Cui, Y., Hada, K., Kawashima, T., et al. 2023, Nature, 621, 711

Cunningham, C. T., & Bardeen, J. M. 1973, ApJ, 183, 237

de Gasperin, F., Orrú, E., Murgia, M., et al. 2012, A&A, 547, A56

Deller, A. T., Brisken, W. F., Phillips, C. J., et al. 2011, PASP, 123, 275

Dexter, J., McKinney, J. C., & Agol, E. 2012, MNRAS, 421, 1517

Doeleman, S. S., Fish, V. L., Schenck, D. E., et al. 2012, Science, 338, 355

EHT MWL Science Working Group (Algaba, J. C., et al.) 2021, ApJ, 911, L11

Emami, R., Ricarte, A., Wong, G. N., et al. 2023, ApJ, 950, 38

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2019a, ApJ, 875, L1

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2019b, ApJ, 875, L2

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2019c, ApJ, 875, L3

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2019d, ApJ, 875, L4

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2019e, ApJ, 875, L5

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2019f, ApJ, 875, L6

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2021a, ApJ, 910, L12

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2021b, ApJ, 910, L13

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2022a, ApJ, 930, L12
Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2022b, ApJ, 930, L13

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2022c, ApJ, 930, L14

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2022d, ApJ, 930, L15

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2022e, ApJ, 930, L16

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2022f, ApJ, 930, L17

Event Horizon Telescope Collaboration (Akiyama, K., et al.) 2023, ApJ, 957, L20

Falcke, H., Melia, F., & Agol, E. 2000, ApJ, 528, L13

Fanaroff, B. L., & Riley, J. M. 1974, MNRAS, 167, 31P

Ferreira, P. G. 2019, ARA&A, 57, 335

Gammie, C. F., McKinney, J. C., & Tóth, G. 2003, ApJ, 589, 444

Gebhardt, K., Adams, J., Richstone, D., et al. 2011, ApJ, 729, 119

Georgiev, B., Pesce, D. W., Broderick, A. E., et al. 2022, ApJ, 930, L20

Goddi, C., Martí-Vidal, I., Messias, H., et al. 2019, PASP, 131, 075003

Goddi, C., Martí-Vidal, I., Messias, H., et al. 2021, ApJ, 910, L14

GRAVITY Collaboration (Abuter, R., et al.) 2018, A&A, 615, L15

Greene, J. E., Strader, J., & Ho, L. C. 2020, ARA&A, 58, 257

Greisen, E. W. 2003, Astrophys. Space Sci. Lib., 285, 109

Hada, K., Kino, M., Doi, A., et al. 2016, ApJ, 817, 131

Hilbert, D. 1917, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen – Mathematisch-physikalische Klasse (Berlin: Weidmannsche Buchhandlung), 53

Hoffman, M. D., & Gelman, A. 2014, J. Mach. Learn. Res., 15, 1593

Högbom, J. A. 1974, A&AS, 15, 417

Hovatta, T., Aller, M. F., Aller, H. D., et al. 2014, AJ, 147, 143

Inoue, M., Algaba-Marcos, J. C., Asada, K., et al. 2014, Radio Sci., 49, 564

Janssen, M., Goddi, C., van Bemmel, I. M., et al. 2019a, A&A, 626, A75

Janssen, M., Blackburn, L., Issaoun, S., et al. 2019b, A Priori Calibration of EHT Stations, Tech. Rep.

Janssen, M., Radcliffe, J. F., & Wagner, J. 2022, Universe, 8, 527

Johnson, M. D., Lupsasca, A., Strominger, A., et al. 2020, Sci. Adv., 6, eaaz1310

Junor, W., Biretta, J. A., & Livio, M. 1999, Nature, 401, L35

Kim, J. Y., Krichbaum, T. P., Lu, R. S., et al. 2018a, A&A, 616, A188

Kim, J.-Y., Lee, S.-S., Hodgson, J. A., et al. 2018b, A&A, 610, L5

Koay, J. Y., Romero-Cañizales, C., Matthews, L. D., et al. 2023a, ArXiv e-prints [arXiv:2312.03505]

Koay, J. Y., Asada, K., Matsushita, S., et al. 2023b, ArXiv e-prints [arXiv:2312.02759]

Kormendy, J., & Ho, L. C. 2013, ARA&A, 51, 511

Lee, S.-S., Wajima, K., Algaba, J.-C., et al. 2016, ApJS, 227, 8

Liepold, E. R., Ma, C.-P., & Walsh, J. L. 1999, ApJ, 945, L3

Liu, D. C., & Nocedal, J. 1989, Math. Program., 45, 503

Lockhart, W., & Gralla, S. E. 2022, MNRAS, 509, 3643

Lu, R., Asada, K., Krichbaum, T. P., et al. 2023, Nature, 616, 686

Luminet, J.-P. 1979, A&A, 75, 228

Martí-Vidal, I., Roy, A., Conway, J., & Zensus, A. J. 2016, A&A, 587, A143

Matthews, L. D., Crew, G. B., Doeleman, S. S., et al. 2018, PASP, 130, 015002

Mertens, F., Lobanov, A. P., Walker, R. C., & Hardee, P. E. 2016, A&A, 595, A54

Miyoshi, M., Kato, Y., & Makino, J. 2022, ApJ, 933, 36

Mogensen, P. K., & Riseth, A. N. 2018, J. Open Source Softw., 3, 615

Nakamura, M., Asada, K., Hada, K., et al. 2018, ApJ, 868, 146

Owen, F. N., Hardee, P. E., & Cornwell, T. J. 1989, ApJ, 340, 698

Park, J., Hada, K., Kino, M., et al. 2019, ApJ, 887, 147

Pordes, R., Petravick, D., Kramer, B., et al. 2007, J. Phys. Conf. Ser., 78, 012057

Readhead, A. C. S., Walker, R. C., Pearson, T. J., & Cohen, M. H. 1980, Nature, 285, 137

Reid, M. J., Biretta, J. A., Junor, W., Muxlow, T. W. B., & Spencer, R. E. 1989, ApJ, 336, 112

Rogers, A. E. E., Hinteregger, H. F., Whitney, A. R., et al. 1974, ApJ, 193, 293

Rogers, A. E. E., Doeleman, S. S., & Moran, J. M. 1995, AJ, 109, 1391

Satapathy, K., Psaltis, D., Özel, F., et al. 2022, ApJ, 925, 13

Schwarzschild, K. 1916, Abh. Konigl. Preuss. Akad. Wissenschaften Jahre, 189

Sfiliagoi, I., Bradley, D. C., Holzman, B., et al. 2009, WRI World Congress on Computer Science and Information Engineering, 428

Shepherd, M. 2011, Astrophysics Source Code Library [record ascl:1103.001]

Shepherd, M. C. 1997, ASP Conf. Ser., 125, 77

Simon, D. A., Cappellari, M., & Hartke, J. 2024, MNRAS, 527, 2341

Skilling, J. 2006, Bayesian Anal., 1, 833

Speagle, J. S. 2020, MNRAS, 493, 3132

Thompson, A. R., Moran, J. M., & Swenson, G. W. 2017, Interferometry and Synthesis in Radio Astronomy, 3rd edn. (Springer)

Tiede, P. 2022, J. Open Source Softw., 7, 4457

Tiede, P., Broderick, A. E., & Palumbo, D. C. M. 2022a, ApJ, 925, 122

Tiede, P., Johnson, M. D., Pesce, D. W., et al. 2022b, Galaxies, 10, 111

Tiede, P., Broderick, A. E., Palumbo, D. C. M., & Chael, A. 2022c, ApJ, 940, 182

van Bemmel, I. M., Kettenis, M., Small, D., et al. 2022, PASP, 134, 114502

Vertatschitsch, L., Primiani, R., Young, A., et al. 2015, PASP, 127, 1226

von Laue, M. 1921, Relativitätstheorie (Braunschweig: Friedrich Vieweg & Sohn)

Walker, R. C., Hardee, P. E., Davies, F. B., Ly, C., & Junor, W. 2018, ApJ, 855, 128

Wielgus, M. 2021, Phys. Rev. D, 104, 124058

Wielgus, M., Blackburn, L., Issaoun, S., et al. 2019, EHT Data Set Validation and Characterization of Errors, Tech. Rep. 2019-CE-02, EHT Memo Series

Wielgus, M., Akiyama, K., Blackburn, L., et al. 2020, ApJ, 901, 67

Xu, K., Ge, H., Tebbutt, W., et al. 2020, Symposium on Advances in Approximate Bayesian Inference, PMLR, 1

Zhu, C., Byrd, R. H., Lu, P., & Nocedal, J. 1997, ACM Trans. Math. Softw., 23, 550

<footnote n="1">¹ Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA</footnote>

<footnote n="2">² National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan</footnote>

<footnote n="3">³ Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="4">⁴ Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, 18008 Granada, Spain</footnote>

<footnote n="5">⁵ Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany</footnote>

<footnote n="6">⁶ Department of Physics, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, Malaysia</footnote>

<footnote n="7">⁷ Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="8">⁸ Department of Physics & Astronomy, The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA</footnote>

<footnote n="9">⁹ Institute of Astronomy and Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building, AS/NTU No. 1, Sec. 4, Roo-sevelt Rd., Taipei 10617, Taiwan ROC</footnote>

<footnote n="10">¹⁰ Departament d'Astronomia i Astrofísica, Universitat de València, C. Dr. Moliner 50, 46100 Burjassot, València, Spain</footnote>

<footnote n="11">¹¹ Observatori Astronòmic, Universitat de València, C. Catedrático José Beltrán 2, 46980 Paterna, València, Spain</footnote>

<footnote n="12">¹² Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, 43992 Onsala, Sweden</footnote>

<footnote n="13">¹³ Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA</footnote>

<footnote n="14">¹⁴ Yale Center for Astronomy & Astrophysics, Yale University, 52 Hillhouse Avenue, New Haven, CT 06511, USA</footnote>

<footnote n="15">¹⁵ Astronomy Department, Universidad de Concepción, Casilla 160, Concepción, Chile</footnote>

<footnote n="16">¹⁶ Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA</footnote>

<footnote n="17">¹⁷ Fermi National Accelerator Laboratory, MS209, PO Box 500, Batavia, IL 60510, USA</footnote>

<footnote n="18">¹⁸ Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA</footnote>

<footnote n="19">¹⁹ East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA</footnote>

<footnote n="20">²⁰ James Clerk Maxwell Telescope (JCMT), 660 N. A'ohoku Place, Hilo, HI 96720, USA</footnote>

<footnote n="21">²¹ California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA</footnote>

<footnote n="22">²² Institute of Astronomy and Astrophysics, Academia Sinica, 645 N. A'ohoku Place, Hilo, HI 96720, USA</footnote>

<footnote n="23">²³ Department of Physics and Astronomy, University of Hawaii at Manoa, 2505 Correa Road, Honolulu, HI 96822, USA</footnote>

<footnote n="24">²⁴ Department of Physics, McGill University, 3600 Rue University, Montréal, QC H3A 2T8, Canada</footnote>

<footnote n="25">²⁵ Trottier Space Institute at McGill, 3550 Rue University, Montréal, QC H3A 2A7, Canada</footnote>
26 Institut de Radioastronomie Millimétrique (IRAM), 300 Rue de la Piscine, 38406 Saint-Martin-d'Hères, France

27 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada

28 Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada

29 Waterloo Centre for Astrophysics, University of Waterloo, ON N2L 3G1, Canada

30 Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, PO Box 9010, 6500 GL Nijmegen, The Netherlands

31 Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA

32 Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

33 Department of Physics, University of Chicago, 5720 South Ellis Avenue, Chicago, IL 60637, USA

34 Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

35 Princeton Gravity Initiative, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA

36 Data Science Institute, University of Arizona, 1230 N. Cherry Ave., Tucson, AZ 85721, USA

37 Program in Applied Mathematics, University of Arizona, 617 N. Santa Rita, Tucson, AZ 85721, USA

38 Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA

39 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, PR China

40 Key Laboratory of Radio Astronomy and Technology, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing 100101, PR China

41 Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea

42 Department of Astronomy, Yonsei University, Yonsei-ro 50, Seodaemun-gu 03722, Seoul, Republic of Korea

43 Physics Department, Fairfield University, 1073 North Benson Road, Fairfield, CT 06824, USA

44 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA

45 Instituto de Astronomía, Universidad Nacional Autónoma de México (UNAM), Apdo Postal 70-264, Ciudad de México, Mexico

46 Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany

47 Research Center for Intelligent Computing Platforms, Zhejiang Laboratory, Hangzhou 311100, PR China

48 Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shengong Road 520, Shanghai 201210, PR China

49 Department of Astronomy and Columbia Astrophysics Laboratory, Columbia University, 500 W. 120th Street, New York, NY 10027, USA

50 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA

51 Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, 80126 Napoli, Italy

52 INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, 80126 Napoli, Italy

53 Wits Centre for Astrophysics, University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein, Johannesburg 2050, South Africa

54 Department of Physics, University of Pretoria, Hatfield, Pretoria 0028, South Africa

55 Centre for Radio Astronomy Techniques and Technologies, Department of Physics and Electronics, Rhodes University, Makhanda 6140, South Africa

56 ASTRON, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

57 LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 Place Jules Janssen, 92195 Meudon, France

58 JILA and Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA

59 National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100101, PR China

60 Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA

61 Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA

62 National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA

63 Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 32-D476, 77 Massachusetts Ave., Cambridge, MA 02142, USA

64 Google Research, 355 Main St., Cambridge, MA 02142, USA

65 Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Emil-Fischer-Str. 31, 97074 Würzburg, Germany

66 Department of History of Science, Harvard University, Cambridge, MA 02138, USA

67 Department of Physics, Harvard University, Cambridge, MA 02138, USA

68 NCSA, University of Illinois, 1205 W. Clark St., Urbana, IL 61801, USA

69 Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, R. do Matão, 1226, São Paulo, SP 05508-090, Brazil

70 Dipartimento di Fisica, Università degli Studi di Cagliari, SP Monserrato-Sestu km 0.7, 09042 Monserrato, CA, Italy

71 INAF – Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius, CA, Italy

72 INFN, Sezione di Cagliari, 09042 Monserrato, CA, Italy

73 CP3-Origins, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark

74 Instituto Nacional de Astrofísica, Óptica y Electrónica, Apartado Postal 51 y 216, 72000 Puebla Pue., Mexico

75 Consejo Nacional de Humanidades, Ciencia y Tecnología, Av. Insurgentes Sur 1582, 03940 Ciudad de México, Mexico

76 Key Laboratory for Research in Galaxies and Cosmology, Chinese Academy of Sciences, Shanghai 200030, PR China

77 Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan

78 Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

79 NOVA Sub-mm Instrumentation Group, Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands

80 Department of Astronomy, School of Physics, Peking University, Beijing 100871, PR China

81 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, PR China

82 Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

83 The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan

84 Department of Statistical Science, The Graduate University for Advanced Studies (SOKENDAI), 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan

85 Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8583, Japan

86 Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, The Netherlands

87 ASTRAVEO LLC, PO Box 1668, Gloucester, MA 01931, USA

88 Applied Materials Inc., 35 Dory Road, Gloucester, MA 01930, USA

89 Institute for Astrophysical Research, Boston University, 725 Commonwealth Ave., Boston, MA 02215, USA

90 University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea
91 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan

92 Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

93 Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu 702-701, Republic of Korea

94 Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

95 Kogakuin University of Technology & Engineering, Academic Support Center, 2665-1 Nakano, Hachioji, Tokyo 192-0015, Japan

96 Graduate School of Science and Technology, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata 950-2181, Japan

97 Physics Department, National Sun Yat-Sen University, No. 70, Lien-Hai Road, Kaosiung City 80424, Taiwan ROC

98 School of Astronomy and Space Science, Nanjing University, Nanjing 210023, PR China

99 Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Nanjing 210023, PR China

100 INAF-Istituto di Radioastronomia, Via P. Gobetti 101, 40129 Bologna, Italy

101 INAF-Istituto di Radioastronomia & Italian ALMA Regional Centre, Via P. Gobetti 101, 40129 Bologna, Italy

102 Department of Physics, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan ROC

103 Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Morelia 58089, Mexico

104 Yunnan Observatories, Chinese Academy of Sciences, 650011 Kunming, Yunnan Province, PR China

105 Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, PR China

106 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 650011 Kunming, PR China

107 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

108 Gravitation and Astroparticle Physics Amsterdam (GRAPPA) Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

109 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA

110 School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, PR China

111 Institut de Radioastronomie Millimétrique (IRAM), Avenida Divina Pastora 7, Local 20, 18012 Granada, Spain

112 National Institute of Technology, Hachinohe College, 16-1 Uwanotai, Tamonoki, Hachinohe City, Aomori 039-1192, Japan

113 Research Center for Astronomy, Academy of Athens, Soranou Efessiou 4, 115 27 Athens, Greece

114 Department of Physics, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA

115 Physics Department, Washington University, CB 1105, St. Louis, MO 63130, USA

116 School of Physics, Georgia Institute of Technology, 837 State St NW, Atlanta, GA 30332, USA

117 Department of Astronomy and Space Science, Kyung Hee University, 1732, Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 17104, Republic of Korea

118 Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada

119 Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada

120 Canadian Institute for Advanced Research, 180 Dundas St West, Toronto, ON M5G 1Z8, Canada

121 Department of Physics, National Taiwan Normal University, No. 88, Sec. 4, Tingzhou Rd., Taipei 116, Taiwan ROC

122 Center of Astronomy and Gravitation, National Taiwan Normal University, No. 88, Sec. 4, Tingzhou Road, Taipei 116, Taiwan ROC

123 Finnish Centre for Astronomy with ESO, University of Turku, 20014 Turku, Finland

124 Aalto University Metsähovi Radio Observatory, Metsähovintie 114, 02540 Kylmälä, Finland

125 Gemini Observatory/NSF's NOIRLab, 670 N. A'ohōkū Place, Hilo, HI 96720, USA

126 Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt, Germany

127 School of Mathematics, Trinity College, Dublin 2, Ireland

128 Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada

129 Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

130 Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan

131 Aalto University Department of Electronics and Nanoengineering, PL 15500, 00076 Aalto, Finland

132 Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, UK

133 National Biomedical Imaging Center, Peking University, Beijing 100871, PR China

134 College of Future Technology, Peking University, Beijing 100871, PR China

135 Tokyo Electron Technology Solutions Limited, 52 Matsunagane, Iwayado, Esashi, Oshu, Iwate 023-1101, Japan

136 Department of Physics and Astronomy, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada

137 Netherlands Organisation for Scientific Research (NWO), Postbus 93138, 2509 AC Den Haag, The Netherlands

138 Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan

139 Astronomical Institute, Tohoku University, Sendai 980-8578, Japan

140 Department of Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea

141 University of New Mexico, Department of Physics and Astronomy, Albuquerque, NM 87131, USA

142 Physics Department, Brandeis University, 415 South Street, Waltham, MA 02453, USA

143 Tuorla Observatory, Department of Physics and Astronomy, University of Turku, Turku, Finland

144 Radboud Excellence Fellow of Radboud University, Nijmegen, The Netherlands

145 School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

146 School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China

147 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK

148 School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, PR China

149 Astronomy Department, University of Science and Technology of China, Hefei 230026, PR China

150 Department of Physics and Astronomy, Michigan State University, 567 Wilson Rd, East Lansing, MI 48824, USA

151 University of Vermont, Burlington, VT 05405, USA

152 National Chung-Shan Institute of Science and Technology, No. 566, Ln. 134, Longyuan Rd., Longtan Dist., Taoyuan City 325, Taiwan ROC

153 Western University, 1151 Richmond Street, London, Ontario N6A 3K7, Canada

154 Department of Physics, Graduate School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan

155 Department of Physics, National Cheng Kung University, Tainan 701401, Taiwan ROC

156 Coordinación Mexicana de Investigación en Materiales S.A. de C.V. Ciencia y Tecnología 790, Colonia Saltillo 400, Saltillo, Coahuila CP 25290, Mexico

157 Dipartimento di Fisica, Università di Trieste, 34127 Trieste, Italy

158 INFN Sez. di Trieste, 34127 Trieste, Italy

159 Ground Station for Deep Space Exploration and Telecommunication (GREAT) Project, Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA), 3-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan
# Appendix A: Flux density calibration parameters and uncertainties

The visibility amplitudes need to be converted from correlation coefficients to units of flux density, considering the different telescope sensitivities across the array. This is achieved by multiplying the normalized visibility amplitudes with the geometric mean of the derived system equivalent flux densities (SEFDs) of the two stations of each respective baseline. The SEFD is given by (e.g., M 87* 2017 III):

$$SEFD = \frac{1}{\eta_{ph}} \frac{T_{sys}^*}{DPFU \times g_E}, \qquad (A.1)$$

where $T_{sys}^*$ is the effective system noise temperature, corrected for atmospheric attenuation. The degrees per flux density unit, DPFU, is a conversion factor from units of temperature (K) to units of flux density (Jy), and $g_E$ is the normalized elevation-dependent gain curve of the telescope. $\eta_{ph}$ is the time-dependent phasing efficiency for a phased array, and has a value of unity for single dish telescopes. The DPFU can be estimated from the antenna aperture efficiency ($\eta_A$) as:

$$DPFU = \frac{\eta_A A_{geom}}{2k}, \qquad (A.2)$$

<table><caption>**Table A.1.** EHT station flux density calibration parameters and their uncertainties for the 2018 observing campaign.</caption><thead><tr><th rowspan="2">Station</th><th colspan="5">DPFU (K/Jy)</th><th colspan="2">g<sub>E</sub></th></tr><tr><th>LSB-RCP</th><th>LSB-LCP</th><th>USB-RCP</th><th>USB-LCP</th><th>uncertainties</th><th>B</th><th>E<sub>0</sub></th></tr></thead><tbody><tr><td>ALMA<sup>a</sup></td><td>1.19</td><td>1.19</td><td>1.19</td><td>1.19</td><td>± 10%</td><td>0</td><td>0</td></tr><tr><td>APEX<sup>b</sup></td><td>0.0253</td><td>0.0262</td><td>0.0259</td><td>0.0270</td><td>± 5.5%</td><td>0.00002 ± 3.6%</td><td>36.6 ± 1.0%</td></tr><tr><td>GLT<sup>c</sup></td><td>-</td><td>-</td><td>0.00885</td><td>0.00885</td><td>± 7%</td><td>0</td><td>0</td></tr><tr><td>LMT</td><td>0.188</td><td>0.126</td><td>0.188</td><td>0.126</td><td>± 22%</td><td>0</td><td>0</td></tr><tr><td>SMT</td><td>0.0188</td><td>0.0188</td><td>0.0182</td><td>0.0182</td><td>± 6%</td><td>0.000082 ± 10.4%</td><td>57.6 ± 2.0%</td></tr><tr><td>JCMT<sup>d</sup></td><td>0.0296</td><td>-</td><td>0.0296</td><td>-</td><td>±(11–14)%</td><td>0</td><td>0</td></tr><tr><td>IRAM-30m</td><td>0.144</td><td>0.138</td><td>0.142</td><td>0.137</td><td>± 19%</td><td>0.00018 ± 5.3%</td><td>43.7 ± 1.3%</td></tr><tr><td>SMA<sup>e</sup></td><td>0.040</td><td>0.040</td><td>0.040</td><td>0.040</td><td>±(5–15)%</td><td>0</td><td>0</td></tr><tr><td>SPT</td><td>0.00698</td><td>0.00731</td><td>0.00698</td><td>0.00731</td><td>± 10%</td><td>0</td><td>0</td></tr></tbody></table>

**Note.** DPFU values are quoted for both lower (LSB) and upper (USB) sidebands, and both polarizations (RCP and LCP). For phased arrays (ALMA and SMA), the DPFUs represent the combined sensitivity of all phased dishes. The gain curve parameters B and $E_0$ as a function of elevation E are given based on a $g_E = 1 - B(E - E_0)^2$ parameterization (see Janssen et al. 2019b; Koay et al. 2023a, for further details). <sup>a</sup>The ALMA DPFU uncertainty is based on the overall 10% systematic uncertainties associated with the QA2 flux calibration tables. <sup>b</sup>APEX measures DPFUs for each of the four bands separately. LSB and USB DPFUs shown here are the mean of the bands 1 and 2 DPFUs, and bands 3 and 4 DPFUs, respectively. <sup>c</sup>GLT DPFU uncertainties are based on the scatter of three measurements, inclusive of flux density model errors, but may be larger due to significant antenna astigmatism. There is also a ~15% $T_{sys}^*$ uncertainty contributing to the total SEFD uncertainty. <sup>d</sup>For the JCMT DPFU, the nighttime value is shown. <sup>e</sup>The SMA DPFU uncertainty is based on the dominant 5–15% uncertainty on the phasing efficiency.

where $A_{geom}$ is the geometrical area of the antenna dish in units of m², and $k = 1.38 \times 10^3$ is the Boltzmann constant in units of Jy m²/K.

A summary of the station DPFUs and gain curve parameters for the EHT array in 2018 is provided in Table A.1, together with their estimated uncertainties. Details on their derivation for each station are described by Koay et al. (2023a). We expect the DPFU uncertainties to be the dominant source of systematic errors in the flux density calibration of EHT data. $T_{sys}^*$ values are not expected to vary much within the typical scan duration of a few minutes, so their uncertainties are expected to be no larger than a few percent. The GLT is an exception, due to sub-optimal performance in 2018 when it was still only partially commissioned, where $T_{sys}^*$ uncertainties are also significant (Koay et al. 2023b).

We note in particular that the quoted errors for the GLT and LMT in Table A.1 are very likely to be underestimated, due to insufficient a priori data collected by the stations, as described in Appendix D. We therefore obtain an independent set of constraints on the amplitude calibration uncertainties for these two stations in Appendix E.
## Appendix B: Non-closing systematic error budget

The closure phases ($\psi_C$), log closure amplitudes ($\ln A_C$), closure trace phases and closure trace log amplitudes $\mathcal{T}$, the latter two of which are novel diagnostics described in detail by Broderick & Pesce (2020), are independent of station-based gain errors, but are sensitive to systematic non-closing systematic errors. Figure B.1 shows the normalized distributions of the differences in the closure quantities between bands (band 1 – band 2, band 3 – band 4), polarizations (RR–LL), as well as the distributions of the trivial closure quantities derived from triangles or quadrangles with at least two co-located stations.

In an ideal scenario where polarimetric leakages are absent, the differences in closure quantities between bands and polarizations, as well as the trivial closure quantities, is expected to follow a normal distribution with a mean value of zero. The presence of non-closing systematic errors can lead to deviations from the expected normal distribution, as can be seen for the blue histograms in Fig. B.1.

The non-closing systematic error, $s$, can thus be determined by solving for its value while enforcing the condition of:

$$ \text{mad}_0\left(\frac{X}{\sigma}\right) = \text{mad}_0\left(\frac{X}{\sqrt{\sigma_{\text{th}}^2 + s^2}}\right) = 1, \qquad (\text{B.1}) $$

where $\sigma$ is the total uncertainty associated with the random variable $X$, and $\sigma_{\text{th}}$ is the known a priori thermal uncertainty. In this construction, $X$ is a random variable representing the closure products being examined, for instance band 1 – band 2 closure phase difference, RR–LL log closure amplitude difference, etc. $\text{mad}_0$ is the modified median absolute deviation given by:

$$ \text{mad}_0(Y) = 1.4826 \text{ med}(|Y|), \qquad (\text{B.2}) $$

where med denotes the median, and the factor 1.4826 scales the result so that it acts as a robust estimator of standard deviation for a normally distributed random variable $Y$ with zero mean.

The non-closing systematic errors estimated for M87* and 3C 279 using the various tests are presented in Table B.1. To avoid biases arising from low $S/N$, only closure quantities above the $S/N > 6$ threshold are used.

<table><caption>**Table B.1.** Non-closing systematic uncertainties, $s$ (and in units of thermal noise, $s/\sigma_{\text{th}}$), for M87* and its calibrator 3C 279 estimated using various statistical tests on both CASA and HOPS products.</caption><thead><tr><th rowspan="2">Source</th><th rowspan="2">Test</th><th colspan="3">CASA</th><th colspan="3">HOPS</th></tr><tr><th>s</th><th>s/σ<sub>th</sub></th><th>n</th><th>s</th><th>s/σ<sub>th</sub></th><th>n</th></tr></thead><tbody><tr><td rowspan="14">M87*</td><td>RR – LL closure phases</td><td>0.0°</td><td>0.0</td><td>339</td><td>0.0°</td><td>0.0</td><td>446</td></tr><tr><td>band 1 – band 2 closure phases</td><td>2.6°</td><td>0.3</td><td>229</td><td>0.0°</td><td>0.0</td><td>300</td></tr><tr><td>band 3 – band 4 closure phases</td><td>1.5°</td><td>0.2</td><td>192</td><td>0.0°</td><td>0.0</td><td>251</td></tr><tr><td>trivial closure phases</td><td>0.0°</td><td>0.0</td><td>436</td><td>1.3°</td><td>0.2</td><td>500</td></tr><tr><td>RR – LL log closure amplitudes</td><td>13.2%</td><td>0.7</td><td>158</td><td>4.0%</td><td>0.2</td><td>294</td></tr><tr><td>band 1 – band 2 log closure amplitudes</td><td>5.4%</td><td>0.3</td><td>139</td><td>2.2%</td><td>0.1</td><td>255</td></tr><tr><td>band 3 – band 4 log closure amplitudes</td><td>0.0%</td><td>0.0</td><td>116</td><td>0.0%</td><td>0.0</td><td>171</td></tr><tr><td>trivial log closure amplitudes</td><td>0.0%</td><td>0.0</td><td>60</td><td>0.0%</td><td>0.0</td><td>119</td></tr><tr><td>band 1 – band 2 closure trace phases</td><td>1.2°</td><td>0.2</td><td>43</td><td>0.0°</td><td>0.0</td><td>103</td></tr><tr><td>band 3 – band 4 closure trace phases</td><td>3.4°</td><td>0.5</td><td>60</td><td>3.8°</td><td>0.5</td><td>114</td></tr><tr><td>trivial closure trace phases</td><td>0.0°</td><td>0.0</td><td>118</td><td>3.5°</td><td>0.5</td><td>152</td></tr><tr><td>band 1 – band 2 log closure trace amplitudes</td><td>0.0%</td><td>0.0</td><td>43</td><td>2.4%</td><td>0.2</td><td>103</td></tr><tr><td>band 3 – band 4 log closure trace amplitudes</td><td>0.0%</td><td>0.0</td><td>60</td><td>5.3%</td><td>0.4</td><td>114</td></tr><tr><td>trivial log closure trace amplitudes</td><td>4.8%</td><td>0.4</td><td>118</td><td>0.0%</td><td>0.0</td><td>152</td></tr><tr><td rowspan="14">3C 279</td><td>RR – LL closure phases</td><td>1.8°</td><td>0.5</td><td>680</td><td>1.9°</td><td>0.6</td><td>687</td></tr><tr><td>band 1 – band 2 closure phases</td><td>1.7°</td><td>0.5</td><td>404</td><td>1.8°</td><td>0.6</td><td>407</td></tr><tr><td>band 3 – band 4 closure phases</td><td>1.9°</td><td>0.5</td><td>416</td><td>1.8°</td><td>0.5</td><td>420</td></tr><tr><td>trivial closure phases</td><td>0.9°</td><td>0.6</td><td>503</td><td>0.8°</td><td>0.6</td><td>503</td></tr><tr><td>RR – LL log closure amplitudes</td><td>5.4%</td><td>0.7</td><td>833</td><td>5.4%</td><td>0.7</td><td>846</td></tr><tr><td>band 1 – band 2 log closure amplitudes</td><td>5.6%</td><td>0.7</td><td>604</td><td>6.5%</td><td>0.9</td><td>605</td></tr><tr><td>band 3 – band 4 log closure amplitudes</td><td>5.1%</td><td>0.7</td><td>619</td><td>6.4%</td><td>0.8</td><td>638</td></tr><tr><td>trivial log closure amplitudes</td><td>3.2%</td><td>0.7</td><td>487</td><td>2.9%</td><td>0.7</td><td>491</td></tr><tr><td>band 1 – band 2 closure trace phases</td><td>3.0°</td><td>0.9</td><td>378</td><td>3.8°</td><td>1.2</td><td>393</td></tr><tr><td>band 3 – band 4 closure trace phases</td><td>3.4°</td><td>1.0</td><td>359</td><td>3.3°</td><td>0.9</td><td>412</td></tr><tr><td>trivial closure trace phases</td><td>0.5°</td><td>0.3</td><td>316</td><td>0.5°</td><td>0.3</td><td>320</td></tr><tr><td>band 1 – band 2 log closure trace amplitudes</td><td>8.5%</td><td>1.4</td><td>378</td><td>9.0%</td><td>1.6</td><td>393</td></tr><tr><td>band 3 – band 4 log closure trace amplitudes</td><td>7.0%</td><td>1.1</td><td>359</td><td>6.3%</td><td>1.0</td><td>412</td></tr><tr><td>trivial log closure trace amplitudes</td><td>2.7%</td><td>1.0</td><td>316</td><td>2.4%</td><td>0.8</td><td>320</td></tr></tbody></table>

**Note.** Only data from Apr 21 and Apr 25 have been used. $n$ is the number of closure quantities used for each test.
<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /><figcaption>**Fig. B.1.** Diagnostic plots showing the normalized distributions of various closure quantities of M87\* and 3C 279 data from both the CASA (top four rows) and HOPS (bottom four rows) reduction pipelines. For each block corresponding to each pipeline, the first two rows show band 1 – band 2 (first column), band 3 – band 4 (second column), RR–LL (third column), and trivial (fourth column) closure quantities. The bottom two rows show band 1 – band 2 (first column), band 3 – band 4 (second column), and trivial (third column) closure trace quantities. Only data from April 21 and 25 have been used. The distributions prior to (blue) and after (red) accounting for the estimated systematic uncertainties, *s*, are shown. The values of *s* for each source and reduction pipeline are given in Table B.1. In the top left corner of each distribution, the number of >3σ outliers are given considering thermal noise only, followed by the number of outliers considering thermal plus systematic noise for *σ* in parenthesis. These numbers are followed by the total number of data points after a slash.</figcaption></figure>
# Appendix C: Temporal coherence

<figure><img src="image_26.png" /><figcaption>**Fig. C.1.** Cumulative histograms of the amplitude ratios between coherent averaging for entire scans ($A_{scan}$), and coherent averaging for 2s before incoherent averaging over scans ($A_{2s}$), for both HOPS (blue) and CASA (orange) M87* data on both April 21 and April 25. The gray histogram shows the corresponding ratios from the HOPS data on the same days with no atmospheric phase corrections applied. For each pipeline, the fraction of data with coherence above 90% (i.e., with decoherence losses of no more than 10%) when averaged over the full scan is indicated.</figcaption></figure>

We quantify the level of decoherence loss of the data calibrated using the HOPS and CASA pipelines using the ratio $A_{scan}/A_{2s}$ for each scan (M87* 2017 III). $A_{scan}$ is the debiased amplitude of the coherently averaged visibilities over the entire scan, and $A_{2s}$ is the amplitude derived from data coherently averaged over 2s, then incoherently averaged over the entire scan.

Figure C.1 shows the cumulative distribution of the quantity $A_{scan}/A_{2s}$ for 2342 unique scans (per band, polarization, and baseline) of the M87* April 21 and 25 data with $S/N > 7$, common to both the HOPS and CASA pipelines. We find that 92.9% of HOPS and 85.6% of CASA scan-averaged data show decoherence losses of less than 10%. When selecting April 21 data only, 97.4% and 87.9% of HOPS and CASA scan-averaged data show better than 90% coherence. For April 25, 86.3% and 82.3% of HOPS and CASA scan-averaged data show better than 90% coherence.

The level of coherence in the 2018 data is slightly lower than that of 2017 M87* 2017 III, due to (i) overall poorer weather at some sites, including Chile, Hawai'i, and at the SMT (especially on April 25), as well as (ii) the poor sensitivity levels of the GLT, as described in Appendix E. Overall, CASA data have lower coherence levels than HOPS data, mostly for baselines with lower $S/N$ such as those including the GLT. For example, when we exclude GLT baselines, the percentage of CASA scan-averaged data showing decoherence losses of less than 10% increases by a few percent, and becomes comparable to that of HOPS.

# Appendix D: Data issues

In this appendix, we provide details on data issues that were identified in the 2018 L1 data package pertaining to M87* (and 3C 279), and steps taken to address these issues. Our analysis

indicated that the flux densities on APEX baselines are underestimated after a priori amplitude calibration, when compared to the flux densities of ALMA baselines to other stations. We also found 25% differences between RCP and LCP amplitudes on all APEX baselines in band 1, and a smaller but still significant 13% difference between the RCP and LCP amplitudes in band 4. Follow-up diagnostics found this issue to be caused by low power levels of the intermediate frequency (IF) signal and too large block-down converter attenuation settings for the ROACH2 Digital Back End (R2DBE; Vertatschitsch et al. 2015) associated with RCP in bands 1 and 4. These issues are corrected at the network calibration stage. There are also large amplitude drops on APEX baselines after UTC 01:00 on April 25, found to be caused by 180° peak-to-peak phase modulations whose origin is not yet determined to date. We therefore flagged and excised all the affected APEX data on April 25. The APEX baselines also show large amplitude jumps in several scans, typically at the beginning and end of a scan, which we also excised during post-processing.

While the above problematic L1 data have either been corrected or excised during post-processing, there remain a number of issues in the data used in our analyses. The GLT was still under commissioning and participated on a best effort basis in 2018. Due to astigmatism and a low ~22% aperture efficiency, the visibilities on GLT baselines have lower $S/N$ compared to that of the other stations. The telescope sensitivity was also not well characterized in 2018, so large uncertainties in the amplitude calibration are expected (Koay et al. 2023b). Due to the low telescope sensitivity, pointing sources could not be tracked down to elevations of < 10°, resulting in a poorly constrained pointing model at those low elevations and additional amplitude losses for sources observed at those elevations. This does not significantly affect M87* which was observed at higher elevations, but severely compromises the quality of GLT data for 3C 279 which was observed between 6° to 8°.

The abrupt halt of the LMT observations in the middle of the 2018 campaign meant that there were no antenna sensitivity measurements recorded at the time. The amplitude calibration applied uses sensitivity measurements obtained in 2020, based on the assumption that the antenna characteristics have not changed in the intervening period. Also there were focusing and pointing issues being corrected on the fly while observing M87* and 3C 279 on April 21, possibly leading to larger amplitude calibration errors. No elevation dependent gain curves were applied owing to a lack of aperture efficiency measurements at low elevations, where large gain errors of 20% (or possibly higher) are expected. This is not expected to significantly affect the M87* and 3C 279 observations, since these targets were observed at high elevations of 50° to 77°.

The SMA phasing efficiencies on April 25 are poor (mostly below 50% between UTC 03:00 and 05:30) and have a large scatter, resulting in a large scatter in amplitudes on SMA baselines. The PV station also experienced poor weather on April 25 after UTC 02:00, leading to a large scatter in amplitudes and phases on associated baselines. There are large drops in amplitudes on all ALMA baselines on April 21 between UTC 04:12 to 04:17, attributed to phasing issues on ALMA. These are partly corrected during a priori amplitude calibration and network calibration, but a residual 40% scatter or drop in amplitude remains on the corresponding scan, which can be fixed during model-dependent self-calibration downstream.
**Appendix E: Additional LMT and GLT gain constraints from overlapping visibilities**

As mentioned in Appendix D, the GLT and LMT station sensitivities are not well constrained (see also Appendix A), and their residual gain uncertainties are expected to be larger than the amplitude calibration uncertainties provided in Table A.1 in Appendix A. To obtain additional a priori constraints on the amplitude calibration uncertainties for baselines to these two stations, we examine and compare amplitudes on the baselines that overlap with that of the LMT and GLT in (u, v) space.

The ALMA–LMT and ALMA–SPT baselines overlap for the calibrator source 3C 279, (Fig. E.1 left panel). Assuming that the SPT sensitivity is well-characterized and that the amplitude calibration uncertainties are relatively small in comparison, we find the amplitudes on LMT baselines to be under-calibrated by ~34% in bands 1 and 2, ~23% in band 3 (Fig. E.1, right panel), and ~16% in band 4.

For the GLT, there are no overlapping baselines for M87*. Amplitudes on GLT baselines for 3C 279 cannot be used to constrain its amplitude uncertainties for M87* due to the large pointing offsets (see Appendix D). The closest baselines in (u, v) space where the amplitudes can be compared for M87* are between ALMA–LMT and GLT–PV, where we find the amplitudes on the GLT–PV baselines are lower than ALMA–LMT amplitudes by up to ~50% in bands 3 and 4. Considering that the LMT itself is under-calibrated, the amplitude errors for GLT are likely to be larger than 50%.

These a priori constraints are very crude, zeroth order estimates, but provide useful consistency checks with downstream station-based gains derived from model-dependent calibration, that is self-calibration during the imaging analysis. They are also important to better characterize the station uncertainties used in the generation of the synthetic data in Appendix G, so that they are reasonably consistent with that of the actual M87* data.

<figure><img src="image_27.png" /><figcaption>**Fig. E.1.** Left: (u, v) coverage for 3C 279 observations in band 3 on 2018 April 21 (colored points), showing the overlap between ALMA–LMT and ALMA–SPT baselines (highlighted with a box). Right: band 3 flux densities on ALMA–LMT (blue) and ALMA–SPT (orange) as a function of the projected baseline length in the east-west direction, in units of wavelength, demonstrating the under-calibration of amplitudes on LMT baselines. HOPS data are shown in this figure, but are consistent with that from CASA.</figcaption></figure>
# Appendix F: Derivation of constraints on the total compact flux density and source size

We estimate constraints on the total compact flux density by using the EHT data themselves, and by comparing the flux density from quasi-simultaneous multi-wavelength observations. These two methods are similar to those used in M87* 2017 IV and Sgr A* 2017 II. We provide the details below.

## F.1. Constraints from EHT observations

Here, we derive constraints on the total compact flux density and the source size. We use the procedure described in Appendix B.1 of M87* 2017 IV, which was updated in Sgr A* 2017 II to incorporate our uncertainties for the antenna gains using the DPFU uncertainties. This procedure included four constraints for the previous M87* analysis. However, the 2018 data do not contain crossing points in the (u, v) space for the SMT-PV and LMT-PV baselines; therefore we cannot apply the 4th constraint. Moreover, only the observations on April 21 and 22 contain the short intra-site baseline, SMT-LMT, which is necessary for this analysis. Thus we can derive the three constraints summarized below only for the first two observing days. For reference, the total flux density $F_{tot}$ estimated from the ALMA interferometric data is 1.13 Jy for both April 21 and 22. This value is an average of the total flux values derived for each of the bands 1–4.

**Constraint 1.** The first constraint is based on the fact that the visibility amplitudes on short baselines can be approximated by a circular Gaussian visibility function,

$$V_G(\mathbf{u}; V_0, \theta) = V_0 \exp\left(-\frac{(\pi\theta|\mathbf{u}|)^2}{4\ln 2}\right), \qquad (F.1)$$

where $V_0$ is the total flux density of the Gaussian source, $|\mathbf{u}|$ is the length of the baseline, and $\theta$ is its FWHM in radians. The intermediate-to-long baselines tend to measure larger correlated flux density than what is expected for an equivalent Gaussian source.

We can deduce that the measured amplitude ratio of the ALMA–LMT over SMT–LMT baselines will be larger than the corresponding ratio from a circular Gaussian source model. Consequently, the FWHM size of a circular Gaussian determined by the amplitude ratio between SMT–LMT and ALMA–LMT baselines provides an estimate of the minimum compact source size $\theta_{cpcpt}$ that is not significantly affected by the intrinsic fine-scale source structure:

$$\theta_{cpcpt} \geq \sqrt{\frac{4 \ln 2 \ln\left(\frac{|V_{SMT-LMT}|}{|V_{ALMA-LMT}|}\right)}{\pi^2\left(|\mathbf{u}_{ALMA-LMT}|^2 - |\mathbf{u}_{SMT-LMT}|^2\right)}}. \qquad (F.2)$$

Here, $V_{i-j}$ denotes the true visibility on the baseline $i-j$. Considering the gain uncertainties, the ratio of the true visibility amplitudes is lower-bounded by

$$\frac{|V_{SMT-LMT}|}{|V_{ALMA-LMT}|} \geq \left(1 - \sqrt{\Delta g_{ALMA}^2 + \Delta g_{SMT}^2}\right) \frac{|V_{SMT-LMT}|}{|V_{ALMA-LMT}|}, \qquad (F.3)$$

where $V_{i-j}$ and $\Delta g_i$ denote the calibrated measured visibility and the gain uncertainty, respectively. We take the median of the visibility amplitude ratio from the collection of constraints derived for each single VLBI scan to derive a robust estimate of the minimum source size.

**Constraint 2.** The second constraint comes from the curvature of visibility amplitudes between the intra-site baselines and

the LMT–SMT baseline. Since the compact flux density should not exceed the total flux density measured with the intra-site baselines, the amplitude drop from the intra-site to LMT–SMT baselines gives the maximum limit of that from the compact flux density to LMT–SMT baseline. Therefore, it gives the maximum limit of the source FWHM size with an equivalent circular Gaussian as

$$\theta_{cpcpt} \leq \frac{2\sqrt{\ln 2}}{\pi|\mathbf{u}|} \sqrt{\ln \frac{F_{tot}}{|V_{LMT-SMT}|}}. \qquad (F.4)$$

When we consider the gain uncertainties,

$$\frac{1}{|V_{LMT-SMT}|} \leq \frac{1 + \sqrt{\Delta g_{LMT}^2 + \Delta g_{SMT}^2 + \Delta g_{tot}^2}}{|V_{LMT-SMT}|}. \qquad (F.5)$$

We assumed the uncertainty of the total flux $\Delta g_{tot} = 0.1$. Similarly to Constraint 1, the median of the ratio is adopted to mitigate the effects of statistical errors.

**Constraint 3.** The minimum compact flux density can be derived by the maximum amplitudes of LMT–SMT baseline, since the visibility amplitudes are maximum at zero baseline length and the a priori calibration should not underestimate the station sensitivity. Therefore the compact flux density is constrained as

$$F_{cpcpt} \geq \left(1 - \sqrt{\Delta g_{LMT}^2 + \Delta g_{SMT}^2}\right) |V_{LMT-SMT}|. \qquad (F.6)$$

The equivalent circular Gaussian gives a stronger constraint with the minimum source size (Constraint 1) extrapolated from the LMT–SMT baselines, described by,

$$F_{cpcpt} \geq |V_{LMT-SMT}| \exp\left(\frac{(\pi|\mathbf{u}|\theta_{cpcpt})^2}{4\ln 2}\right). \qquad (F.7)$$

These constraints are validated using the various synthetic models described in Appendix G. The upper and lower limits on the source size obtained from Constraints 1 and 2 are along the directions of LMT–SMT and LMT–ALMA baselines.

In summary, Constraints 1–3 lead to the conclusion that $0.30 \text{ Jy} \leq F_{cpcpt} \leq 1.13 \text{ Jy}$ on April 21 and 22. If we use Eq. F.7, the minimum compact flux is 0.38 Jy. The compact source size has an equivalent Gaussian FWHM ranging between 39 and 98 µas on the first two observing days.

## F.2. Constraints from quasi-simultaneous multi-wavelength observations

Supplementary, we also make use of quasi-simultaneous VLBI data at longer wavelengths obtained through part of a large MWL observing campaign coordinated with the EHT 2018 observations. Our approach assumes that the total compact emission of M87* at $\nu \approx 230 \text{ GHz}$ ($\lambda \approx 1.3 \text{ mm}$) $F_{tot,230}$ originates from the combination of (i) a compact emitter with unknown flux density $F_{cpcpt,230}$ and spectral index at 230 GHz, and (ii) a large and diffuse (milliarcsecond-scale) jet, which can be characterized by its total flux density $F_{jet,230} = F_{tot,230} - F_{cpcpt,230}$ and a steep spectral index of $\alpha_{jet} \sim -(0.7 - 1.0)$, as measured at cm- and mm-wavelengths (with typical errors of $\sim 0.3$; see Hovatta et al. 2014; Hada et al. 2016). Then, by measuring $F_{jet}$ and $F_{tot}$ at lower frequencies, and extrapolating them to 230 GHz using a power-law model, we can estimate the total compact flux density $F_{cpcpt}$ at 230 GHz: $F_{cpcpt,230} = F_{tot,230} - F_{jet,230}$.

We utilize the data obtained with the East Asian VLBI Network (EAVN; An et al. 2018; Cui et al. 2021; Cho et al.
2022) at 22 and 43 GHz, the Korean VLBI Network (KVN; Lee et al. 2016) at 43, 86, and 129 GHz, and the Global Mil- limeter VLBI Array (GMVA)+ALMA at 86 GHz. All of these observations were conducted between 2018 March 25 and April 21. For details of data reduction and calibration, see Cho et al. (2017), Cui et al. (2021) for EAVN, Lee et al. (2016), Kim et al. (2018b) for KVN, and Lu et al. (2023) for GMVA+ALMA, respectively.

The total flux densities of the nuclear region (i.e., $F_{\text{tot}} = F_{\text{jet}} + F_{\text{cpct}}$) at 22, 43, and 86 GHz are measured from the EAVN and KVN images by integrating the flux densities over a large window of $500 \times 500 \ \mu\text{as}^2$ centered on the radio core. We obtain total flux densities of $F_{\text{tot,22}} \approx 1.1$ Jy, $F_{\text{tot,43}} \approx 0.9-1.1$ Jy, $F_{\text{tot,86}} \approx 0.9$ Jy and $F_{\text{tot,129}} \approx 0.6-0.8$ Jy at 22, 43, 86, and 129 GHz, respectively. The KVN 129 GHz data suffer from larger uncertainties due to a stronger influence from weather. We fit a power-law model to the total flux densities (i.e., $F_{\text{tot}} \propto v^{\alpha_{\text{tot}}}$). We find $\alpha_{\text{tot}} \approx -0.17$ when 129 GHz data are included, while $\alpha_{\text{tot}} \approx -0.11$ when they are excluded. By extrapolating a power-law model with a range $\alpha_{\text{tot}} \approx -(0.11-0.17)$ to 230 GHz, we estimate $F_{\text{tot,230}} = 0.72-0.80$ Jy. While this range of values is lower than the ALMA interferometric measurements during the EHT observations (~1.11–1.18 Jy; Table 3 in Goddi et al. 2021), it only includes the total flux density within the inner $500 \times 500 \ \mu\text{as}^2$, which is nearly three orders of magnitude below the resolution limit of ALMA.

The compact flux density $F_{\text{cpct,86}}$ at 86 GHz is estimated from a GMVA+ALMA image that has a higher angular resolu- tion than the KVN images (Lu et al. 2023). We obtain $F_{\text{cpct,86}} \approx 0.51$ Jy by integrating over a smaller window of $100 \times 100 \ \mu\text{as}^2$ centered on the radio core. We can then derive a jet flux density at 86 GHz to be $F_{\text{jet,86}} = F_{\text{tot,86}} - F_{\text{cpct,86}} \approx 0.39$ Jy.

Finally, we compute the expected $F_{\text{cpct,230}}$ by estimating $F_{\text{jet,230}}$ and subtracting it from the previously estimated $F_{\text{tot,230}}$. As explained above, we assume that $F_{\text{jet,230}}$ follows a power law that can be extrapolated up to 230 GHz (i.e., $F_{\text{jet,230}} \propto v^{\alpha_{\text{jet}}}$). For a range of $\alpha_{\text{jet}} = -(0.7-1.0)$, we find $F_{\text{jet,230}} = 0.15-0.20$ Jy. This value gives a range of $F_{\text{cpct,230}} \approx 0.5-0.7$ Jy within the central $100 \times 100 \ \mu\text{as}^2$ region. We emphasize that this estimate is based on lower-frequency data and assumptions on the jet spec- tral index. Therefore it should not be regarded as a tight con- straint on $F_{\text{cpct,230}}$ but more as a useful reference.

# Appendix G: Imaging validation

Here we describe the synthetic data generation process and sub- sequent reconstructions, as well as the resulting distribution of Top Set images for the real M87* data. We also investigate the properties of the image reconstructions when removing individ- ual baselines.

## G.1. Synthetic data

In order to test the efficacy of the imaging pipelines, we generate synthetic data sets using both geometric models and snapshots of GRMHD simulations. All of the geometric models contain 0.6 Jy in compact flux density and 0.5 Jy in a larger extended jet mod- eled by three Gaussians. We show the ring model as an example in Fig. G.1. The sizes and locations of these three Gaussians are identical to those reported in Table 10 of M 87* 2017 IV, with the flux densities scaled down to total 0.5 Jy rather than 0.6 Jy. Of the geometric models we generated, four of them were utilized in the procedure to generate the Top Set images. These four geometric

models are very similar to those used in M 87* 2017 IV and are as follows:

1. cres180: An asymmetric ring model with $r_0 = 23 \ \mu\text{as}$, a brightness position angle oriented south, and blurred by a circular Gaussian beam of FWHM $10 \ \mu\text{as}$.

2. ring: A thin uniform ring of radius $r_0 = 23 \ \mu\text{as}$ blurred by a circular Gaussian beam of FWHM $10 \ \mu\text{as}$.

3. db1src: Two circular Gaussian components each with FWHM of $20 \ \mu\text{as}$. One is located at the origin with a flux density of 0.27 Jy, while the second is positioned at $\Delta_{\text{R.A.}} = 30 \ \mu\text{as}$ and $\Delta_{\text{decl.}} = -12 \ \mu\text{as}$ with a flux density of 0.33 Jy.

4. disk: A uniform disk of radius $r_0 = 35 \ \mu\text{as}$ blurred by a circular Gaussian beam of FWHM $10 \ \mu\text{as}$.

Ground-truth images of these models are shown in Fig. G.2. Aside from these training data, we also generated seven valida- tion data sets to ensure good performance from the CLEAN and RML method Top Sets. These data, along with reconstructions from each pipeline, are shown in Fig. G.3. The model descrip- tions are as follows

1. cres-90, cres0, cres90: Three asymmetric ring models with $r_0 = 23 \ \mu\text{as}$, and brightness position angles oriented east, north, and west, blurred by a circular Gaussian beam of FWHM $10 \ \mu\text{as}$.

2. edisk: A uniform elliptical disk model with a major axis of $66 \ \mu\text{as}$, a minor to major axis ratio of 0.65, a major axis position angle of 60°, and a blurring of $10 \ \mu\text{as}$.

3. point+disk: A point source plus symmetric disk model containing a $10 \ \mu\text{as}$ point source centered in $100 \ \mu\text{as}$ diam- eter disk. The point source to disk flux ratio is chosen to be 0.192.

4. point+edisk: A point source plus elliptical disk model containing a $10 \ \mu\text{as}$ point source centered in an ellipse of major axis of $96 \ \mu\text{as}$, a minor to major axis ratio of 0.8, a major axis position angle of 60°, and a blurring of $10 \ \mu\text{as}$. The point source to disk flux ratio is 0.16.

5. GRMHD: A snapshot from a GRMHD simulation of M87*. The synthetic visibilities were generated using eht-imaging and all include random thermal noise, station- based gain and phase errors, and polarimetric leakage. For a more in-depth explanation of synthetic visibility generation, we refer readers to Appendix C.2 of M 87* 2017 IV.

Figures G.2 and G.3 show the image reconstructions for four training and seven validation data sets including the GRMHD model from each imaging pipeline, respectively. The image reconstructed with the fiducial parameters is displayed for DIFMAP, eht-imaging, and SMILI. A single image ran- domly selected from the posterior is displayed for THEMIS and Comrade. The THEMIS reconstruction of the GRMHD syn- thetic data uses a Hybrid Themage model. This figure clearly shows that a very wide range of morphology types can be recov- ered by all imaging pipelines. These simulations are based on the same $(u, v)$ coverage of the 2018 EHT observation, and thus have a common point spread function (PSF). This ensures that the recovered structures by imaging pipelines are not artificially caused by the $(u, v)$ coverage and/or the PSF, but rather come from the real structure imprinted in the data.

Based on $\rho_{\text{NX}}$ values and ring parameters, we again confirmed that most of the Top Set parameters can reconstruct the correct morphology. For instance, 95, 96, 97% of cres-90 April 21 band 3 images reconstructed with the Top Set param- eters from DIFMAP, eht-imaging, SMILI passed the $\rho_{\text{NX}}$ crite- ria imposed on the training data sets though some images have deviated structure as seen in the training data sets. Figure G.4 shows the significance of the deviated structure in the Top Set of
<figure><img src="image_28.png" /><figcaption>**Fig. G.1.** Example geometric ring model used to generate the synthetic data. The left panel exhibits the morphological characteristics of the ring on a linear scale, encompassing a FOV of 130 µas. The right panel depicts the logarithmic scale representation of the extended jet model (FOV = 2900 µas).</figcaption></figure>

synthetic data reconstructions with SMILI. Above a certain dynamic range threshold (e.g., 10% of peak intensity), artificial structures which deviate from the ground truth are shown but its fraction in the Top Set is minor, especially the repeated central structure. In addition, the fiducial images recover the station gains as presented in Fig. G.5. This shows the multiplicative gains at each telescope, in comparison with true gains from the cres180 model on April 21 band 3. In a same manner, the station gains are derived for M87* which are consistent across different imaging approaches, as presented in Fig. G.6. Certain imaging pipelines may perform poorly at certain epochs and bands. In case of point+disk April 21 band 3, only 40% of DIFMAP images reconstructed with the Top Set parameters passed the ρ$_{NX}$ criteria.

## G.2. The distribution of M87* Top Set images

As shown in the main text, all Top Set M87* images have consistent ring features. However, there are slight differences among the Top Set reconstructions. Figure G.7 shows 25 randomly selected Top Set reconstructions from HOPS band 3 April 21 data for DIFMAP, eht-imaging, and SMILI. We see the ~40 µas ring for all images, and most images have a similar brightness feature, but the ring width seems to have relatively large

uncertainty depending on the image assumptions of each imaging method. Also, a few images from eht-imaging and SMILI show double-ring structures although they are a minority in the Top Sets. As discussed above, we see similar minor double-ring structures in ring and crescent synthetic data reconstructions, so the second ring is unlikely to be real and we anticipate this does not affect our conclusion. These trends are also seen on the other observational dates and bands. Image variations among Top Sets are relatively larger, especially for data with poor coverage.

## G.3. Image dependence on sites

In addition to analyzing the dependence of M87* images on frequency bands and changes in coverage from different epochs, it is necessary to understand how much the reconstructed image relies on data from single antenna sites. For this purpose, we imaged M87* excluding all the data from baselines connected to one site, and repeated the imaging for all six sites. The test was performed on April 21 band 3 data, using the eht-imaging and Comrade pipelines as representative for the RML and Bayesian methods. The first row of Fig. G.8 shows the mean image reconstructed by Comrade, the second row shows eht-imaging images reconstructed with fiducial parameters, while the third one presents eht-imaging fiducial images of the 2017 April 11 data, reconstructed with the 2018 pipeline. We see that in most cases, the removal of one antenna in 2018 affects the quality of the image reconstruction, both for the Comrade and the eht-imaging pipelines.

The strongest image degradation happens when removing the Chile sites, which are fundamental to reconstruct the basic image morphology, but also imaging without the SMT or the Hawai'i sites results in the appearance of a tessellation pattern for eht-imaging and in a loss of contrast for Comrade. Removing PV baselines results in an elongation of the ring-like structure for both pipelines, while the absence of either the GLT or the LMT stations only introduces minor artifacts, without changing the overall image morphology.

Compared to 2017, the 2018 eht-imaging reconstructions without one antenna are affected by a stronger tessellation pattern. However, the morphology of the ring-like structure is more robust to the removal of a single site in 2018 compared to 2017. For example, in 2018, a ring-like structure is still distinguishable even without the Chile site. Also the removal of the LMT antenna does not affect the orientation of the ring, and the removal of PV does not strongly affect the morphology.
<figure><img src="image_29.png" /><figcaption>**Fig. G.2.** Four training geometric models as imaged by each imaging method. The first row shows the visibility amplitudes of the model (orange) compared to the visibility amplitudes measured for M87+ (blue). The second row shows the ground-truth images. The DIFMAP, eht-imaging, and SMILI rows show a fiducial image made from the same parameter sets as the images shown in Fig. 17. The THEMIS and Comrade rows show a random draw from the posterior. The circle in the DIFMAP panels represent a Gaussian blurring kernel of 20 µas. The white lines in the THEMIS and Comrade panels represent the size of FWHM for the Gaussian blurring kernel needed to match the DIFMAP effective resolution.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>**Fig. G.3.** Seven geometric validation models plus one GRMHD snapshot as imaged by each method. The first row shows the visibility amplitudes of the model (orange) compared to the visibility amplitudes measured for M87* (blue). The second row shows the ground-truth images. The DIFMAP, eht-imaging, and SMILI rows show a fiducial image made from the same parameter sets as the images shown in Fig. 17. The THEMIS and Comrade rows show a random draw from the posterior. The THEMIS reconstruction of the GRMHD synthetic data uses a Hybrid Themage model. The circle in the DIFMAP panels represent a Gaussian blurring kernel of 20 µas. The white lines in the THEMIS and Comrade panels represent the size of FWHM for the Gaussian blurring kernel needed to match the DIFMAP effective resolution.</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>**Fig. G.4.** Stacked images with a dynamic range cutoff used to investigate the structural deviations in the Top Set (from SMILI as a representative). First, each Top Set image is normalized with its peak intensity, and the pixels with a flux larger than a certain threshold (e.g., 0.1) are converted to unity (unless 0). Then all the images across the Top Set are stacked and normalized by the maximum number of Top Set images so that the fraction of reliable features over the entire Top Set is emphasized. The ground-truth structure of each synthetic data model is shown in gray contours.</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>**Fig. G.5.** Multiplicative gain correction factors at each station as a function of time. The ground truth gains (gray circles) are used to generate the synthetic data with the *cres180* model for April 21, band 3. Model gains from each pipeline are derived from the fiducial images or posterior samples of the *cres180* synthetic data reconstructions.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>**Fig. G.6.** Multiplicative gain correction factors at each station as a function of time from the fiducial images or posterior samples of the image reconstructions for the M87* HOPS band 3 data.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>**Fig. G.7.** Twenty five randomly selected Top Set images from HOPS band 3 April 21 data for DIFMAP (upper left), eht-imaging (upper right), and SMILI (bottom).</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>**Fig. G.8.** Example reconstructions of M87* for 2018 April 21 and 2017 April 11 after omitting visibilities to each geographical site. The images presented show reconstructions that exclude all baselines to the indicated site (i.e., mimicking an observation without that site). Top and middle rows show the reconstructions for Comrade and eht-imaging respectively for band 3 on 2018 April 21. The images for the eht-imaging pipeline were reconstructed using the eht-imaging fiducial parameters (see Sect. 5.2). The images shown for Comrade are the mean images from the posterior. The bottom row shows the reconstructions for 2017 April 11 using the 2018 eht-imaging pipeline but 2017 fiducial parameters (M87* 2017 IV). The ellipse in each panel shows the corresponding synthesized beam with uniform weighting, but the image is not convolved with this beam.</figcaption></figure>
## Appendix H: Imaging with DIFMAP using a point source model for initial self-calibration

This Appendix reports the results of the DIFMAP parameter survey, which used an initial phase-only self-calibration process assuming a point source starting model. Unlike the parameter survey discussed in the main text (Sect. 5.1.1), which seeks to determine the best geometric model for initial self-calibration, this survey requires a careful consideration of the CLEAN window location. This is due to the fact that the self-calibration process results in the brightest part of the source being placed at the map's coordinate origin, requiring a shift in the position of the window center's position for the subsequent CLEAN and self-calibration procedures.

To ensure optimal performance, the ideal window center location should be specified for each source model and not treated as a global free parameter. In this analysis, we employed an automated optimal mask search procedure during the parameter survey using a point source model for initial self-calibration. The procedure involved a systematic search of the parameter space by shifting the window center from the map origin in increments of 5 µas along the right ascension and declination directions. The shift is limited to a maximum value equal to the size of the CLEAN window. To maintain consistency, the five imaging parameters presented in Sect. 5.1.1 were kept constant at their fiducial parameters which were obtained from the CLEAN imaging of the 2017 EHT M87* data (M87* 2017 IV). These parameters included a compact flux density of 0.5 Jy, stopping CLEANing when the required compact flux density was reached, an ALMA weight re-scaling factor of 0.1, a window diameter of 60 µas, and a (u, v) weight exponent of -1. The pro-

cedure computed the $\chi^2_{CP}$ value for each image in the survey, as described in Sect. 5.2, and determined the optimal window position based on the image with the lowest $\chi^2_{CP}$ value. This approach enabled the determination of the optimal window position for each source model and window diameter.

Next, we conducted a parameter survey by varying the five imaging parameters (Sect. 5.1.1), assuming the optimal window position obtained during the automatic mask search procedure. We followed the image selection criteria outlined in Sect. 5.2 to select both Top Set and fiducial images from the survey. The fiducial images resulting from this survey are displayed in Fig. H.1. Our findings indicate that the survey was able to accurately reconstruct the majority of the synthetic data models. For the real M87* data, the Top Set images displayed a ring-like morphology, consistent with the results from the survey discussed in Sect. 5.1.1. This result demonstrates that the ring-like structure for M87*, reconstructed by DIFMAP, is stable independently of the geometrical model selected for the first phase of self-calibration. However, the overall performance of this survey was lower than that of the survey that optimized the geometric models for initial phase-only self-calibration, resulting in a smaller number of Top Set images. This can be attributed to the fact that the point source model used as initial phase-only self-calibration model in this survey is overly simplistic for many geometric models and actual M87* source structure. Despite encountering challenges with the uniform disk model, resulting in a lower number of Top Set images obtained from the survey, successful reconstruction of both the ring and crescent synthetic models was achieved. These models were found to closely resemble the true source structure.

<figure><img src="image_36.png" /><figcaption>**Fig. H.1.** Fiducial images produced by the DIFMAP pipeline reconstructed through the implementation of an initial phase-only self-calibration process using a point source model. The reconstructed images exhibit ring-like structures, which are comparable to the results obtained from the pipeline that utilized closure phases to select the optimal geometric model for the initial phase-only self-calibration, as demonstrated in Fig. 7. The circle represents a Gaussian blurring kernel with a FWHM of 20 µas.</figcaption></figure>
# Appendix I: Testing the backward compatibility of the DIFMAP pipelines on the 2017 data

The DIFMAP pipelines presented in Sect. 5.1.1 are updated as compared to the DIFMAP pipelines used in previous EHT imaging analyses (M87* 2017IV; Sgr A* 2017III). We conducted a backward compatibility test of these pipelines on the 2017 EHT

M87* data, including the real data from the HOPS pipeline (April 11, low-band) and the synthetic data sets for the training. Our analysis revealed that the resulting images align well with those previously reported in (M87* 2017IV). Our fiducial images of the M87* data and synthetic data can be found in Fig. 1.1.

<figure><img src="image_37.png" /></figure>

**Fig. 1.1.** Fiducial images of the 2017 April 11 low-band data obtained from the 2018 DIFMAP pipelines discussed in Sect. 5.1.1, as a backward compatibility check. The top row of images is obtained using the pipeline that conducts initial phase-only self-calibration by employing a point source model, whereas the bottom row of images is obtained from the pipeline that identifies the optimal geometric model for self-calibration. The circles represent a Gaussian blurring kernel with a FWHM of 20 µas.
## Appendix J: THEMIS imaging model details and priors

The THEMIS image reconstructions are composed of three different primary model components, the splined raster model, a large-scale asymmetric Gaussian, and a thin slashed crescent. A more detailed description of the large-scale Gaussian and crescent models, as well as details about their technical implementation, are described in Broderick et al. (2020a). A much more detailed discussion of the raster model can be found in Broderick et al. (2020b).

The splined raster model is defined by a rectilinear set of control points which may independently vary in intensity, $I_{M,N}$. The intensity map is produced using an approximate cubic spline interpolation between the control points. The field of view $FOV_x$ and $FOV_y$ are permitted to vary, along with the orientation of the grid $\phi$ and the raster center $(x, y)$. The large-scale Gaussian is asymmetric, and is characterized by its total flux $I_G$, a symmetric standard deviation $\sigma_G$, asymmetry parameter $A_G$, position angle $\phi_G$, and position $(x_G, y_G)$. The slashed crescent is based on the xs-ringauss model also described in Broderick et al. (2020b), which also serves as the core of the THEMIS xs-ringauss geometric model featured in Sect. 6.2 and in M 87* 2017 VI.

As a part of the Hybrid Themage model, we eliminate the contributions from the asymmetry and pinned Gaussian parameters by appropriately restricting their priors. We also restrict the priors on the fractional width parameter $\psi$ to range from 0% to 5%. This ensures the ring width is well below that of the raster spacing, to avoid excessive flux trading between the ring and raster components. This leaves the ring flux $I_X$, outer radius $R_{out}$, linear brightness gradient $f_X$, brightness gradient position angle $\phi_X$, and position $(x_X, y_X)$ as searchable parameters.

We use a deterministic even-odd (DEO) swap tempering scheme with a Hamiltonian Monte-Carlo sampling kernel based on the Stan package. Chain convergence is assessed based on traditional criteria such as the split $R$, auto-correlation time, and visual inspection of the parameter traces. The parameters and priors for each model component are listed in Table J.1.

<table><caption>Table J.1. THEMIS Raster Model Priors.</caption><thead><tr><th>Component</th><th>Parameter</th><th>Units</th><th>Prior<sup>a</sup></th></tr></thead><tbody><tr><td rowspan="6">Raster</td><td>$I_{M,N}$</td><td>Jy &mu;as<sup>-2</sup></td><td>L(10<sup>-14</sup>, 3 &times; 10<sup>3</sup>)</td></tr><tr><td>$FOV_x$</td><td>&mu;as</td><td>U(0, 200)</td></tr><tr><td>$FOV_y$</td><td>&mu;as</td><td>U(0, 200)</td></tr><tr><td>$\phi$</td><td>rad</td><td>U(-0.25&pi;, 0.25&pi;)</td></tr><tr><td>$x$</td><td>&mu;as</td><td>U(-40, 40)</td></tr><tr><td>$y$</td><td>&mu;as</td><td>U(-40, 40)</td></tr><tr><td rowspan="6">Gaussian</td><td>$I_G$</td><td>Jy</td><td>U(0, 10)</td></tr><tr><td>$\sigma_G$</td><td>mas</td><td>U(0.1, 10<sup>4</sup>)</td></tr><tr><td>$A_G$</td><td>...</td><td>U(0, 1)</td></tr><tr><td>$\phi_G$</td><td>rad</td><td>U(0, &pi;)</td></tr><tr><td>$x_G$</td><td>mas</td><td>U(-2, 2)</td></tr><tr><td>$y_G$</td><td>mas</td><td>U(-2, 2)</td></tr><tr><td rowspan="7">Ring</td><td>$I_X$</td><td>Jy</td><td>U(0, 2)</td></tr><tr><td>$R_{out}$</td><td>&mu;as</td><td>U(0, 10<sup>2</sup>)</td></tr><tr><td>$\psi$</td><td>...</td><td>U(0, 0.05)</td></tr><tr><td>$f_X$</td><td>...</td><td>U(0, 1)</td></tr><tr><td>$\phi_X$</td><td>rad</td><td>U(-&pi;, &pi;)</td></tr><tr><td>$x_X$</td><td>&mu;as</td><td>U(-40, 40)</td></tr><tr><td>$y_X$</td><td>&mu;as</td><td>U(-40, 40)</td></tr></tbody></table>

<footnote><sup>a</sup> Logarithmic priors from *a* to *b* are represented by $\mathcal{L}(a, b)$, and uniform (linear) priors from *a* to *b* are represented by $\mathcal{U}(a, b)$.</footnote>
# Appendix K: THEMIS Raster dimensions

<table><caption>**Table K.1.** THEMIS raster size ($N_x \times N_y$) survey for April 21 Band 3.</caption><thead><tr><th>Raster Dims.</th><th>red. χ²</th><th>Δ log(Z)</th></tr></thead><tbody><tr><td>4 × 4</td><td>1.15</td><td>-6</td></tr><tr><td>5 × 5</td><td>1.14</td><td>-</td></tr><tr><td>6 × 6</td><td>1.17</td><td>-21</td></tr><tr><td>7 × 7</td><td>1.20</td><td>-20</td></tr></tbody></table>

The best fit raster dimensions $N_x \times N_y$ for the THEMIS Raster models in principle depend on the (u, v) coverage and source complexity. To construct the best model, one should survey the raster dimensions for each unique data set. In order to provide easier comparisons across data sets, we only survey the raster dimensions on our primary science data set, April 21 band 3, and use those best fit raster dimensions for the other data sets. We present the results of this survey in Table K.1, where we survey a 4 × 4, 5 × 5, 6 × 6, and 7 × 7 raster. We rank the models in terms of the difference in the logarithm of the Bayesian Evidence (Δlog(Z)), where positive values are more preferred. The 5 × 5 raster is most preferred by this data, followed by the 4 × 4 raster. The 5 × 5 raster is also the same dimensions as the model used in Broderick et al. (2022a), which applied the Hybrid Themage model to the 2017 data. Thus, the image structures we show in this analysis should be directly comparable to the image structures in the previous paper. We also report the best fit reduced χ² for the surveyed raster sizes, and note that all models provide good fits to the data.

<table><caption>**Table K.2.** Complex visibility reduced χ² ($\chi_V^2$) for the THEMIS Raster and Hybrid Themage model best fits to the HOPS pipeline data.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="1">THEMIS Raster</th><th colspan="2">Hybrid Themage</th></tr><tr><th>Red. χ<sup>2</sup><sub>V</sub></th><th>Red. χ<sup>2</sup><sub>V</sub></th><th>Δ log(Z)</th></tr></thead><tbody><tr><td rowspan="4">April 21</td><td>band 1</td><td>1.41</td><td>1.43</td><td>-9</td></tr><tr><td>band 2</td><td>1.1</td><td>1.17</td><td>+1</td></tr><tr><td>band 3</td><td>1.14</td><td>1.07</td><td>+26</td></tr><tr><td>band 4</td><td>1.19</td><td>1.14</td><td>+23</td></tr><tr><td rowspan="4">April 25</td><td>band 1</td><td>2.20</td><td>2.7</td><td>+6</td></tr><tr><td>band 2</td><td>1.71</td><td>2.05</td><td>+8</td></tr><tr><td>band 3</td><td>1.17</td><td>1.17</td><td>+18</td></tr><tr><td>band 4</td><td>0.96</td><td>0.97</td><td>+14</td></tr></tbody></table>

We can also compare the fit quality between the standard THEMIS raster and Hybrid Themage models by similarly comparing the Δlog(Z). In Table K.2 we show the reduced χ² and difference in Bayesian evidence between the Hybrid Themage and raster-only model for both days and all bands. Here we see that the Hybrid Themage model is generally preferred in nearly all data sets, except for the April 21 band 1 data.

# Appendix L: Comrade image model and prior

For all Comrade image reconstructions, we construct our image model in the following steps. We begin with a rasterized image model convolved with a B-spline kernel of order 3 (see Eqs. L.1

and L.2) to generate flux densities that smoothly vary in all directions. We set the raster points by ($l_i, m_j$) given by Eq. L.3, for a given FOV and number of pixels $N_x$ and $N_y$ for each side. The FOV and number of pixels are the hyperparameters of the model:

$$I(l, m) = \sum_{i,j} B\left(\frac{l-l_i}{s_x}\right) B\left(\frac{m-m_j}{s_y}\right) I_{ij}, \qquad (L.1)$$

where $B(x)$ is a B-spline kernel of order 3 given by,

$$B(x) = \begin{cases} x^2/2 & 0 \le x < 1 \\ (-2x^2 + 6x - 3)/2 & 1 \le x < 2 \\ (3 - x)^2/2 & 2 \le x < 3 \end{cases} \qquad (L.2)$$

and raster points are given by,

$$\begin{aligned} l_i &= -\text{FOV}/2 + s_x/2 + s_x(i-1) \\ m_j &= -\text{FOV}/2 + s_y/2 + s_y(j-1) \end{aligned} \qquad (L.3)$$

where $s_x = \text{FOV}/N_x$ and $s_y = \text{FOV}/N_y$. We restrict the images considered in this paper to having $\tilde{N}_x = N_y$, such that the images are square and $s_x = s_y$. We add a circular Gaussian with $\text{FWHM} = 2\sqrt{2\ln}\sigma = 1$ mas ($G_{1\text{mas}}$) to the raster to model the amplitudes and station gains of short intra-site (ALMA–APEX, JCMT–SMA) baselines. We form our final image model,

$$M(I_{ij}, f, f_g) = f(1 - f_g) \times I(I_{ij}) + f_g f \times G_{1\text{mas}} \qquad (L.4)$$

where $f$ is the total flux density of the raster (also described as compact flux) and $f_g$ is the fraction of the total flux density that corresponds to the flux density of the Gaussian. We take the Fourier transform of $M(I_{ij}, f, f_g)$ to get model Fourier transform $\tilde{I}_{ab}$ for each baseline $ab$. The Fourier amplitudes are then corrupted to model the individual station gain amplitudes $|g|$ to construct the visibility amplitudes, $|\hat{V}_{ab}| = |g_a||g_b||\tilde{I}_{ab}|$

We form the visibility amplitude likelihood and closure phase likelihood as described in Appendix F of Sgr A* 2017 IV. Next, we form our prior distribution by using a uniform distribution $U(0.0, 1.5)$ Jy for the prior on the total flux density $f$ and $U(0.0, 1.0)$ for the fraction of the total flux density $f_g$ for the flux density of the large-scale Gaussian. For the raster pixels fluxes, $I_i$, we use a symmetric Dirichlet distribution

$$p(\mathbf{I}|\xi) = \frac{\Gamma(K\xi)}{\Gamma(\xi)^K} \prod_{i=1}^{K} I_i^{\xi_i-1}, \qquad (L.5)$$

where $\mathbf{I}$ is the flattened vector of pixel fluxes, $K$ is the total number of pixels in the image, $\Gamma$ is gamma distribution, and $\xi$ is the concentration parameter that controls the sparsity and smoothness of the image. The Dirichlet support is on the simplex, meaning that $\sum_i I_i = 1$ and $0 \le I_i \le 1$. For this work, we set $\xi = 1$, which is equivalent to a uniform distribution on the simplex.

From the analysis of the crossing baseline tracks and the EHT station flux density calibration parameters (Table A.1), for station gain priors, we used a Normal distribution for the log-gain amplitudes for each station. For the April 21 data, we imposed prior widths of 10% or all stations, that is a $N(0.0, 0.1)$ prior, except 30% and 100% prior widths for LMT and GLT, respectively. We used the same gain priors for the April 25 data except for PV which used a 100% prior width, since it was found that PV exhibited large amplitude fluctuations after UTC 02:00 due to poor weather as mentioned in Appendix D.
# Appendix M: Impact of compact flux density on image reconstruction

The non-imaging constraints on compact flux density permit a wide range of allowable values, and a comparison of the compact flux densities measured by the Bayesian methods suggests that the current data set cannot be used to constrain the compact flux on horizon scales. In this appendix, we test the robustness of the ring reconstructions from the RML and CLEAN methods against different assumptions of the compact flux density.

Firstly, we investigated the dependence of the assumed total compact flux density ($F_{cpct}$) on the estimated ring parameters of M87* from reconstructed images following the approach adopted in our previous EHT analysis (M87* 2017 IV). In this analysis, we only change $F_{cpct}$, while all other imaging parameters are kept at their fiducial values. We select acceptable images based on two criteria: (1) normalized $\chi^2_{CP}$, $\chi^2_{log CA} < 2$ (with a 10% systematic uncertainty for DIFMAP), and (2) a corresponding self-calibration solution with $0.9 \le \text{median}(1/|g_{SMT}|) \le 1.1$, where $g_{SMT}$ is the gain for the SMT station. Figure M.1 illustrates how the estimated ring parameters vary with the assumed total compact flux density. While the ring width for all pipelines and fractional central brightness for RML methods slightly

increase with $F_{cpct}$, the ring diameter and position angle values of the acceptable images do not exhibit significant dependence on $F_{cpct}$. Therefore, the ring diameter and position angle are robust against different assumptions on the compact total flux density.

Secondly, We tested if our DIFMAP imaging script could distinguish different morphologies for different values of the total compact flux density using synthetic data. We prepared a series of synthetic data with the same morphologies of four geometric models used for Top Set selection in the imaging survey, but with the assumed compact total flux density in range from 0.4 to 1.1 Jy. Here, we used the total flux density, which is the sum of the assumed compact total flux density and flux density of a larger extended jet modeled by three Gaussians, to be 1.1 Jy for all synthetic data. We conduct image reconstruction using the DIFMAP imaging script with the fiducial parameter sets, but adopting the compact flux densities to be the same as that of the assumed values of synthetic data. In Fig. M.2, we show the reconstructed images of geometric models with different values of the total compact flux density. It is clearly demonstrated that our DIFMAP imaging script can distinguish ring-like morphologies from disk or double source morphologies if we properly choose the expected total compact flux density.

<figure><img src="image_38.png" /></figure>

**Fig. M.1.** Measured diameter *d*, width *w*, position angle, and fractional central brightness *f*<sub>C</sub> of M87*, measured from image reconstructions assuming different total compact flux densities, *F*<sub>cpct</sub> for DIFMAP (orange), eht-imaging (blue), and SMILI (green) (see Sect. 5.3). All measurements are made using REx. All other imaging parameters were set to the fiducial parameters of the corresponding pipeline. DIFMAP values were measured after restoring with a 20 µas FWHM Gaussian beam. The solid lines indicate the measured value, and the shaded regions give the ±1σ uncertainty of the REx measurement. The darker colored regions correspond to values of *F*<sub>cpct</sub> that produces images that have (1) normalized *χ*<sup>2</sup><sub>CP</sub>, *χ*<sup>2</sup><sub>log CA</sub> < 2 (with 10% systematic uncertainty for DIFMAP), and (2) a corresponding self-calibration solution with 0.9 ≤ median(1/|g<sub>SMT</sub>|) ≤ 1.1.
<figure><img src="image_39.png" /><figcaption>**Fig. M.2.** Reconstructed DIFMAP images of the four geometric models with different values of the total compact flux density. We use the fiducial parameters from the 2018 Top Set, but with different assumed total compact flux densities. The top label above each column indicates the assumed total compact flux density in units of Jy.</figcaption></figure>
## Appendix N: Geometric modeling details

<figure><img src="image_40.png" /><figcaption>**Fig. N.1.** Schematic summary of the xs-ringauss and mF-ring model parameters. The xs-ringauss is defined to be a disk with a circular hole removed, that is allowed to be offset from the center of the disk and rotated. A circular Gaussian is pinned to the center of the disk as an emission floor, and an additional elliptical Gaussian is pinned to the inner edge of the offset hole. The mF-ring is an m-ring of order 2 that has an elliptical stretch which is allowed to be rotated. A flat emission floor is included to match the interior of the m-ring. Both models have an additional blur defined on them with a Gaussian blurring kernel. The intensity profiles of the fiducial models along two axes are indicated in red and green. An angular intensity profile is also shown for the mF-ring in blue.</figcaption></figure>

Image reconstructions of M87* show strong support for a ring-like structure. We quantify this support through the Bayesian evidence as a robust metric for the goodness of fit.

The Bayesian approach is reliant on making an appropriate estimate of the posterior distribution, $P(\Theta|M, D)$, for some parameters $\Theta$ of a chosen model, $M$, that is conditioned on some data, $D$. The posterior is defined by the relationship,

$$P(\Theta|M, D) = \frac{P(D|M, \Theta) P(\Theta|M)}{P(D|M)} \equiv \frac{\mathcal{L}(\Theta|M) \pi(\Theta|M)}{Z(M)}, \quad (N.1)$$

which also defines the likelihood of the data for a chosen model and parameter,

$$\mathcal{L}(\Theta|M) \equiv P(D|M, \Theta) \quad (N.2)$$

the prior probability on model parameters,

$$\pi(\Theta|M) = P(\Theta|M) \quad (N.3)$$

and the Bayesian evidence for a chosen model,

$$Z(M) \equiv P(D|M) = \int \mathcal{L}(\Theta|M) \pi(\Theta|M) d\Theta. \quad (N.4)$$

The Bayesian evidence allows the relative support between two models to be determined through a Bayes factor,

$$R(\mathbf{M}_1, \mathbf{M}_2) \equiv \frac{Z(\mathbf{M}_1) \pi(\mathbf{M}_2)}{Z(\mathbf{M}_1) \pi(\mathbf{M}_2)} \quad (N.5)$$

where $\pi(M)$ is the probability prior defined on the set of models. We use the Bayesian evidence with a constant prior to define the goodness of fit for a model $M \in \{M\}$ as,

$$\Delta \log(Z) = \log Z(M) - \text{maximum} [\log Z(\{M\})], \quad (N.6)$$

where the argument of $\Delta \log(Z)$ is the Bayes factor between $M$ and the best performing model in the set.

The best performing models are selected with this metric to construct fiducial models for feature comparison; namely, a stretched m-ring of order 2 with a flat emission floor and two elliptical Gaussians (mF-ring), and an xs-ringauss with two elliptical Gaussians an a large scale circular Gaussian. The schematic summary of defining parameters of these models is shown in Fig. N.1. See Sect. 6.2 for details on these models.

The shared geometric parameters of interest between both the mF-ring and the xs-ringauss model are their diameters, widths, central brightnesses, and brightness asymmetry position angles. The diameter of the mF-ring is defined to be the sum of the debiased semi-major and semi-minor axis, while that of the xs-ringauss is taken to be the sum of its inner and outer radii,

$$\hat{d} = \begin{cases} \hat{r}_0 + \hat{r}_1, & \text{mF-ring} \\ R_{\text{out}} + R_{\text{in}}, & \text{xs-ringauss.} \end{cases} \quad (N.7)$$

Here, the debiased radius ($\hat{r}_i$) is linked to the defining radius of the mF-ring ($r_i$), and the FWHM of the blurring Gaussian kernel ($\sigma^*$) through:

$$\hat{r}_i = r_i - \frac{\sigma^*}{16 \ln(2)} r_i. \quad (N.8)$$

We define the fractional widths of the mF-ring and the xs-ringauss respectively to be,

$$f_w = \begin{cases} \frac{\sigma^*}{d}, & \text{mF-ring} \\ \frac{R_{\text{out}} - R_{\text{in}} + \sigma^*}{d}, & \text{xs-ringauss.} \end{cases} \quad (N.9)$$
<table><caption>**Table N.1.** THEMIS xs-ringauss model parameters and priors.</caption><thead><tr><th>Parameter</th><th>Description</th><th>Units</th><th>Prior<sup>a</sup></th></tr></thead><tbody><tr><td>I<sub>0</sub> + I<sub>1</sub></td><td>Flux Density of Crescent and Fixed Gaussian</td><td>Jy</td><td>U(0, 2)</td></tr><tr><td>R<sub>out</sub></td><td>Outer Radius of Crescent</td><td>μas</td><td>U(0, 100)</td></tr><tr><td>φ<sub>X</sub></td><td>Position Angle</td><td>rad</td><td>U(-π, π)</td></tr><tr><td>ψ</td><td>Fractional Thickness</td><td>...</td><td>U(0.0001, 0.9999)</td></tr><tr><td>τ</td><td>Structural Asymmetry</td><td>...</td><td>U(0.0001, 0.9999)</td></tr><tr><td>f<sub>X</sub></td><td>Flux Asymmetry</td><td>...</td><td>U(0, 1)</td></tr><tr><td>σ<sup>*</sup></td><td>Width of Gaussian Smoothing Kernel</td><td>μas</td><td>U(0, 100)</td></tr><tr><td>I<sub>f</sub></td><td>Flux Density of Central Gaussian Emission Floor</td><td>Jy</td><td>U(0, 2)</td></tr><tr><td>σ<sub>f</sub></td><td>Width of Central Gaussian Emission Floor</td><td>μas</td><td>U(0, 25)</td></tr><tr><td>I<sub>G</sub></td><td>Flux Density of Large-Scale Gaussian</td><td>Jy</td><td>U(0, 10)</td></tr><tr><td>σ<sub>G</sub></td><td>Width of Large-Scale Gaussian</td><td>mas</td><td>L<sub>10</sub>(-2, 10)</td></tr><tr><td>I<sub>1</sub>/I<sub>0</sub></td><td>Ratio of Fixed Gaussian Flux Density to Crescent Flux Density</td><td>...</td><td>U(0, 1)</td></tr><tr><td>σ<sub>x</sub>/R<sub>out</sub></td><td>Ratio of Fixed Gaussian Width to Outer Radius of Crescent</td><td>...</td><td>U(0, 3)</td></tr><tr><td>σ<sub>y</sub>/σ<sub>x</sub></td><td>Axis Ratio of Fixed Gaussian</td><td>...</td><td>U(0, 100)</td></tr><tr><td>I<sub>g</sub></td><td>Flux Density of Additional Gaussian Component</td><td>Jy</td><td>U(0, 2)</td></tr><tr><td>x<sub>0</sub></td><td>Central x-coordinate of Additional Gaussian</td><td>μas</td><td>U(-200, 200)</td></tr><tr><td>y<sub>0</sub></td><td>Central y-coordinate of Additional Gaussian</td><td>μas</td><td>U(-200, 200)</td></tr><tr><td>σ<sub>g</sub></td><td>Width of Additional Gaussian</td><td>μas</td><td>U(0, 100)</td></tr><tr><td>A<sub>g</sub></td><td>Anisotropy of Additional Gaussian</td><td>...</td><td>U(0, 0.99)</td></tr><tr><td>φ<sub>g</sub></td><td>Position Angle of Additional Gaussian</td><td>rad</td><td>U(0, π)</td></tr></tbody></table>

<table><caption>**Table N.2.** mF-ring parameters and priors.</caption><thead><tr><th>Component</th><th>Parameter</th><th>Description</th><th>Units</th><th>Prior</th></tr></thead><tbody><tr><td rowspan="9">mF-ring</td><td>r<sub>1</sub></td><td>Semi-major axis length</td><td>μas</td><td>U(10, 30)</td></tr><tr><td>r<sub>0</sub>/r<sub>1</sub></td><td>Semi-major/Semi-major length ratio</td><td>...</td><td>U(0.1, 1)</td></tr><tr><td>φ<sub>m</sub></td><td>Semi-major axis orientation</td><td>rad</td><td>U(0, π)</td></tr><tr><td>f̂<sub>C</sub></td><td>Relative central brightness</td><td>...</td><td>U(0.0, 1.0)</td></tr><tr><td>A<sub>i</sub></td><td>Amplitude of the i<sup>th</sup> mode</td><td>...</td><td>U(0.0, 0.5)</td></tr><tr><td>β<sub>i</sub></td><td>Phase of the i<sup>th</sup> mode</td><td>...</td><td>U(0.0, 0.5)</td></tr><tr><td>σ<sup>*</sup> / (2√2 ln(2))</td><td>Blurring kernel standard deviation</td><td>μas</td><td>U(1.0, 50)</td></tr><tr><td>f</td><td>mF-ring/(all nuisance Gaussians) flux ratio</td><td>...</td><td>U(0, 1)</td></tr><tr><td rowspan="10">Nuisance Gaussians</td><td>f<sub>g</sub></td><td>Primary/secondary Gaussian flux ratio</td><td>...</td><td>U(0.51, 1)</td></tr><tr><td>σ<sub>gi</sub></td><td>Standard deviation of i<sup>th</sup> nuisance Gaussian</td><td>μas</td><td>U(2, 40)</td></tr><tr><td>τ<sub>i</sub></td><td>Semi-major/semi-minor length ratio of i<sup>th</sup> nuisance Gaussian</td><td>...</td><td>U(0.1, 1)</td></tr><tr><td>ξ<sub>gi</sub></td><td>Stretch angle of i<sup>th</sup> nuisance Gaussian</td><td>rad</td><td>U(0, π)</td></tr><tr><td>x<sub>1</sub></td><td>Horizontal displacement of primary Gaussian from m-ring center</td><td>μas</td><td>U(-50, 50)</td></tr><tr><td>x<sub>2</sub></td><td>Horizontal displacement of secondary Gaussian from primary center</td><td>μas</td><td>U(-50, 50)</td></tr><tr><td>y<sub>1</sub></td><td>Vertical displacement of primary Gaussian from m-ring center</td><td>μas</td><td>U(-50, 50)</td></tr><tr><td>y<sub>2</sub></td><td>Vertical displacement of secondary Gaussian from primary center</td><td>μas</td><td>U(-50, 50)</td></tr></tbody></table>

and the brightness asymmetry position angle to be,

$$ \hat{\phi} = \begin{cases} \beta_1 + \phi_m, & \text{mF-ring} \\ \phi_X, & \text{xs-ringauss.} \end{cases} $$

(N.10)

which is given by,

$$ \hat{f}_C = \begin{cases} \text{brightness of emission floor} \\ \text{mean brightness of m-ring}, \end{cases}, \quad \text{mF-ring} \\ \begin{cases} \text{brightness of central Gaussian} \\ \text{mean brightness of crescent}, \end{cases}, \quad \text{xs-ringauss.} \quad (N.11) $$

The last shared parameter shown between the two models is the fractional central flux depression, or relative central brightness,

As described above in the main text, the THEMIS xs-ringauss model used to fit the data from the 2018 observations

<footnote><sup>a</sup> Logarithmic priors in base x spanning a range x<sup>a</sup> to x<sup>b</sup> are represented by $\mathcal{L}_x(a, b)$, and uniform (linear) priors spanning a to b are represented by $\mathcal{U}(a, b)$.</footnote>
<figure><img src="image_41.png" /><figcaption>**Fig. N.2.** Representative images for each observing band from random posterior samples for the Hybrid Themage, xs-ringauss, and mF-ring models, for the April 21 and April 25 HOPS data.</figcaption></figure>

is the same model used to fit the data from the 2017 observations. We list the model parameters and their priors in Tables N.1 and N.2.

Even though we construct and fit the Hybrid Themage, xs-ringauss, and mF-ring in the visibility domain, we can also produce conventional image domain representations of the models. Figure N.2 shows the on-sky representations of the Hybrid Themage, xs-ringauss, and mF-ring models. We find that these models reproduce many of the same features seen in the more agnostic models shown in Sect. 5, such as a consistent diam-

eter and position angle of the brightest part of the ring in the southwest. The xs-ringauss and mF-ring models use a pair of asymmetric Gaussians as nuisance components, designed to help fit the residual data that cannot be captured by the ring component. The raster component in the Hybrid Themage serves a similar purpose. We note that for April 21 band 3 and band 4, the non-ring components settle in the west and southwest part of the image, but these components occupy more varied locations in the band 1, band 2, and April 25 reconstructions.