From 1b84a940ab3834602028a2afdb9c113caa3afa34 Mon Sep 17 00:00:00 2001 From: Claude Date: Sun, 18 Oct 2026 12:58:51 +0000 Subject: [PATCH 1/3] Lensing note: SDP.81's second lump, and the optional fixes from #30's re-review - SDP.81 had two reported dark lumps, Hezaveh et al. 2016 and Inoue et al. 2016 (MNRAS 457, 2936), and the note named only one. Stacey et al. find no evidence for either and conclude "these previous results are not robust": Hezaveh et al.'s lump most likely spurious (poorly fitted multipoles), Inoue et al.'s explained by limits of the earlier method and the lens's angular structure. Checked in Stacey et al.'s LaTeX (lines 421 and 452). The summary, the table row and section 5 now say so, and Inoue et al. joins the Sources. - The PJ+PM point mass is 1.9e5 Msun in both places (was "2e5" once). - "which the data disfavour" for Delta ln E = -30, closer to Vegetti et al.'s own "significantly disfavored" than "fits less well". - Section 4: central images "bound" lens-galaxy black holes; they gave one-sided bounds, not masses (section 5 already said so). Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01XfgH6WcwTueUFZKb6fS2s2 --- notes/bh-lensing.md | 47 ++++++++++++++++++++++++--------------------- 1 file changed, 25 insertions(+), 22 deletions(-) diff --git a/notes/bh-lensing.md b/notes/bh-lensing.md index 6c0fb5a..e9fa97e 100644 --- a/notes/bh-lensing.md +++ b/notes/bh-lensing.md @@ -21,9 +21,9 @@ the opposite question: can a galaxy's central black hole shape its wave dark mat central image can. - **So the test changes shape.** As of 18 October 2026 I found no study that tests lensing anomalies against the lens galaxy's black hole mass or spin (section 4). No lens I found has both a small-scale anomaly and a bound on - its black hole. SDP.81 had both until its dark lump was found most likely spurious (Stacey et al. 2025). Masses - read off the M–σ relation would only restate the galaxy's velocity dispersion, which also sets the size of the - wave granules. Section 6 gives testable versions. + its black hole. SDP.81 had both until Stacey et al. (2025) found its two reported dark lumps not robust (section + 5). Masses read off the M–σ relation would only restate the galaxy's velocity dispersion, which also sets the + size of the wave granules. Section 6 gives testable versions. ## 1. The gravitational atom, step by step @@ -158,16 +158,16 @@ source quasar's, not the lens's: microlensing light curves give accretion disc s extended mass of almost constant surface density out to 139 pc, an edge they flag as possibly an artefact of their model. Oguri and Kubo take the point mass to be the black hole. Other readings need no wave dark matter. Vegetti et al. note that, if the lump is dark matter, its structure is incompatible with cold dark matter but may - fit a self-interacting halo whose centre has collapsed into a black hole. Zhang and Yu (2026) match instead - Vegetti et al.'s pseudo-Jaffe plus point mass model, which fits less well (Δln E = −30 against the best fit) and - has a 1.9×10⁵ M☉ point mass: with a self-interacting halo deep in collapse, or with the tidally stripped remnant - of a cold dark matter halo hosting a black hole of that mass, the halo having lost about 5 orders of magnitude of - its mass. The two black hole masses, 4×10⁵ and 2×10⁵ M☉, come from different fits to the same data. In Zeng et - al. (2026, a preprint) a black hole seeded by the collapse grows to 5 to 12% of the halo's mass within 2 Gyr and - depletes the inner halo, which matches the lump qualitatively. + fit a self-interacting halo whose centre has collapsed into a black hole. Zhang and Yu (2026) instead match + Vegetti et al.'s pseudo-Jaffe plus point mass model, which the data disfavour (Δln E = −30 against the best fit) + and has a 1.9×10⁵ M☉ point mass: with a self-interacting halo deep in collapse, or with the tidally stripped + remnant of a cold dark matter halo hosting a black hole of that mass, the halo having lost about 5 orders of + magnitude of its mass. The two black hole masses, 4×10⁵ and 1.9×10⁵ M☉, come from different fits to the same + data. In Zeng et al. (2026, a preprint) a black hole seeded by the collapse grows to 5 to 12% of the halo's mass + within 2 Gyr and depletes the inner halo, which matches the lump qualitatively. - **Davies and Mocz (2020)** and **Bar et al. (2019)**: how a black hole squeezes, and may swallow, the core. Bar et al. use Sgr A* and M87* to bound the core's mass. -- **Central images** weigh lens-galaxy black holes directly: below 2×10⁸ M☉ in PMN J1632−0033 (Winn, Rusin and +- **Central images** bound lens-galaxy black holes directly: below 2×10⁸ M☉ in PMN J1632−0033 (Winn, Rusin and Kochanek 2004), above 3×10⁸ M☉ in SDP.81 (Tamura et al. 2015; Wong et al. 2015 find log M ≳ 8.5). The SDP.81 bound trades off against the assumed size of the galaxy's stellar core (Tamura et al. 2015). - **Two wave dark matter models of the main lens** mention its central black hole, in passing. Chan et al. (2020) @@ -197,23 +197,25 @@ source quasar's, not the lens's: microlensing light curves give accretion disc s | System | Anomaly measurement | Wave dark matter analysis | Lens galaxy's black hole | |---|---|---|---| | HS 0810+2554 | Radio jets with VLBI (Hartley et al. 2019); narrow-line fluxes (Nierenberg et al. 2020) | Amruth et al. 2023 and Zhou et al. 2026: reproduced at ~10⁻²² eV; cold dark matter also fits (section 4) | Not measured. M–σ estimate ~6×10⁸ M☉ (section 3) | -| SDP.81 | A 10^(8.96±0.12) M☉ dark lump at 6.9σ (Hezaveh et al. 2016), most likely spurious: poorly fitted multipoles (Stacey et al. 2025) | None found | Above 3×10⁸ M☉ from the missing central image (Tamura et al. 2015; Wong et al. 2015) | +| SDP.81 | A 10^(8.96±0.12) M☉ dark lump at 6.9σ (Hezaveh et al. 2016), most likely spurious: poorly fitted multipoles; a second reported lump (Inoue et al. 2016) is not robust either (Stacey et al. 2025) | None found | Above 3×10⁸ M☉ from the missing central image (Tamura et al. 2015; Wong et al. 2015) | | PMN J1632−0033 | None found | None found | Below 2×10⁸ M☉ from the central image (Winn, Rusin and Kochanek 2004) | | JVAS B1938+666 | A 10⁶ M☉ dark lump at 26σ (Powell et al. 2025) | Oguri and Kubo 2026: soliton plus a 4×10⁵ M☉ black hole inside the lump; self-interacting and cold dark matter readings too (section 4) | Not measured | | MG J0751+2716 | Extended arc at milliarcsecond resolution | Powell et al. 2023: up to 4.4×10⁻²¹ eV ruled out | Not measured | | B1422+231, HE0435−1223, WGD J0405−3308, RX J0911+0551, PS J1606−2333, WFI 2026−4536, WFI 2033−4723, WGD 2038−4008 | Narrow-line fluxes, as compiled by Gilman et al. 2020 | Laroche et al. 2022 bound the particle mass with an 11-quad sample of this kind | Not measured | | 17 cusp quads, e.g. J1042+1641 | Microlensing-free fluxes, many from JWST | Hou et al. 2026: FDM preferred | Not measured | -No system I found has both a small-scale anomaly and a bound on its black hole. SDP.81 had both until its dark lump -was found most likely spurious (Stacey et al. 2025). "Not measured" means I found no direct measurement, and none -is expected from stars alone: at these redshifts a black hole's sphere of influence (18 to 63 pc) spans 2 to 14 -milliarcseconds, well inside the 36 milliarcsecond core of the Hubble image blur that a stellar-dynamical mass must -resolve (Kormendy and Ho 2013). Lensing does better. Central images have so far given bounds (the table above); -lensed images near the centre have measured two black holes, of (3.3 ± 2.1)×10¹⁰ M☉ (3σ range) in Abell 1201 at z = -0.17 (Nightingale et al. 2023; one model without a black hole fits comparably, which the authors argue is -unphysical) and 3.6×10¹⁰ M☉ in the Cosmic Horseshoe at z = 0.44, with stellar kinematics (Melo-Carneiro et al. -2025): both about 10¹⁰ M☉, ten times or more heavier than those in section 1. Neither is among the anomaly lenses -above. I found no spin measurement for any lens galaxy's black hole. +No system I found has both a small-scale anomaly and a bound on its black hole. SDP.81 had both until Stacey et al. +(2025) found its two reported dark lumps (Hezaveh et al. 2016; Inoue et al. 2016) not robust: Hezaveh et al.'s most +likely spurious, from poorly fitted multipoles, and Inoue et al.'s explained by limits of the earlier method and +the lens's angular structure. "Not measured" means I found no direct measurement, and none is expected from stars +alone: at these redshifts a black hole's sphere of influence (18 to 63 pc) spans 2 to 14 milliarcseconds, well +inside the 36 milliarcsecond core of the Hubble image blur that a stellar-dynamical mass must resolve (Kormendy and +Ho 2013). Lensing does better. Central images have so far given bounds (the table above); lensed images near the +centre have measured two black holes, of (3.3 ± 2.1)×10¹⁰ M☉ (3σ range) in Abell 1201 at z = 0.17 (Nightingale et +al. 2023; one model without a black hole fits comparably, which the authors argue is unphysical) and 3.6×10¹⁰ M☉ in +the Cosmic Horseshoe at z = 0.44, with stellar kinematics (Melo-Carneiro et al. 2025): both about 10¹⁰ M☉, ten +times or more heavier than those in section 1. Neither is among the anomaly lenses above. I found no spin +measurement for any lens galaxy's black hole. ## 6. How the question changes shape @@ -275,6 +277,7 @@ at 10⁻²² eV. - P. Hartley, N. Jackson, D. Sluse, H. R. Stacey and H. Vives-Arias, *Strong lensing reveals jets in a sub-microJy radio-quiet quasar*, MNRAS 485, 3009 (2019). https://doi.org/10.1093/mnras/stz510 (https://arxiv.org/abs/1901.05791) - Y. D. Hezaveh et al., *Detection of lensing substructure using ALMA observations of the dusty galaxy SDP.81*, ApJ 823, 37 (2016). https://doi.org/10.3847/0004-637X/823/1/37 (https://arxiv.org/abs/1601.01388) - S. Hou et al., *Flux-ratio anomalies in cusp quasars reveal dark matter beyond CDM* (2026). https://arxiv.org/abs/2601.16818 +- K. T. Inoue, T. Minezaki, S. Matsushita and M. Chiba, *ALMA imprint of intergalactic dark structures in the gravitational lens SDP.81*, MNRAS 457, 2936 (2016). https://doi.org/10.1093/mnras/stw168 (https://arxiv.org/abs/1510.00150) - J. Kormendy and L. C. Ho, *Coevolution (or not) of supermassive black holes and host galaxies*, ARA&A 51, 511 (2013). https://doi.org/10.1146/annurev-astro-082708-101811 (https://arxiv.org/abs/1304.7762) - A. Laroche, D. Gilman, X. Li, J. Bovy and X. Du, *Quantum fluctuations masquerade as haloes: bounds on ultra-light dark matter from quadruply imaged quasars*, MNRAS 517, 1867 (2022). https://doi.org/10.1093/mnras/stac2677 (https://arxiv.org/abs/2206.11269) - K. Mehrgan et al., *A 40 billion solar-mass black hole in the extreme core of Holm 15A, the central galaxy of Abell 85*, ApJ 887, 195 (2019). https://doi.org/10.3847/1538-4357/ab5856 (https://arxiv.org/abs/1907.10608) From 508e41d510d4029feb740cdf9fcc9e06a754fa23 Mon Sep 17 00:00:00 2001 From: Claude Date: Sun, 18 Oct 2026 13:07:38 +0000 Subject: [PATCH 2/3] Lensing note: Inoue et al.'s anomaly as they report it; #31 review fixes From #31's independent review (non-blocking, all upheld, all in text this PR added): - F2: Inoue et al. 2016 report a 2-3 sigma flux-ratio and astrometric anomaly that "can be explained by intergalactic dark structures in the line of sight", and argue against a sub-halo in the lens galaxy (their LaTeX, checked). Calling it SDP.81's "second reported dark lump" put it beside Hezaveh et al.'s 6.9 sigma lens sub-halo. The summary, table row and section 5 now speak of SDP.81's two reported anomalies and describe Inoue et al.'s as a 2-3 sigma line-of-sight clump. - F3: Stacey et al. say Inoue et al.'s anomaly "can be explained" by the earlier method's limits and the lens's angular structure; the note no longer states it as settled. - F1: section 4 names all three models of SDP.81's anomalies. Inoue et al.'s model (SIE, shear and a compensated clump) has no black hole mass either, so the negative claim holds. - F4: the Zhang and Yu sentence is grammatical again. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01XfgH6WcwTueUFZKb6fS2s2 --- notes/bh-lensing.md | 46 ++++++++++++++++++++++----------------------- 1 file changed, 23 insertions(+), 23 deletions(-) diff --git a/notes/bh-lensing.md b/notes/bh-lensing.md index e9fa97e..3026c18 100644 --- a/notes/bh-lensing.md +++ b/notes/bh-lensing.md @@ -21,9 +21,9 @@ the opposite question: can a galaxy's central black hole shape its wave dark mat central image can. - **So the test changes shape.** As of 18 October 2026 I found no study that tests lensing anomalies against the lens galaxy's black hole mass or spin (section 4). No lens I found has both a small-scale anomaly and a bound on - its black hole. SDP.81 had both until Stacey et al. (2025) found its two reported dark lumps not robust (section - 5). Masses read off the M–σ relation would only restate the galaxy's velocity dispersion, which also sets the - size of the wave granules. Section 6 gives testable versions. + its black hole. SDP.81 had both until Stacey et al. (2025) found both reported anomalies not robust (section 5). + Masses read off the M–σ relation would only restate the galaxy's velocity dispersion, which also sets the size of + the wave granules. Section 6 gives testable versions. ## 1. The gravitational atom, step by step @@ -144,12 +144,12 @@ I found no such study. A test here means a study that fits a small-scale anomaly astrometric anomaly, or a detected dark lump) with a model that includes the lens galaxy's own black hole, or compares such anomalies with that black hole's mass or spin. Models that use the black hole only in the smooth macro-model or for a central image's flux, as Tamura et al. (2015) and Wong et al. (2015) do for SDP.81, do not -count; the models of SDP.81's lump (Hezaveh et al. 2016; Stacey et al. 2025) do not include its black hole. The -search used the arXiv API, INSPIRE full-text search, citation chasing from this note's sources and web searches; -ADS was not queried. Microlensing and X-ray work on lensed quasars does involve black hole mass and spin, but the -source quasar's, not the lens's: microlensing light curves give accretion disc sizes that scale as M_BH^(0.80±0.17) -(Morgan et al. 2010), and X-ray spectra brightened by lensing give the spin of the quasar in RX J1131−1231, a = -0.87 (Reis et al. 2014). The closest work: +count; the models of SDP.81's anomalies (Hezaveh et al. 2016; Inoue et al. 2016; Stacey et al. 2025) do not include +its black hole. The search used the arXiv API, INSPIRE full-text search, citation chasing from this note's sources +and web searches; ADS was not queried. Microlensing and X-ray work on lensed quasars does involve black hole mass +and spin, but the source quasar's, not the lens's: microlensing light curves give accretion disc sizes that scale +as M_BH^(0.80±0.17) (Morgan et al. 2010), and X-ray spectra brightened by lensing give the spin of the quasar in RX +J1131−1231, a = 0.87 (Reis et al. 2014). The closest work: - **Oguri and Kubo (2026)**: a soliton whose centre is deepened by a black hole makes a dense, hard-to-strip lump. The 10⁶ M☉ dark lump in JVAS B1938+666 fits a particle of about 4×10⁻²¹ eV with a 4×10⁵ M☉ black hole. That black @@ -159,8 +159,8 @@ source quasar's, not the lens's: microlensing light curves give accretion disc s their model. Oguri and Kubo take the point mass to be the black hole. Other readings need no wave dark matter. Vegetti et al. note that, if the lump is dark matter, its structure is incompatible with cold dark matter but may fit a self-interacting halo whose centre has collapsed into a black hole. Zhang and Yu (2026) instead match - Vegetti et al.'s pseudo-Jaffe plus point mass model, which the data disfavour (Δln E = −30 against the best fit) - and has a 1.9×10⁵ M☉ point mass: with a self-interacting halo deep in collapse, or with the tidally stripped + Vegetti et al.'s pseudo-Jaffe plus point mass model, a model the data disfavour (Δln E = −30 against the best + fit), with a 1.9×10⁵ M☉ point mass: with a self-interacting halo deep in collapse, or with the tidally stripped remnant of a cold dark matter halo hosting a black hole of that mass, the halo having lost about 5 orders of magnitude of its mass. The two black hole masses, 4×10⁵ and 1.9×10⁵ M☉, come from different fits to the same data. In Zeng et al. (2026, a preprint) a black hole seeded by the collapse grows to 5 to 12% of the halo's mass @@ -197,7 +197,7 @@ source quasar's, not the lens's: microlensing light curves give accretion disc s | System | Anomaly measurement | Wave dark matter analysis | Lens galaxy's black hole | |---|---|---|---| | HS 0810+2554 | Radio jets with VLBI (Hartley et al. 2019); narrow-line fluxes (Nierenberg et al. 2020) | Amruth et al. 2023 and Zhou et al. 2026: reproduced at ~10⁻²² eV; cold dark matter also fits (section 4) | Not measured. M–σ estimate ~6×10⁸ M☉ (section 3) | -| SDP.81 | A 10^(8.96±0.12) M☉ dark lump at 6.9σ (Hezaveh et al. 2016), most likely spurious: poorly fitted multipoles; a second reported lump (Inoue et al. 2016) is not robust either (Stacey et al. 2025) | None found | Above 3×10⁸ M☉ from the missing central image (Tamura et al. 2015; Wong et al. 2015) | +| SDP.81 | A 10^(8.96±0.12) M☉ dark lump at 6.9σ (Hezaveh et al. 2016), most likely spurious: poorly fitted multipoles; a line-of-sight dark clump suggested at 2–3σ (Inoue et al. 2016) is not robust either (Stacey et al. 2025) | None found | Above 3×10⁸ M☉ from the missing central image (Tamura et al. 2015; Wong et al. 2015) | | PMN J1632−0033 | None found | None found | Below 2×10⁸ M☉ from the central image (Winn, Rusin and Kochanek 2004) | | JVAS B1938+666 | A 10⁶ M☉ dark lump at 26σ (Powell et al. 2025) | Oguri and Kubo 2026: soliton plus a 4×10⁵ M☉ black hole inside the lump; self-interacting and cold dark matter readings too (section 4) | Not measured | | MG J0751+2716 | Extended arc at milliarcsecond resolution | Powell et al. 2023: up to 4.4×10⁻²¹ eV ruled out | Not measured | @@ -205,17 +205,17 @@ source quasar's, not the lens's: microlensing light curves give accretion disc s | 17 cusp quads, e.g. J1042+1641 | Microlensing-free fluxes, many from JWST | Hou et al. 2026: FDM preferred | Not measured | No system I found has both a small-scale anomaly and a bound on its black hole. SDP.81 had both until Stacey et al. -(2025) found its two reported dark lumps (Hezaveh et al. 2016; Inoue et al. 2016) not robust: Hezaveh et al.'s most -likely spurious, from poorly fitted multipoles, and Inoue et al.'s explained by limits of the earlier method and -the lens's angular structure. "Not measured" means I found no direct measurement, and none is expected from stars -alone: at these redshifts a black hole's sphere of influence (18 to 63 pc) spans 2 to 14 milliarcseconds, well -inside the 36 milliarcsecond core of the Hubble image blur that a stellar-dynamical mass must resolve (Kormendy and -Ho 2013). Lensing does better. Central images have so far given bounds (the table above); lensed images near the -centre have measured two black holes, of (3.3 ± 2.1)×10¹⁰ M☉ (3σ range) in Abell 1201 at z = 0.17 (Nightingale et -al. 2023; one model without a black hole fits comparably, which the authors argue is unphysical) and 3.6×10¹⁰ M☉ in -the Cosmic Horseshoe at z = 0.44, with stellar kinematics (Melo-Carneiro et al. 2025): both about 10¹⁰ M☉, ten -times or more heavier than those in section 1. Neither is among the anomaly lenses above. I found no spin -measurement for any lens galaxy's black hole. +(2025) found both reported anomalies not robust. Hezaveh et al.'s (2016) dark lump in the lens was most likely +spurious, from poorly fitted multipoles; Inoue et al.'s (2016) 2–3σ line-of-sight dark clump can be explained by +limits of the earlier method and the lens's angular structure. "Not measured" means I found no direct measurement, +and none is expected from stars alone: at these redshifts a black hole's sphere of influence (18 to 63 pc) spans 2 +to 14 milliarcseconds, well inside the 36 milliarcsecond core of the Hubble image blur that a stellar-dynamical +mass must resolve (Kormendy and Ho 2013). Lensing does better. Central images have so far given bounds (the table +above); lensed images near the centre have measured two black holes, of (3.3 ± 2.1)×10¹⁰ M☉ (3σ range) in Abell +1201 at z = 0.17 (Nightingale et al. 2023; one model without a black hole fits comparably, which the authors argue +is unphysical) and 3.6×10¹⁰ M☉ in the Cosmic Horseshoe at z = 0.44, with stellar kinematics (Melo-Carneiro et al. +2025): both about 10¹⁰ M☉, ten times or more heavier than those in section 1. Neither is among the anomaly lenses +above. I found no spin measurement for any lens galaxy's black hole. ## 6. How the question changes shape From 023c60dc090f431baa3d32671864c28c1583dae6 Mon Sep 17 00:00:00 2001 From: Claude Date: Sun, 18 Oct 2026 13:12:45 +0000 Subject: [PATCH 3/3] Lensing note: three wording fixes from #31's re-review MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - "its two reported anomalies" where "both" had meant two different things in one sentence. - "which the data disfavour ... and which has", so the later "with" phrases attach to "match", not to the point mass. - "2 to 3σ", the repo's style for ranges. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01XfgH6WcwTueUFZKb6fS2s2 --- notes/bh-lensing.md | 24 ++++++++++++------------ 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/notes/bh-lensing.md b/notes/bh-lensing.md index 3026c18..b2b5680 100644 --- a/notes/bh-lensing.md +++ b/notes/bh-lensing.md @@ -21,9 +21,9 @@ the opposite question: can a galaxy's central black hole shape its wave dark mat central image can. - **So the test changes shape.** As of 18 October 2026 I found no study that tests lensing anomalies against the lens galaxy's black hole mass or spin (section 4). No lens I found has both a small-scale anomaly and a bound on - its black hole. SDP.81 had both until Stacey et al. (2025) found both reported anomalies not robust (section 5). - Masses read off the M–σ relation would only restate the galaxy's velocity dispersion, which also sets the size of - the wave granules. Section 6 gives testable versions. + its black hole. SDP.81 had both until Stacey et al. (2025) found its two reported anomalies not robust (section + 5). Masses read off the M–σ relation would only restate the galaxy's velocity dispersion, which also sets the + size of the wave granules. Section 6 gives testable versions. ## 1. The gravitational atom, step by step @@ -159,12 +159,12 @@ J1131−1231, a = 0.87 (Reis et al. 2014). The closest work: their model. Oguri and Kubo take the point mass to be the black hole. Other readings need no wave dark matter. Vegetti et al. note that, if the lump is dark matter, its structure is incompatible with cold dark matter but may fit a self-interacting halo whose centre has collapsed into a black hole. Zhang and Yu (2026) instead match - Vegetti et al.'s pseudo-Jaffe plus point mass model, a model the data disfavour (Δln E = −30 against the best - fit), with a 1.9×10⁵ M☉ point mass: with a self-interacting halo deep in collapse, or with the tidally stripped - remnant of a cold dark matter halo hosting a black hole of that mass, the halo having lost about 5 orders of - magnitude of its mass. The two black hole masses, 4×10⁵ and 1.9×10⁵ M☉, come from different fits to the same - data. In Zeng et al. (2026, a preprint) a black hole seeded by the collapse grows to 5 to 12% of the halo's mass - within 2 Gyr and depletes the inner halo, which matches the lump qualitatively. + Vegetti et al.'s pseudo-Jaffe plus point mass model, which the data disfavour (Δln E = −30 against the best fit) + and which has a 1.9×10⁵ M☉ point mass: with a self-interacting halo deep in collapse, or with the tidally + stripped remnant of a cold dark matter halo hosting a black hole of that mass, the halo having lost about 5 + orders of magnitude of its mass. The two black hole masses, 4×10⁵ and 1.9×10⁵ M☉, come from different fits to the + same data. In Zeng et al. (2026, a preprint) a black hole seeded by the collapse grows to 5 to 12% of the halo's + mass within 2 Gyr and depletes the inner halo, which matches the lump qualitatively. - **Davies and Mocz (2020)** and **Bar et al. (2019)**: how a black hole squeezes, and may swallow, the core. Bar et al. use Sgr A* and M87* to bound the core's mass. - **Central images** bound lens-galaxy black holes directly: below 2×10⁸ M☉ in PMN J1632−0033 (Winn, Rusin and @@ -197,7 +197,7 @@ J1131−1231, a = 0.87 (Reis et al. 2014). The closest work: | System | Anomaly measurement | Wave dark matter analysis | Lens galaxy's black hole | |---|---|---|---| | HS 0810+2554 | Radio jets with VLBI (Hartley et al. 2019); narrow-line fluxes (Nierenberg et al. 2020) | Amruth et al. 2023 and Zhou et al. 2026: reproduced at ~10⁻²² eV; cold dark matter also fits (section 4) | Not measured. M–σ estimate ~6×10⁸ M☉ (section 3) | -| SDP.81 | A 10^(8.96±0.12) M☉ dark lump at 6.9σ (Hezaveh et al. 2016), most likely spurious: poorly fitted multipoles; a line-of-sight dark clump suggested at 2–3σ (Inoue et al. 2016) is not robust either (Stacey et al. 2025) | None found | Above 3×10⁸ M☉ from the missing central image (Tamura et al. 2015; Wong et al. 2015) | +| SDP.81 | A 10^(8.96±0.12) M☉ dark lump at 6.9σ (Hezaveh et al. 2016), most likely spurious: poorly fitted multipoles; a line-of-sight dark clump suggested at 2 to 3σ (Inoue et al. 2016) is not robust either (Stacey et al. 2025) | None found | Above 3×10⁸ M☉ from the missing central image (Tamura et al. 2015; Wong et al. 2015) | | PMN J1632−0033 | None found | None found | Below 2×10⁸ M☉ from the central image (Winn, Rusin and Kochanek 2004) | | JVAS B1938+666 | A 10⁶ M☉ dark lump at 26σ (Powell et al. 2025) | Oguri and Kubo 2026: soliton plus a 4×10⁵ M☉ black hole inside the lump; self-interacting and cold dark matter readings too (section 4) | Not measured | | MG J0751+2716 | Extended arc at milliarcsecond resolution | Powell et al. 2023: up to 4.4×10⁻²¹ eV ruled out | Not measured | @@ -205,8 +205,8 @@ J1131−1231, a = 0.87 (Reis et al. 2014). The closest work: | 17 cusp quads, e.g. J1042+1641 | Microlensing-free fluxes, many from JWST | Hou et al. 2026: FDM preferred | Not measured | No system I found has both a small-scale anomaly and a bound on its black hole. SDP.81 had both until Stacey et al. -(2025) found both reported anomalies not robust. Hezaveh et al.'s (2016) dark lump in the lens was most likely -spurious, from poorly fitted multipoles; Inoue et al.'s (2016) 2–3σ line-of-sight dark clump can be explained by +(2025) found its two reported anomalies not robust. Hezaveh et al.'s (2016) dark lump in the lens was most likely +spurious, from poorly fitted multipoles; Inoue et al.'s (2016) 2 to 3σ line-of-sight dark clump can be explained by limits of the earlier method and the lens's angular structure. "Not measured" means I found no direct measurement, and none is expected from stars alone: at these redshifts a black hole's sphere of influence (18 to 63 pc) spans 2 to 14 milliarcseconds, well inside the 36 milliarcsecond core of the Hubble image blur that a stellar-dynamical