Recognition: 2 theorem links
· Lean TheoremLocating the missing large-scale emission in the jet of M87* with short EHT baselines
Pith reviewed 2026-05-16 12:42 UTC · model grok-4.3
The pith
Closure phases from nearly co-located stations measure the centroid of large-scale jet emission in M87* relative to the compact ring.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
A Taylor expansion of closure phases about co-located stations isolates the non-zero contributions from large-scale diffuse emission when the source also contains a bright compact component. This isolates the centroid offset and higher moments with few model assumptions. When applied to 2017 and 2018 EHT data of M87*, it yields a weak preference for extended emission along the jet direction; the 2021 data place the centroid about 1 mas northwest of the compact ring, consistent with the jet observed at lower frequencies.
What carries the argument
Taylor expansion of closure phases about co-located stations, which converts the otherwise trivial zero phases into direct measurements of the source centroid and image moments induced by large-scale structure.
If this is right
- The technique locates missing large-scale emission in VLBI data without requiring full imaging reconstructions.
- For M87*, the detected offset aligns with the known jet direction at lower frequencies.
- Application to 2021 EHT data specifically measures a ~1 mas northwest shift of the centroid.
- The same expansion can flag data issues or reveal structure in other sources observed with short baselines.
Where Pith is reading between the lines
- The method could be applied to other EHT targets to search for jet bases or extended emission near the event horizon.
- If the offset traces the jet launch region, it would tighten constraints on models connecting the compact ring to the large-scale jet.
- Arrays with denser short-baseline coverage might allow measurement of higher-order moments beyond the centroid.
Load-bearing premise
The source consists of a bright compact component plus a large-scale diffuse component, and the Taylor expansion accurately captures the non-zero closure phases produced by that structure.
What would settle it
A high-fidelity image or simulation at EHT frequencies in which the measured 1 mas northwest centroid offset vanishes once the large-scale jet component is removed from the model.
read the original abstract
In Very-Long Baseline Interferometric arrays, nearly co-located stations probe the largest scales and typically cannot resolve the observed source. In the absence of large-scale structure, closure phases constructed with these stations are zero and, since they are independent of station-based errors, they can be used to probe data issues. Here, we show with an expansion about co-located stations, how these trivial closure phases become non-zero with brightness distribution on smaller scales than their short baseline would suggest. When applied to sources that are made up of a bright compact and large-scale diffuse component, the trivial closure phases directly measure the centroid relative to the compact source and higher-order image moments. We present a technique to measure these image moments with minimal model assumptions and validate it on synthetic Event Horizon Telescope (EHT) data. We then apply this technique to 2017 and 2018 EHT observations of M87* and find a weak preference for extended emission in the direction of the large-scale jet. We also apply it to 2021 EHT data and measure the source centroid about 1 mas northwest of the compact ring, consistent with the jet observed at lower frequencies.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a first-order Taylor expansion of closure phases for nearly co-located VLBI stations to show that non-zero phases arise from brightness structure on scales smaller than the baseline length. For sources consisting of a bright compact component plus large-scale diffuse emission, the expansion isolates the centroid offset of the diffuse component relative to the compact core (plus higher moments). The method is validated on synthetic EHT data and applied to 2017/2018 M87* observations (weak preference for jet-aligned extended emission) and 2021 data (measured centroid ~1 mas northwest of the ring, consistent with lower-frequency jet).
Significance. If the expansion and validation hold, the technique supplies a low-assumption route to locate missing large-scale flux using only short-baseline closure phases that are immune to station-based errors. This would be useful for EHT analyses of M87* and similar sources where full imaging of extended jet emission remains challenging.
major comments (3)
- [§2] §2 (Taylor expansion derivation): the first-order truncation is presented as sufficient to isolate the centroid shift, but the manuscript does not quantify the size of second-order terms (proportional to baseline separation squared times image second moments) for the actual 2021 EHT array geometry and a jet model whose extent reaches the claimed 1 mas scale.
- [Synthetic validation section] Synthetic validation section: the tests use generic EHT-like arrays but do not reproduce the precise 2021 station positions, baseline lengths, and reported thermal noise levels against a diffuse component whose centroid offset is 1 mas; this leaves open whether the recovered offset is biased by higher-order contributions or by the compact ring itself on those baselines.
- [2021 data application] 2021 data application: the reported 1 mas northwest centroid is stated without error bars, without explicit propagation of station-based gain errors (even though closure phases are used), and without a quantitative assessment of how post-hoc data selection choices affect the result.
minor comments (2)
- [Abstract] Abstract: the 2021 centroid offset is given as 'about 1 mas' without uncertainty; adding a quantitative error estimate would improve clarity.
- [§2] Notation: the expansion is introduced with co-located stations, but the transition to actual EHT station separations is not accompanied by a clear statement of the validity regime (maximum baseline length or maximum source size).
Simulated Author's Rebuttal
We thank the referee for their thoughtful and constructive comments on our manuscript. We have carefully considered each point and made revisions to strengthen the paper accordingly. Our point-by-point responses are provided below.
read point-by-point responses
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Referee: [§2] §2 (Taylor expansion derivation): the first-order truncation is presented as sufficient to isolate the centroid shift, but the manuscript does not quantify the size of second-order terms (proportional to baseline separation squared times image second moments) for the actual 2021 EHT array geometry and a jet model whose extent reaches the claimed 1 mas scale.
Authors: We agree that an explicit quantification of the second-order terms would improve the rigor of the derivation. In the revised version, we have added an estimate in §2 using the actual 2021 EHT array geometry and a representative jet model extending to 1 mas. For the relevant short baselines, the second-order terms contribute less than 15% to the closure phase signal for the measured centroid offset, justifying the first-order approximation. We also provide the scaling with baseline length and image moments for general use. revision: yes
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Referee: [Synthetic validation section] Synthetic validation section: the tests use generic EHT-like arrays but do not reproduce the precise 2021 station positions, baseline lengths, and reported thermal noise levels against a diffuse component whose centroid offset is 1 mas; this leaves open whether the recovered offset is biased by higher-order contributions or by the compact ring itself on those baselines.
Authors: We have extended the synthetic validation to include simulations with the exact 2021 EHT station configuration, baseline lengths, and thermal noise levels corresponding to the 2021 observations. Using a model with a compact ring plus a diffuse component offset by 1 mas northwest, the recovered centroid from the closure phase expansion matches the input within 0.1 mas, with no significant bias from higher-order terms or the ring structure. These new results are incorporated into the validation section. revision: yes
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Referee: [2021 data application] 2021 data application: the reported 1 mas northwest centroid is stated without error bars, without explicit propagation of station-based gain errors (even though closure phases are used), and without a quantitative assessment of how post-hoc data selection choices affect the result.
Authors: We have revised the 2021 data application section to include error bars on the centroid measurement, derived from the uncertainties in the measured closure phases via standard error propagation. As closure phases are insensitive to station-based gain errors, we explicitly demonstrate that any residual calibration effects are below the thermal noise level. Furthermore, we performed a quantitative assessment by varying the data selection criteria (e.g., different flagging thresholds) and show that the 1 mas offset remains consistent within the error bars. These additions address the concerns directly. revision: yes
Circularity Check
Minor self-citation present; centroid measurement extracted directly from closure-phase expansion on data
full rationale
The paper derives a Taylor expansion of closure phases for nearly co-located stations and applies it to extract the source centroid offset from 2021 EHT observations. This step uses observed data values as direct inputs to the expansion without fitting free parameters to the target result or reducing the offset to a self-cited prior. Synthetic validation tests the method but does not force the reported 1 mas northwest value. Overlapping authors appear in EHT citations, yet these are not load-bearing for the central measurement, which remains independent of the paper's own equations.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Closure phases formed by nearly co-located stations are exactly zero in the absence of source structure on scales smaller than the baseline length.
- domain assumption The observed source consists of a bright compact core plus a large-scale diffuse component.
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AbsoluteFloorClosure.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
ψAA′B≈−2πuAA′·C (Eq. 10); linear fit to trivial closure phases on JCMT-SMA/ALMA-APEX triangles
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
synthetic validation and 2021 data centroid ~1 mas NW of ring
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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