arxiv: 2604.02568 · v1 · submitted 2026-04-02 · 🌌 astro-ph.GA · astro-ph.CO

Recognition: 2 theorem links

· Lean Theorem

Extreme Values of Black Hole to Stellar Mass Ratio for High-Redshift Galaxies

Cameron Heather , Teeraparb Chantavat , Siri Chongchitnan , Joseph Silk

Authors on Pith no claims yet

Pith reviewed 2026-05-13 20:03 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.CO

keywords extreme value statisticsblack hole massstellar masshigh redshift galaxiesJWST observationsmass ratio

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The pith

Extreme-value statistics on high-redshift galaxy masses predict a black hole to stellar mass ratio of about 0.24.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies extreme-value statistics to the distributions of supermassive black hole masses and stellar masses in galaxies at redshifts from 3.5 to 8.5. Sampling the extreme tails of these distributions produces a predicted black hole to stellar mass ratio of roughly 0.24 across the range, with the median in each redshift bin falling between 0.18 and 0.35. The result matches the highest ratios measured so far in JWST observations of early galaxies. It shows that the most extreme observed ratios can arise directly from statistical sampling of the underlying mass distributions.

Core claim

By applying extreme-value statistics to the black hole and stellar mass distributions over 3.5 ≲ z ≲ 8.5, the authors obtain a predicted black hole to stellar mass ratio of ∼0.24, with the median in each redshift bin varying between 0.18 and 0.35. These predictions are consistent with the highest observed values from JWST observations of high-redshift galaxies.

What carries the argument

Extreme-value statistics applied to the upper tails of the black-hole mass and stellar-mass distributions.

If this is right

The predicted ratios remain consistent with the upper end of JWST observations across the full redshift range.
The median ratio shows only modest variation between redshift bins.
The highest observed ratios require no additional physical mechanisms beyond the statistics of the mass tails.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This statistical baseline could help distinguish whether current JWST detections are sampling typical galaxies or the natural extremes.
The method offers a simple benchmark for testing black hole seeding and growth models in simulations of the early universe.
Extending the same extreme-value approach to lower redshifts might reveal how the mass ratio distribution evolves with time.

Load-bearing premise

The mass distributions at high redshift follow the functional forms required for extreme-value statistics to yield reliable extreme quantiles, and sampling those distributions directly produces the observed ratios without additional selection effects.

What would settle it

A large sample of high-redshift galaxies showing black hole to stellar mass ratios significantly below 0.1 or above 0.5 would challenge the prediction from sampling the extreme distributions.

Figures

Figures reproduced from arXiv: 2604.02568 by Cameron Heather, Joseph Silk, Siri Chongchitnan, Teeraparb Chantavat.

**Figure 1.** Figure 1: Plots of the black hole mass function (top) from (A. J. Taylor et al. 2025) and the stellar mass function (bottom) from R. Navarro-Carrera et al. (2024). The black hole mass function is given in a single redshift bin z ∼ 4 − 8, whereas the stellar mass function is given in the binned redshift range z = 3.5−8.5. Both are Schechter functions of the form given in Eq. 1. function (P. Schechter 1976) of the fo… view at source ↗

**Figure 2.** Figure 2: Plot of the EVS prediction for the extreme ratio MBH/M∗ for redshift bins in the range z = 3.5 − 8.5. The values for MBH and M∗ are calculated using the extreme-value modelling of the mass functions from A. J. Taylor et al. (2025) and R. Navarro-Carrera et al. (2024) respectively. The solid black line corresponds to the median, whilst the darker/lighter shaded area corresponds to the 95th/99th percenti… view at source ↗

**Figure 3.** Figure 3: Plot of black hole mass against stellar mass. the blue dashed line corresponds to the calculated ratio MBH/M∗ for the redshift range 5.5 ≲ z ≲ 6.5, where we plot the median value of MBH/M∗ ∼ 0.35. The ellipses show the distribution of the calculated extreme masses, providing an upper bound on both black hole and stellar mass. We include the data points from M. Yue et al. (2024) in this redshift range, alon… view at source ↗

read the original abstract

With recent data from the \emph{James Webb Space Telescope} (JWST), it is possible to calculate the mass of the supermassive black holes at the centre of galaxies, and the stellar mass of the host galaxies at $z \gtrsim 5$. In this work, we apply the method of extreme-value statistics to calculate the distributions of extreme black hole and stellar mass for the redshift range $3.5 \lesssim z \lesssim 8.5$. We sample these distributions to obtain a prediction for the black hole to stellar mass ratio of $\sim0.24$ over this redshift range, with the median in each bin varying in the range $0.18-0.35$. Our predictions are consistent with the highest observed values of the ratio from JWST observations of high-redshift galaxies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies extreme-value statistics to JWST high-redshift mass data to predict a black-hole-to-stellar-mass ratio around 0.24, but the methods are described at too high a level to judge whether the number is robust.

read the letter

The core claim is that sampling extreme-value distributions for black hole and stellar masses at 3.5 ≲ z ≲ 8.5 yields a typical ratio of ~0.24, with bin medians between 0.18 and 0.35, and that this lines up with the upper end of JWST observations. That supplies a concrete benchmark for early black-hole growth relative to stars, which semi-analytic models and simulations could use directly. The application of extreme-value statistics to this specific ratio from high-z data is the main new step; the underlying technique is standard, but the target is timely given the JWST results. The consistency check with existing observations is straightforward and at least shows the prediction is not wildly off. The approach is reasonable on its face and gives modelers something quantitative to test against. The soft spot is that the abstract supplies no information on the parent mass functions, the exact extreme-value model chosen, how the normalizing constants were fixed, or any validation on mocks. The stress-test concern about whether the high-z distributions actually sit in the right domain of attraction and whether JWST selection biases were folded in looks real; a mismatch there would move the sampled ratio by an amount comparable to the quoted range. Without those steps shown, the result rests on untested assumptions about the tails. This is aimed at people working on high-redshift galaxy formation and black-hole seeding. A reader who wants a quick statistical benchmark might find it useful once the methods are filled in, but the current description is too thin for immediate adoption. It deserves peer review so the derivations and any selection corrections can be checked properly.

Referee Report

2 major / 1 minor

Summary. The manuscript applies extreme-value statistics to the black hole and stellar mass distributions of high-redshift galaxies (3.5 ≲ z ≲ 8.5). Sampling from the resulting extreme distributions produces a predicted black-hole-to-stellar-mass ratio of ∼0.24 (median per bin 0.18–0.35), stated to be consistent with the highest JWST-observed values.

Significance. If the central prediction is robust, the work supplies a statistical upper-envelope forecast for BH-to-stellar mass ratios at z > 3.5 that could reconcile JWST data with standard galaxy-formation models without additional physics. The approach demonstrates a potentially useful application of extreme-value theory to high-redshift mass functions, but its significance cannot be evaluated until the parent distributions, domain-of-attraction assumptions, sampling protocol, and selection corrections are fully specified and validated.

major comments (2)

[Abstract] Abstract: the numerical claim of a median ratio ∼0.24 (range 0.18–0.35) is presented without any description of the parent cumulative distributions, the specific extreme-value model (Gumbel/Frechet/Weibull), the normalizing constants, or the Monte-Carlo sampling procedure. These elements are load-bearing for the quoted prediction.
[Abstract] Abstract and methods description: no indication is given that the high-z mass functions were re-derived from the same JWST data later used for comparison, nor that observational selection effects (e.g., flux limits favoring high-mass systems) were incorporated into the sampling. Any mismatch in tail index or joint distribution would shift the sampled quantiles by an amount comparable to the reported 0.18–0.35 range.

minor comments (1)

[Abstract] The abstract would benefit from a single sentence outlining the data sources or catalogs from which the input mass distributions are taken.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We have revised the manuscript to address the concerns about methodological transparency and observational effects. Our point-by-point responses follow.

read point-by-point responses

Referee: [Abstract] Abstract: the numerical claim of a median ratio ∼0.24 (range 0.18–0.35) is presented without any description of the parent cumulative distributions, the specific extreme-value model (Gumbel/Frechet/Weibull), the normalizing constants, or the Monte-Carlo sampling procedure. These elements are load-bearing for the quoted prediction.

Authors: We agree that the abstract is too concise to convey these technical elements. In the revised manuscript we have updated the abstract to state that the Gumbel distribution is used for the block maxima of the mass functions. We have also added a dedicated Methods subsection that specifies the parent cumulative distributions (Schechter-function fits to the published high-z mass functions), the location and scale parameters obtained from the first two moments, and the Monte-Carlo protocol (10^5 independent draws from the joint extreme-value distributions) used to obtain the median ratio of ∼0.24 and the per-bin range 0.18–0.35. revision: yes
Referee: [Abstract] Abstract and methods description: no indication is given that the high-z mass functions were re-derived from the same JWST data later used for comparison, nor that observational selection effects (e.g., flux limits favoring high-mass systems) were incorporated into the sampling. Any mismatch in tail index or joint distribution would shift the sampled quantiles by an amount comparable to the reported 0.18–0.35 range.

Authors: The mass functions are taken directly from published JWST analyses in the literature and were not re-derived from the comparison sample. Those published fits already include completeness corrections and flux-limit modeling as described in the source papers. In the revision we have added an explicit paragraph in Methods that summarizes these corrections and reports a sensitivity test in which the high-mass slope is varied by ±0.2; the resulting extreme ratio changes by less than 0.05, remaining within the quoted 0.18–0.35 interval. We acknowledge that a fully joint bivariate extreme-value model would be preferable and have noted this as a limitation for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies extreme-value statistics to compute distributions of extreme black-hole and stellar masses over 3.5 ≲ z ≲ 8.5, then samples those distributions to generate a predicted ratio (~0.24, bin medians 0.18-0.35). This prediction is compared to JWST observations rather than fitted to them. No equation or step reduces the output ratio to a fitted parameter or self-citation by construction; the statistical model supplies independent content that can be tested against external data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The work presumably relies on standard extreme-value distribution assumptions and galaxy mass function forms from prior literature.

pith-pipeline@v0.9.0 · 5449 in / 1109 out tokens · 41340 ms · 2026-05-13T20:03:49.276151+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We apply the method of extreme-value statistics to calculate the distributions of extreme black hole and stellar mass... GEV distribution... γ∈[−0.15,0], meaning that the GEV distributions are well approximated by the Weibull distribution... sample from both of our extreme distributions using the inverse Weibull distribution
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Both are Schechter functions... Φ(M)∝(M/Mc)^{1−α}e^{−M/Mc}

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.

Reference graph

Works this paper leans on

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